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+DEEP BIOLOGICAL PATHWAY INFORMED PATHOLOGY-
+GENOMIC MULTIMODAL SURVIVAL PREDICTION
+Lin Qiu1, Aminollah Khormali2, Kai Liu1
+1 Division of Research and Early Development, Genentech
+{qiul13, liuk3}@gene.com
+2 Department of Electrical and Computer Engineering, University of Central Florida
+aminollah.khormali@gmail.com
+ABSTRACT
+The integration of multi-modal data, such as pathological images and genomic data,
+is essential for understanding cancer heterogeneity and complexity for personalized
+treatments, as well as for enhancing survival predictions. Despite the progress made
+in integrating pathology and genomic data, most existing methods cannot mine the
+complex inter-modality relations thoroughly. Additionally, identifying explainable
+features from these models that govern preclinical discovery and clinical prediction
+is crucial for cancer diagnosis, prognosis, and therapeutic response studies. We
+propose PONET- a novel biological pathway informed pathology-genomic deep
+model that integrates pathological images and genomic data not only to improve
+survival prediction but also to identify genes and pathways that cause different
+survival rates in patients. Empirical results on six of The Cancer Genome Atlas
+(TCGA) datasets show that our proposed method achieves superior predictive
+performance and reveals meaningful biological interpretations. The proposed
+method establishes insight into how to train biologically informed deep networks on
+multimodal biomedical data which will have general applicability for understanding
+diseases and predicting response and resistance to treatment.
+1
+INTRODUCTION
+Manual examination of hematoxylin and eosin (H&E)-stained slides of tumor tissue by pathologists
+is currently the state-of-the-art for cancer diagnosis (Chan, 2014). The recent advancements in deep
+learning for digital pathology have enabled the use of whole-slide images (WSIs) for computational
+image analysis tasks, such as cellular segmentation (Pan et al., 2017; Hou et al., 2020), tissue
+classification and characterization (Hou et al., 2016; Hekler et al., 2019; Iizuka et al., 2020). While
+H&E slides are important and sufficient to establish a profound diagnosis, genomics data can provide
+a deep molecular characterization of the tumor, potentially offering the chance for prognostic and
+predictive biomarker discovery.
+Cancer prognosis via survival outcome prediction is a standard method used for biomarker discovery,
+stratification of patients into distinct treatment groups, and therapeutic response prediction (Cheng
+et al., 2017; Ning et al., 2020). WSIs exhibit enormous heterogeneity and most approaches adopt a
+two-stage multiple instance learning-based (MIL) approach for the representation learning of WSIs.
+Firstly, instance-level feature representations are extracted from image patches in the WSI, and then
+global aggregation schemes are applied to the bag of instances to obtain a WSI-level representation
+for subsequent modeling purpose (Hou et al., 2016; Courtiol et al., 2019; Wulczyn et al., 2020; Lu
+et al., 2021). Therefore, multimodal survival prediction faces an additional challenge due to the
+large data heterogeneity gap between WSIs and genomics, and many existing approaches use simple
+multimodal fusion mechanisms for feature integration, which prevents mining important multimodal
+interactions (Mobadersany et al., 2018; Chen et al., 2022b;a).
+The incorporation of biological pathway databases in a model takes advantage of leveraging prior
+biological knowledge so that potential prognostic factors of well-known biological functionality can
+be identified (Hao et al., 2018). Moreover, encoding biological pathway information into the neural
+1
+arXiv:2301.02383v1  [q-bio.QM]  6 Jan 2023
+
+Figure 1: Overview of PONET model.
+networks achieved superior predictive performance compared with established models (Elmarakeby
+et al., 2021).
+Based on the current challenges in multimodal fusion of pathology and genomics and the potential
+prognostic interpretation to link pathways and clinical outcomes in pathway-based analysis, we
+propose a novel biological pathway-informed pathology-genomic deep model, PONET, that uses H&E
+WSIs and genomic profile features for survival prediction. The proposed method contains four major
+contributions: 1) PONET formulates a biological pathway-informed deep hierarchical multimodal
+integration framework for pathological images and genomic data; 2) PONET captures diverse and
+comprehensive modality-specific and cross-modality relations among different data sources based
+on the factorized bilinear model and graph fusion network; 3) PONET reveals meaningful model
+interpretations on both genes and pathways for potential biomarker and therapeutic target discovery;
+PONET also shows spatial visualization of the top genes/pathways which has enormous potential
+for novel and prognostic morphological determinants; 4) We evaluate PONET on six public TCGA
+datasets which showed superior survival prediction comparing to state-of-the-art methods. Fig. 1
+shows our model framework.
+2
+RELATED WORK
+Multimodal Fusion. Earlier works on multimodal fusion focus on early fusion and late fusion. Early
+fusion approaches fuse features by simple concatenation which cannot fully explore intra-modality
+dynamics (Wöllmer et al., 2013; Poria et al., 2016; Zadeh et al., 2016). In contrast, late fusion
+fuses different modalities by weighted averaging which fails to model cross-modal interactions
+(Nojavanasghari et al., 2016; Kampman et al., 2018). The exploitation of relations within each
+modality has been successfully introduced in cancer prognosis via bilinear model (Wang et al., 2021b)
+and graph-based model (Subramanian et al., 2021). Adversarial Representation Graph Fusion (ARGF)
+(Mai et al., 2020) interprets multimodal fusion as a hierarchical interaction learning procedure where
+firstly bimodal interactions are generated based on unimodal dynamics, and then trimodal dynamics
+are generated based on bimodal and unimodal dynamics. We propose a new hierarchical fusion
+framework with modality-specific and cross-modality attentional factorized bilinear modules to
+mine the comprehensive modality interactions. Our proposed hierarchical fusion framework is
+different from ARGF in the following ways: 1) We take the sum of the weighted modality-specific
+representation as the unimodal representation instead of calculating the weighted average of the
+modality-specific representation in ARGF; 2) For higher level’s fusion, ARGF takes the original
+embeddings of each modality as input while we use the weighted modality-specific representations;
+3) We argue that ARGF takes redundant information during their trimodal dynamics.
+Multimodal Survival Analysis. There have been exciting attempts on multimodal fusion of pathol-
+ogy and genomic data for cancer survival prediction (Mobadersany et al., 2018; Cheerla & Gevaert,
+2019; Wang et al., 2020). However, these multimodal fusion based methods fail to model the interac-
+tion between each subset of multiple modalities explicitly. Kronecker product considers pairwise
+interactions of two input feature vectors by producing a high-dimensional feature of quadratic ex-
+pansion (Zadeh et al., 2017), and showed its superiority in cancer survival prediction (Wang et al.,
+2021b; Chen et al., 2022b;a). Despite promising results, using Kronecker product in multimodal
+2
+
+Unimodal
+ Gene layer  Pathway layer Hidden layer
+MFB
+>C
+Atten
+um
+hm
+hm
+hm
+↑
+Bimodal
+himz
+Gene
+hg
+Gene
+MFB
+α
+Multimodal
+Atten
+zuuy
+Wm1m2
+Representation
+hm1m2
+r
+hm
+Cox
+Pathway
+Pathology
+Trimodal
+↓
+hp
+hmi
+fe
+MFB
+Atten
+Wm1m2m3
+ hm1m2m3
+CNV + MUT
+hc
+Spatial Pathway
+Data embedding
+Hierarchical multimodal fusion
+Interpretationfusion may introduce a large number of parameters that may lead to high computational cost and risk
+of overfitting (Kim et al., 2017; Liu et al., 2021), thus limiting its applicability and improvement in
+performance. To overcome this drawback, hierarchical factorized bilinear fusion for cancer survival
+prediction (HFBSurv) (Li et al., 2022) uses factorized bilinear model to fuse genomic and image
+features, dramatically reducing computational complexity. PONET differs from HFBSurv in two
+ways: 1) PONET’s multimodal framework has three levels of hierarchical fusion module includ-
+ing unimodal, bimodal, and trimodal fusion while HFBSurv only considers within-modality and
+cross-modality fusion which we argue it is not adequate for mining the comprehensive interactions;
+2) PONET leverages biological pathway informed network for better prediction and meaningful
+interpretation purposes.
+Pathway-associated Sparse Neural Network. The pathway-based analysis is an approach that a
+number of studies have investigated to improve both predictive performance and biological inter-
+pretability (Jin et al., 2014; Cirillo et al., 2017; Hao et al., 2018; Elmarakeby et al., 2021). Moreover,
+pathway-based approaches have shown more reproducible analysis results than gene expression data
+analysis alone (Li et al., 2015; Mallavarapu et al., 2017). These pathway-based deep neural networks
+can only model genomic data which severely inhibits their applicability in current biomedical re-
+search. Additionally, the existing pathway-associated sparse neural network structures are limited
+for disease mechanism investigation: there is only one pathway layer in PASNet (Hao et al., 2018)
+which contains limited biological prior information to deep dive into the hierarchical pathway and
+biological process relationships; P-NET (Elmarakeby et al., 2021) calculates the final prediction by
+taking the average of all the gene and pathway layers’ outputs, and this will bias the learning process
+because it will put more weights for some layers’ outputs while underestimating the others.
+3
+METHODOLOGY
+3.1
+PROBLEM FORMULATION AND NOTATIONS
+The model architecture of PONET is presented in Fig. 1, where three modalities are included as input:
+gene expression g ∈ Rdg, pathological image p ∈ Rdp, and copy number (CNV) + mutation (MUT)
+CNV + MUT ∈ Rdc, with dp being the dimensionality of p and so on. We define a hierarchical
+factorized bilinear fusion model for PONET. We build a sparse biological pathway-informed embed-
+ding network for gene expression. A fully connected (FC) embedding layer for both preprocessed
+pathological image feature (fp) and the copy number + mutation (fc) to map feature into similar
+embedding space for alleviating the statistical property differences between modalities, the three
+network architecture details are in Appendix C.1. We label the three modality embeddings as hm,
+m ∈ {g, p, c}, the superscript/subscript u, b, and t represents unimodal fusion, biomodal fusion and
+trimodal fusion. After that, the embeddings of each modality are first used as input for unimodal fusion
+to generate the modality-specific representation hu
+m = ωmˆhm, ωm represent the modality-specific im-
+portance, the feature vector of the unimodal fusion is the sum of all modality-specific representations
+hu = �
+m hu
+m. In the bimodal fusion, modality-specific representations from the output of unimodal
+fusion are fused to yield cross-modality representations hb
+m1m2 = ωm1m2ˆhm1m2, m1, m2 ∈ {p, c, g}
+and m1 ̸= m2, ωm1m2 represents the corresponding cross-modality importance. Similarly, the feature
+vector of bimodal fusion is calculated as hb = �
+m1,m2 hb
+m1m2. We propose to build a trimodal
+fusion to take each cross-modality representation from the output of bimodal fusion to mine the
+interactions. Similarly to the bimodal fusion architecture, the trimodal fusion feature vector will
+be ht = �
+m1,m2,m3 ωm1m2m3ˆhm1m2m3, m1, m2, m3 ∈ {p, c, g} and m1 ̸= m2 ̸= m3, ωm1m2m3
+represents the corresponding trimodal importance. Finally, PONET concatenates hu, hb, ht to obtain
+the final comprehensive multimodal representation and pass it to the Cox proportional hazards model
+(Cox, 1972; Cheerla & Gevaert, 2019) for survival prediction. In the following sections we will
+describe our hierarchical factorized bilinear fusion framework, l, o, s represents the dimensionality
+of hm, zm, ˆhm1m2.
+3.2
+SPARSE NETWORK
+We design the sparse gene-pathway network consisting of one gene layer followed by three pathway
+layers. A patient sample of e gene expressions is formed as a column vector, which is denoted by
+X = [x1, x2, ..., xe], each node represents one gene. The gene layer is restricted to have connections
+3
+
+73,703 x 50,706 px
+224 x 224 px, mpp: 0.5
+Whole Slide Image
+WSI patching
+Image Augmentation
+𝑔!!(#)
+𝑞!!(#)
+𝑔!"(#)
+𝑓!!(#)
+𝑓!"(#)
+𝑦"
+'𝑦#
+𝑧"
+̂𝑧#
+𝑝"
+𝐿(𝑝, 𝑧)
+Visual representation learning using SSL ViT
+Student Network
+Teacher Network
+Patch features
+𝑣
+𝑢
+Figure 2: Overall framework of the visual representation extraction using pre-trained self-supervised
+vision transformer.
+reflecting the gene-pathway relationships curated by the Reactome pathway dataset (Fabregat et al.,
+2020). The connections are encoded by a binary matrix M ∈ Ra×e, where a is the number of
+pathways and e is the number of genes, an element of M, mij, is set to one if gene j belongs to
+pathway i. The connections that do not exist in the Reactome pathway dataset will be zero-out. For
+the following pathway-pathway layers, a similar scheme is applied to control the connection between
+consecutive layers to reflect the parent-child hierarchical relationships that exist in the Reactome
+dataset. The output of each layer is calculated as
+y = f[(M ∗ W)T X + ϵ]
+(1)
+where f is the activation function, M represents the binary matrix, W is the weights matrix, X is the
+input matrix, ϵ is the bias vector, and ∗ is the Hadamard product. We use tanh for the activation of
+each node. We allow the information flow from the biological prior informed network starting from
+the first gene layer to the last pathway layer, and we label the last layer output embeddings of the
+sparse network for gene expression as hg.
+3.3
+UNIMODAL FUSION
+Bilinear models (Tenenbaum & Freeman, 2000) provide richer representations than linear models.
+Given two feature vectors in different modalities, e.g., the visual features x ∈ Rm×1 for an image and
+the genomic features y ∈ Rn×1 for a genomic profile, the bilinear model uses a quadratic expansion
+of linear transformation considering every pair of features:
+zi = xT Wiy
+(2)
+where Wi ∈ Rm×n is a projection matrix, zi ∈ R is the output of the bilinear model. Bilinear
+models introduce a large number of parameters which potentially lead to high computational cost
+and overfitting risk. To address these issues, Yu et al. (2017) develop the Multi-modal Factorized
+Bilinear pooling (MFB) method, which enjoys the dual benefits of compact output features and robust
+expressive capacity.
+Inspired by the MFB (Yu et al., 2017) and its application in pathology and genomic multimodal
+learning (Li et al., 2022), we propose unimodal fusion to capture modality-specific representations
+and quantify their importance. The unimodal fusion takes the embedding of each modality hm as
+input and factorizes the projection matrix Wi in Eq. (2) as two low-rank matrices:
+zi
+=
+hT
+mWihm =
+k�
+d=1
+hT
+mum,dvT
+m,dhm
+=
+1T (U T
+m,ihm ◦ V T
+m,ihm), m ∈ {p, c, g}
+(3)
+we get the output feature zm:
+zm = SumPooling
+�
+˜U T
+mhm◦ ˜V T
+mhm, k
+�
+, m ∈ {p, c, g}
+(4)
+where k is the latent dimensionality of the factorized matrices. SumPooling (x, k) function performs
+sum pooling over x by using a 1-D non-overlapped window with the size k, ˜Um ∈ Rl×ko and
+˜Vm ∈ Rl×ko are 2-D matrices reshaped from Um and Vm, Um =[Um,1, . . . , Um,h] ∈ Rl×k×o and
+Vm = [Vm,1, . . . , Vm,h] ∈ Rl×k×o. Each modality-specific representation ˆhm ∈ Rl+o is obtained
+as:
+ˆhm = hm©zm, m ∈ {p, c, g}
+(5)
+4
+
+where © denotes vector concatenation. We also introduce a modality attention network Atten ∈
+Rl+o → R1 to determine the weight for each modality-specific representation to quantify its impor-
+tance:
+ωm = Atten(ˆhm; ΘAtten), m ∈ {p, c, g}
+(6)
+where ωm is the weight of modality m. In practice, Atten consists of a sigmoid activated dense layer
+parameterized by ΘAtten. Therefore, the output of each modality in unimodal fusion, hu
+m, is denoted
+as ωmˆhm ∈ Rl+o, m ∈ {p, c, g}. Accordingly, the output of unimodal fusion, hu, is the sum of each
+weighted modality-specific representation ωmˆhm, m ∈ {p, c, g} which is different from ARGF (Mai
+et al., 2020) that used the weighted average of different modalities as the unimodal fusion output.
+3.4
+BIMODAL AND TRIMODAL FUSION
+Bimodal fusion aims to fuse diverse information of different modalities and quantify different
+importance for them. After receiving the modality-specific representations hu
+m from the unimodal
+fusion, we can generate the cross-modality representation ˆhm1m2 ∈ Rs similar to Eq. (4) :
+ˆhm1,m2 = Sum Pooling
+�
+˜U T
+m1hu
+m1◦ ˜V T
+m2hu
+m2, k
+�
+,
+m1, m2 ∈ {p, c, g}, m1 ̸= m2
+(7)
+where
+˜U T
+m1 ∈ R(l+o)×ks and ˜V T
+m2 ∈ R(l+o)×ks are 2-D matrices reshaped from Um1 and Vm2
+and Um1 = [Um1,1, . . . , Um1,s] ∈ R(l+o)×k×s and Vm2 = [Vm2,1, . . . , Vm2,s] ∈ R(l+o)×k×s. We
+leverage a bimodal attention network (Mai et al., 2020) to identify the importance of the cross-
+modality representation. The similarity Sm1m2 ∈ R1 of hu
+m1 and hu
+m2 is first estimated as follows:
+Sm1,m2 =
+l+o
+�
+i=1
+�
+eωm1 hu
+m1,i
+�l+o
+j=1 eωm1hu
+m1,j
+� �
+eωm2hu
+m2,i
+�l+o
+j=1 eωm2hu
+m2,j
+�
+(8)
+where the computed similarity is in the range of 0 to 1. Then, the cross-modality importance ωm1m2
+is obtained by:
+ωm1m2 =
+eˆωmimj
+�
+mi̸=mj eˆωmimj , ˆωm1m2 = ωm1 + ωm2
+Sm1m2 + S0
+(9)
+where S0 represents a pre-defined term controlling the relative contribution of similarity and modality-
+specific importance, and here is set to 0.5. Therefore, the output of bimodal fusion, hb, is the sum of
+each weighted cross-modality representation ωm1m2ˆhm1m2, m1, m2 ∈ {p, c, g} and m1 ̸= m2.
+In trimodal fusion, each bimodal fusion output is fused with the unimodal fusion output that does
+not contribute to the formation of the bimodal fusion. The output for each corresponding trimodal
+representation is ˆhm1m2m3. In addition, trimodal attention was applied to identify the importance of
+each trimodal representation, ωm1m2m3. The output of the trimodal fusion, ht, is the sum of each
+weighted trimodal representation ωm1m2m3ˆhm1m2m3, m1, m2, m3 ∈ {p, c, g} and m1 ̸= m2 ̸= m3.
+3.5
+SURVIVAL LOSS FUNCTION
+We train the model through the Cox partial likelihood loss (Cheerla & Gevaert, 2019) with l1
+regularization for survival prediction, which is defined as:
+ℓ(Θ) = −
+�
+i:Ei=1
+�
+�ˆhΘ (xi) − log
+�
+j:Ti>Tj
+exp
+�
+ˆhΘ (xj)
+�
+�
+� + λ (∥Θ∥1)
+(10)
+where the values Ei, Ti and xi for each patient represent the survival status, the survival time, and the
+feature, respectively. Ei = 1 means event while Ei = 0 represents censor. ˆhΘ is the neural network
+model trained for predicting the risk of survival, Θ is the neural network model parameters, and λ is
+a regularization hyperparameter to avoid overfitting.
+5
+
+4
+EXPERIMENTS
+4.1
+EXPERIMENTAL SETUP
+Datasets. To validate our proposed method, we used six cancer datasets from The Cancer Genome
+Atlas (TCGA), a public cancer data consortium that contains matched diagnostic WSIs and genomic
+data with labeled survival times and censorship statuses. The genomic profile features (mutation
+status, copy number variation, RNA-Seq expression) are preprocessed by Porpoise 1 (Chen et al.,
+2022b). For this study, we used the following cancer types: Bladder Urothelial Carcinoma (BLCA)
+(n = 437), Kidney Renal Clear Cell Carcinoma (KIRC) (n = 350), Kidney Renal Papillary Cell
+Carcinoma (KIRP) (n = 284), Lung Adenocarcinoma (LUAD) (n = 515), Lung Squamous Cell
+Carcinoma (LUSC) (n = 484), Pancreatic adenocarcinoma (PAAD) (n = 180). We downloaded the
+same diagnostic WSIs from the TCGA website 2 that were used in Porpoise study to match the
+paired genomic features and survival times. The feature alignment table for all the cancer types
+is in Appendix A. For each WSI, automated segmentation of tissue was performed. Following
+segmentation, image patches of size 224 × 224 were extracted without overlap at the 20 X equivalent
+pyramid level from all tissue regions identified while excluding the white background and selecting
+only patches with at least 50% tissue regions. Subsequently, a visual representation of those patches
+is extracted with a vision transformer (Wang et al., 2021a) pre-trained on the TCGA dataset through
+a self-supervised constructive learning approach, such that each patch is represented as a 1 × 2048
+vector. Fig. 2 shows the framework for the visual representation extraction by vision transformer
+(VIT). Survival outcome information is available at the patient level, we aggregated the patch-level
+feature into slide level feature representations based on an attention-based method (Lu et al., 2021;
+Ilse et al., 2018).
+Baselines. Using the same 5-fold cross-validation splits for evaluating PONET, we implemented and
+evaluated six state-of-the-art methods for survival outcome prediction. Additionally, we included
+three variations of PONET: a) PONET-O represents only genomic data, and pathway architecture
+for the gene expression are included in the model; b) PONET-OH represents only genomic and
+pathological image data but without pathway architecture in the model; c) PONET is our full model.
+For all methods, we use the same VIT feature extraction pipeline for WSIs, as well as identical
+training hyperparameters and loss functions for supervision. Training details and the parameters
+tuning can be found in Appendix C.2.
+CoxPH (Cox, 1972) represents the standard Cox proportional hazard models.
+DeepSurv (Katzman et al., 2018) is the deep neural network version of the CoxPH model.
+Pathomic Fusion (Chen et al., 2022a) as a pioneered deep learning-based framework for predicting
+survival outcomes by fusing pathology and genomic multimodal data, in which Kronecker product is
+taken to model pairwise feature interactions across modalities.
+GPDBN (Wang et al., 2021b) adopts Kronecker product to model inter-modality and intra-modality
+relations between pathology and genomic data for cancer prognosis prediction.
+HFBSurv (Li et al., 2022) extended GPDBN using the factorized bilinear model to fuse genomic
+and pathology features in a within-modality and cross-modalities hierarchical fusion.
+Porpoise (Chen et al., 2022b) applied the discrete survival model and Kronecker product to fuse
+pathology and genomic data for survival prediction (Zadeh & Schmid, 2020).
+Evaluation. For each cancer dataset, we used the cross-validated concordance index (C-Index)
+(Appendix B.1) (Harrell et al., 1982) to measure the predictive performance of correctly ranking the
+predicted patient risk scores with respect to overall survival.
+4.2
+RESULTS
+Comparison with Baselines. In combing pathology image, genomics, and pathway network via
+PONET, our approach outperforms CoxPH models, unimodal networks, and previous deep learning-
+based approaches on pathology-genomic-based survival outcome prediction (Table 1). The results
+show that deep learning-based approaches generally perform better than the CoxPH model. PONET
+achieves superior C-index values in all six cancer types. All versions of PONET outperform Pathomic
+1https://github.com/mahmoodlab/PORPOISE
+2https://www.cancer.gov/about-nci/organization/ccg/research/structural-genomics/tcga
+6
+
+Table 1: C-Index (mean ± standard deviation) of PONET and ablation experiments in TCGA survival
+prediction. The top two performers are highlighted in bold.
+Model
+TCGA-BLCA
+TCGA-KIRC
+TCGA-KIRP
+TCGA-LUAD
+TCGA-LUSC
+TCGA-PAAD
+CoxPH (Age + Gender) (Cox, 1972)
+0.525 ± 0.130
+0.550 ± 0.070
+0.544 ± 0.050
+0.531 ± 0.082
+0.532 ± 0.094
+0.539 ± 0.092
+DeepSurv (Kampman et al., 2018)
+0.580 ± 0.062
+0.620 ± 0.043
+0.560± 0.063
+0.534 ±0.077
+0.541 ± 0.066
+0.544 ± 0.076
+GPDBN (Wang et al., 2021b)
+0.612 ± 0.042
+0.647 ± 0.073
+0.669 ± 0.109
+0.565 ± 0.057
+0.545 ± 0.063
+0.571 ± 0.060
+HFBSurv (Li et al., 2022)
+0.622 ± 0.043
+0.667 ± 0.053
+0.769 ± 0.109
+0.581 ± 0.017
+0.548 ± 0.049
+0.591 ± 0.052
+Pathomic Fusion (Chen et al., 2022a)
+0.586 ± 0.062
+0.598 ± 0.060
+0.577 ± 0.032
+0.543 ± 0.065
+0.523 ±0.045
+0.545 ± 0.064
+Porpoise (Chen et al., 2022b)
+0.617 ± 0.048
+0.711 ± 0.051
+0.811 ± 0.089
+0.586 ±0.056
+0.527 ± 0.043
+0.591 ± 0.064
+PONET-O (ours)
+0.596 ± 0.056
+0.664 ± 0.056
+0.761 ± 0.093
+0.623 ±0.062
+0.538 ± 0.037
+0.598 ± 0.027
+PONET-OH (ours)
+0.625 ± 0.063
+0.695 ± 0.043
+0.776 ± 0.123
+0.618 ± 0.049
+0.553 ± 0.049
+0.591 ± 0.050
+PONET (ours)
+0.643 ± 0.037
+0.726 ± 0.056
+0.829 ± 0.054
+0.646 ±0.047
+0.567 ± 0.066
+0.639 ± 0.080
+Table 2: Evaluation of PONET on different fusion methods and pathway designs by C-index (mean
+± standard deviation). The best performer is highlighted in bold.
+Methods
+TCGA-BLCA
+TCGA-KIRP
+TCGA-LUAD
+TCGA-LUSC
+TCGA-PAAD
+Single fusion
+Simple concatenation
+0.585 ± 0.045
+0.652 ± 0.049
+0.554 ± 0.065
+0.525 ± 0.066
+0.568 ± 0.075
+Element-wise addition
+0.592 ± 0.047
+0.655 ± 0.055
+0.587 ± 0.065
+0.522 ± 0.046
+0.588 ± 0.055
+Tensor fusion (Zadeh et al., 2017)
+0.605 ± 0.046
+0.775 ± 0.053
+0.595 ± 0.060
+0.545 ± 0.045
+0.592 ± 0.061
+Hierarchical fusion
+Unimodal
+0.596 ± 0.035
+0.783 ± 0.063
+0.611 ± 0.056
+0.553 ± 0.073
+0.595 ± 0.053
+Bimodal
+0.602 ± 0.062
+0.789 ± 0.053
+0.601 ± 0.056
+0.552 ± 0.051
+0.598 ± 0.083
+ARGF (Mai et al., 2020)
+0.597 ± 0.054
+0.792 ± 0.043
+0.614 ± 0.051
+0.556 ± 0.063
+0.602 ± 0.065
+Unimodal + Bimodal
+0.614 ± 0.052
+0.803 ± 0.061
+0.631 ± 0.044
+0.578 ± 0.058
+0.615 ± 0.057
+Pathway design
+PASNet (Hao et al., 2018)
+0.606 ± 0.045
+0.793 ± 0.051
+0.621 ± 0.061
+0.551 ± 0.069
+0.625 ± 0.057
+P-NET (Elmarakeby et al., 2021)
+0.622 ± 0.047
+0.802 ± 0.071
+0.625 ± 0.045
+0.562 ± 0.054
+0.627 ± 0.073
+PONET
+0.643 ± 0.037
+0.829 ± 0.054
+0.641 ± 0.046
+0.567 ± 0.066
+0.639 ± 0.070
+Fusion by a big margin. Pathomic Fusion uses Kronecker product to fuse the two modalities, and
+that’s also the reason why other advanced fusion methods, like GPDBN and HFBSurv, achieve better
+performance. Also, we argue that Pathomic Fusion extracts the region of interest of pathology image
+for feature extraction might limit the understanding of the tumor microenvironment of the whole slide.
+HFBSurv shows better performance than GPDBN and Pathomic Fusion which is consistent with
+their findings, and these results further demonstrate that the hierarchical factorized bilinear model
+can better mine the rich complementary information among different modalities compared to the
+Kronecker product. Porpoise performs similarly with PONET on TCGA-KIRC and TCGA-KIRP and
+outperformed HFBSurv in these two studies, this probably is due to Porpoise partitioned the survival
+time into different non-overlapping bins and parameterized it as a discrete survival model (Zadeh
+& Schmid, 2020) which works better for these two cancer types. In other cases, Porpoise performs
+similarly to HFBSurv. Note: the results of Porpoise are from their paper (Chen et al., 2022b).
+Additionally, we can see that PONET consistently outperforms PONET-O and PONET-OH indi-
+cating the effectiveness of the biological pathway-informed neural network and the contribution of
+pathological image for the overall survival prediction.
+Ablation Studies. To assess whether the impact of hierarchical factorized bilinear fusion strategy
+is indeed effective, we compare PONET with four single-fusion methods: 1) Simple concatenation:
+concatenate each modality embeddings; 2) Element-wise addition: element-wise addition from each
+modality embeddings; 3) Tensor fusion (Zadeh et al., 2017): Kronecker product from each modality
+embeddings. Table 2 shows the C-index values of different methods. We can see that PONET
+achieves the best performance and shows remarkable improvement over single-fusion methods on
+different cancer type datasets. For example, PONET outperforms the Simple concatenation by
+8.4% (TCGA-BLCA), 27% (TCGA-KIRP), 15% (TCGA-LUAD), 8.0% (TCGA-LUSC), and 11.4%
+(TCGA-PAAD), etc.
+Furthermore, we adopted five different configurations of PONET to evaluate each hierarchical
+component of the proposed method: 1) Unimodal: unimodal fusion output as the final feature
+representation; 2) Bimodal: bimodal fusion output as the final feature representation; 3) Unimodal
++ Bimodal: hierarchical (include both unimodal and bimodal feature representation) fusion; 4)
+ARGF: ARGF (Mai et al., 2020) fusion strategy; 5) PONET: our proposed hierarchical strategy by
+incorporating unimodal, bimodal, and trimodal fusion output. As shown in Table 2, Unimodal +
+Bimodal performs better than Unimodal and Bimodal which demonstrates that Unimodal + Bimodal
+can capture the relations within each modality and across modalities. ARGF performs worse than
+Unimodal + Bimodal and far worse than PONET across all the cancer types. PONET outperforms
+7
+
+Figure 3: Inspecting and interpreting PONET on TCGA-KIRP. a: Sankey diagram visualization
+of inner layers of PONET shows the estimated relative importance of different nodes in each layer.
+Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes
+in each layer, H1 presents the gene layer, and H2-H4 represent pathway layers; b: Co-attention
+visualization of top 4 ranked pathways in one case of TCGA-KIRP.
+Unimodal + Bimodal in 4 out of 5 cancer types indicating that three layers of hierarchical fusion can
+mine the comprehensive interactions among different modalities.
+To evaluate our sparse gene-pathway network design, we compare PONET with PASNet (Hao et al.,
+2018) and P-NET (Elmarakeby et al., 2021) pathway architecture, PASNet performs the worst due to
+the fact that it only has one pathway layer in the network, and thus limited prior information was used
+to predict the outcome. PONET constantly outperforms P-NET across all the cancer types, which
+demonstrates that averaging all the intermediate layers’ output for the final prediction cannot fully
+capture the prior information flow among the hierarchical biological structures.
+Model Interpretation. We discuss the model interpretation results for cancer type TCGA-KIRP
+here and the results for other cancer types are included in the Appendix C.3. To understand the
+interactions between different genes, pathways, and biological processes that contributed to the
+predictive performance and to study the paths of impact from the input to the outcome, we visualized
+the whole structure of PONET with the fully interpretable layers after training (Fig. 3 a). To evaluate
+the relative importance of specific genes contributing to the model prediction, we inspected the genes
+layer and used the Integrated Gradients attribution (Sundararajan et al., 2017) method to obtain
+the total importance score of genes, and the modified ranking algorithm details are included in the
+Appendix B.3. Highly ranked genes included KRAS, PSMB6, RAC1, and CTNNB1 which are known
+kidney cancer drivers previously (Yang et al., 2017; Shan et al., 2017; Al-Obaidy et al., 2020; Guo
+et al., 2022). GBN2, a member of the guanine nucleotide-binding proteins family, has been reported
+that the decrease of its expression reduced tumor cell proliferation (Zhang et al., 2019). A recent study
+identified a strong dependency on BCL2L1, which encodes the BCL-XL anti-apoptotic protein, in a
+subset of kidney cancer cells (Grubb et al., 2022). This biological interpretability revealed established
+and novel molecular features contributing to kidney cancer. In addition, PONET selected a hierarchy
+of pathways relevant to the model prediction, including downregulation of TGF-β receptor signaling,
+regulation of PTEN stability and activity, the NLRP1 inflammasome, and noncanonical activation of
+NOTCH3 by PSEN1, PSMB6, and BCL2L1. TGF-β signaling is increasingly recognized as a key
+driver in cancer, and in progressive cancer tissues TGF-β promotes tumor formation, and its increased
+expression often correlates with cancer malignancy (Han et al., 2018). Noncanonical activation of
+NOTCH3 was reported to limit tumor angiogenesis and plays a vital role in kidney disease (Lin et al.,
+2017).
+8
+
+a
+H1
+H2
+H3
+H4
+KRAS
+Downregulation of TGF-beta receptor signaling
+TGF-beta receptor signaling activates SMADs
+Neurodegenerative Diseases
+PSMB6
+Regulation of PTEN stability and activity
+NOTCH3 Activation and Transmission of Signal
+Cellular Senescence
+RAC1
+Calmodulin induced events
+Semaphorin interactions
+Signal amplification
+CTNNB1
+The NLRP1 inflammasome
+Downstream signaling events of B Cell Receptor
+Signaling by FGFR
+BCL2L1
+Noncanonical activation of NOTCH3
+M Phase
+LeadingStrand Synthesis
+GNB2
+Synthesis of PIPs in the nucleus
+Biosynthesis of the N-glycan precursor
+ER to Golgi Anterograde Transport
+outcome
+PSEN1
+Regulation of PTEN gene transcription
+Biosynthesis of DHA-derived SPMs
+Signaling by TGF-beta Receptor Complex in Cancer
+MTMR4
+RPA3
+Viral mRNA Translation
+Export of Viral Ribonucleoproteins from Nucleus
+Interferon Signaling
+HDAC3
+Complex I biogenesis
+ZBP1(DAI) mediated induction of type I IFNs
+Transportofvitaminsandnucleosides
+residual 
+Formation of tubulin folding intermediates by CCT/TriC Amine Oxidase reactions
+Transportofbilesalts,organicacids,andmetalions
+residual
+residual
+residual
+b
+TCGA-Q2-A5QZ
+Downregulation of TGF-beta
+Regulation of PTEN
+Survival Month: 14.06
+receptor signaling
+ stability and activity
+Calmodulin induced events
+The NLRP1 inflammasome
+High Attn
+Low AttnFigure 4: Kaplan-Meier analysis of patient stratification of low and high risk patients via four
+variations of PONET on TCGA-KIRP. Low and high risks are defined by the median 50% percentile
+of hazard predictions via each model prediction. Log-rank test was used to test for statistical
+significance in survival distributions between low and high risk patients.
+To further inspect the pathway spatial association with the WSI slide we adopted the co-attention
+survival method MCAT (Chen et al., 2021) between WSIs and genomic features on the top
+pathways of the second layer, visualized as a WSI-level attention heatmap for each pathway
+genomic embedding in Fig.
+3 b (algorithm details are included in the Appendix B.4).
+We
+used the gene list from the top 4 pathways as the genomic features and trained MCAT on the
+TCGA-KIRP dataset for survival prediction. Overall, we observe that high attention in different
+pathways showed different spatial pattern associations with the slide. This heatmap can reflect
+genotype-phenotype relationships in cancer pathology. The high attention regions (red) of dif-
+ferent pathways in the heatmap have positive associations with the predicted death risk while
+the low attention regions (blue) have negative associations with the predicted risk. By further
+checking the cell types in high attention patches we can gain insights of prognostic morpho-
+logical determinants and have a better understanding of the complex tumor microenvironment.
+Table 3: Comparison of model complexity
+Methods
+Number of Parameters
+FLOPS
+Pathomic Fusion
+175M
+168G
+GPDBN
+82M
+91G
+HFBSurv
+0.3M
+0.5G
+PONET
+2.8M
+3.1G
+Patient Stratification.
+In visualizing
+the Kaplan-Meier survival curves of pre-
+dicted high risk and low risk patient
+populations, we plot four variations of
+PONET in Fig. 4. PONET-ARGF rep-
+resents the model that we use the hier-
+archical fusion strategy of ARGF in our
+pathway-informed PONET model. From
+the results, PONET enables easy sepa-
+ration of patients into low and high risk
+groups with remarkably better stratifica-
+tion (P-Value = 6.60e-7) in comparison to the others.
+Complexity Comparison. We compared PONET with Pathomic Fusion, GPDBN, and HFBSurv
+since both Pathomic Fusion and GPDBN are based on Kronecker product to fuse different modalities
+while GPDBN and HFBSurv modeled inter-modality and intra-modality relations which have similar
+consideration to our method. As illustrated in Table 3, PONET has 2.8M (M = Million) trainable
+parameters, which is approximately 1.6%, 3.4%, and 900% of the number of parameters of Pathomic
+Fusion, GPDBN, and HFBSurv. To assess the time complexity of PONET and the competitive
+methods, we calculate each method’s floating-point operations per second (FLOPS) in testing. The
+results in Table 3 show that PONET needs 3.1G during testing, compared with 168G, 91G, and 0.5G
+in Pathomic Fusion, GPDBN, and HFBSurv. The main reason for fewer trainable parameters and the
+number of FLOPS lies in that PONET and HFBSurv perform multimodal fusion using the factorized
+bilinear model, and can significantly reduce the computational complexity and meanwhile obtain
+more favorable performance. PONET has one additional trimodal fusion which explains why it has
+more trainable parameters than HFBSurv.
+5
+CONCLUSION
+In this study, we pioneer propose a novel biological pathway-informed hierarchical multimodal
+fusion model that integrates pathology image and genomic profile data for cancer prognosis. In
+comparison to previous works, PONET deeply mines the interaction from multimodal data by
+conducting unimodal, bimodal and trimodal fusion step by step. Empirically, PONET demonstrates
+9
+
+PONET-O
+PONET-OH
+PONET-ARGF
+PONET
+1.0 -
+1.0
+1.0
+P-Value =1.90e-3
+P-Value =7.49e-4
+P-Value =4.27e-5
+ P-Value =6.60e-7
+0.9
+0.9
+0.9
+0.9
+0.8
+0.8
+0.8
+0.8
+0.7
+0.7
+Proportion s
+0.7
+0.7
+0.6
+0.6
+0.6
+0.6
+0.5
+0.5
+0.5
+0.5
+Low risk
+0.4
+Low risk
+0.4
+Low risk
+Low risk
+High risk
+0.4
+High risk
+High risk
+High risk
+0.3
+0
+25
+50
+75
+100
+125
+0
+25
+50
+75
+100
+125
+0
+25
+50
+75
+100
+125
+0
+25
+50
+75
+100
+125
+Time (months)the effectiveness of the model architecture and the pathway-informed network for superior predictive
+performance. Specifically, PONET provides insight on how to train biologically informed deep
+networks on multimodal biomedical data for biological discovery in clinic genomic contexts which
+will be useful for other problems in medicine that seek to combine heterogeneous data streams for
+understanding diseases and predicting response and resistance to treatment.
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+13
+
+Table 4: TCGA Data Feature Alignment Summary
+WSI
+CNV
+MUT
+RNA
+WSI+CNV+MUT
+WSI+MUT+RNA
+ALL
+Cancer Type
+BLCA
+454
+443
+452
+450
+441
+448
+437
+KIRC
+517
+509
+357
+514
+352
+355
+350
+KIRP
+294
+291
+286
+293
+284
+285
+284
+LUAD
+528
+522
+523
+522
+519
+519
+515
+LUSC
+505
+502
+489
+503
+486
+487
+484
+PAAD
+208
+201
+187
+195
+187
+180
+180
+A
+DATA
+Table 3 in Appendix A shows the number of patients with matched different data modalities: WSI
+(Whole slide image), CNV (Copy number), MUT (Mutation), RNA (RNA-Seq gene expression). For
+each TCGA dataset and each patient we have preprocessed data dimensions dg ∈ R1×2000 (RNA),
+dc ∈ R1×227 (CNV + MUT), and dp ∈ R1×32 (WSI) which will be used for our multimodal fusion.
+B
+METHODS
+B.1
+C-INDEX
+We use concordance-index (C-index) (Harrell et al., 1982) to measure the performance of survival
+models. It evaluates the model by measuring the concordance of the ranking of predicted harzards
+with the true survival time of patients. The range of the C-index is [0, 1], and larger values indicate
+better performance with a random guess leading to a C-index of 0.5.
+B.2
+WSI REPRESENTATION LEARNING
+It has been shown that the WSI visual representations extracted by self-supervised learning methods
+on histopathological images are more accurate and transferable than the supervised baseline models
+on domain-irrelevant datasets such as ImageNet. In this work, a pre-trained Vision Transformer (ViT)
+model (Wang et al., 2021a) that is trained on a large histopathological image dataset has been utilized
+for tile feature extraction. The model is composed of two main neural networks that learn from each
+other, i.e., student and teacher networks. Parameters of the teacher model θt are updated using the
+student network with parameter θs using the update rule represented in Eq. (11).
+θt ← τθt + (1 − τ)θs
+(11)
+Two different views of a given input H&E image x, uniformly selected from the training set I, are
+generated using random augmentations, i.e., u, v. Then, student and teacher models generate two
+different visual representations according to u and v as y1 = f θs (u) and ˆy2 = f θt (v), respectively.
+Finally, the generated visual representations are transformed into latent space using linear projection as
+p1 = gθs �
+gθs (y1)
+�
+and ˆz2 = gθt (ˆy2) for student and teacher networks, respectively. Similarly, feed-
+ing v and u to student and teacher networks leads to y2 = f θs (v) , ˆy1 = f θt (u) , p2 = gθs �
+gθs (y2)
+�
+and ˆz1 = gθt ( ˆy1). Finally, the symmetric objective function Lloss is optimized through minimizing
+the ℓ2 − norm distance between student and teacher as Eq. (12)
+Lloss = 1
+2L (p1, ˆz2) + 1
+2L (p2, ˆz1)
+(12)
+where L(p, z) = −
+p
+∥p∥2 ·
+z
+∥z∥2 and ∥ · ∥2 represents ℓ2 − norm.
+14
+
+B.3
+SPARSE NETWORK FEATURE INTERPRETATION
+We use the Integrated Gradients attribution algorithm to rank the features in all layers. Inspired by
+PNET (Elmarakeby et al., 2021), to reduce the bias introduced by over-annotation of certain nodes
+(nodes that are members of too many pathways), we adjusted the Integrated Gradients scores using a
+graph informed function f that considers the connectivity of each node. The importance score of
+each node i, Cl
+i is divided by the node degree dl
+i if the node degree is larger than the mean of node
+degrees plus 5σ where σ is the standard deviation of node degrees.
+dl
+i = fan − inl
+i + fan − outl
+i
+adjusted Cl
+i = f(x) =
+� Cl
+i
+dl
+i ,
+dl
+i > µ + 5σ
+Cl
+i,
+otherwise
+B.4
+CO-ATTENTION BASED PATHWAY VISUALIZATION
+After we got the ranking of top genes and pathways, we adopted the co-attention survival model
+(MCAT) (Chen et al., 2021) to show the spatial visualization of genomic features. We trained MACT
+on all our TCGA datasets, and MACT learns how WSI patches attend to genes when predicting
+patient survival. We define each WSI patch representation and pathway genomic features as Hbag
+and Gbag. The genomic features are the gene list values from the top pathways of each TCGA dataset.
+The model uses Gbag ∈ RN×dg to guide the feature aggregation of Hbag ∈ RN×dp into a clustered
+set of gene-guided visual concepts �Hbag ∈ RN×dp , dg and dp represents the dimension for the
+pathway (number of genes involved in the pathway) and patch. Through the following mapping:
+CoAttnG→H(G, H) = softmax
+�
+QK⊤
+�
+dp
+�
+= softmax
+�
+WqGH⊤W⊤
+s
+�
+dp
+�
+WvH → Acoattn WvH → �H
+where Wq, Ws, Wv ∈ Rdp×dp are trainable weight matrices multiplied to the queries Gbag and
+key-value pair (Hbag , Hbag ), and Acoattn ∈ RN×M is the co-attention matrix for computing the
+weighted average of Hbag . Here, M represents the number of patches in one slide, and N represents
+the number of pathways (We trained the top four pathways, so N = 4 in our study).
+C
+EXPERIMENTS
+C.1
+NETWORK ARCHITECTURE
+Sparse network for gene: The final gene expression embedding is hg ∈ R1×50.
+Pathology network: The slide level image feature representation is passed through an image embed-
+ding layer and encodes the embedding as hp ∈ R1×50.
+CNV + MUT network: Similarly as the pathology network, the patient level CNV + MUT feature
+representation is passed through an FC embedding layer and encodes the embedding as hc ∈ R1×50.
+C.2
+EXPERIMENTAL DETAILS
+PONET. The latent dimensionality of the factorized matrices k is a very important tuning parameter.
+We tune k = [3, 5, 10, 20, 30, 50] based on the testing C-index value (Appendix Fig. 5) and the loss
+of training and testing plot (Appendix Fig. 6) for each dataset. We choose k to maximize the C-index
+value and also it should have stable convergence in both training and testing loss. For example, we
+choose k = 10 in TCGA-KIRP for the optimized results. We can see that in Appendix Fig. 5 the
+testing loss is quite volatile when k is less than 10. Similarly, we choose k = [20, 10, 20, 20, 10] for
+TCGA-BLCA, TCGA-KIRC, TCGA-LUAD, TCGA-LUSC, and TCGA-PAAD, respectively.
+15
+
+Figure 5: C-Index value under K = 3, 5, 10, 20, 30, 50 for TCGA-KIRP. The mean value and standard
+deviation for 5-fold cross-validation are plotted.
+The learning rate and the regularization hyperparameter λ for the Cox partial likelihood loss are
+also tunable parameters. The model is trained with Adam optimizer. For each training/testing pair,
+we first empirically preset the learning rate to 1.2e-4 as a starting point for a grid search during
+training, the optimal learning rate is determined through the 5-fold cross-validation on the training
+set, C-index was used for the performance metric. After that, the model is trained on all the training
+sets and evaluated on the testing set. We use 2e-3 through the experiments for λ. The batch size is
+set to 16, and the epoch is 100. During the training process, we carefully observe the training and
+testing loss for convergence (Figure 4 in Appendix C.2). The server used for experiments is NVIDIA
+GeForce RTX 2080Ti GPU.
+CoxPH. We only include the age and gender for the survival prediction. Using CoxPHFitter from
+lifelines 3.
+DeepSurv 4. We concatenate preprocessed pathological image features, gene expression, and copy
+number + mutant data in a vector to train the DeepSurv model. L2 reg = 10.0, dropout = 0.4, hidden
+layers sizes = [25, 25], learning rate = 1e-05, learning rate decay = 0.001, momentum = 0.9.
+Pathomic Fusion 5. We use the pathomicSurv model which takes our preprocessed image feature,
+gene expression, and copy number + mutation as model input. k = 20, Learning rate is 2e-3, weight
+decay is 4e-4. The batch size is 16, and the epoch is 100. Drop out rate is 0.25.
+GPDBN 6. The learning rate is 2e-3, the batch size is 16, the weight decay is 1e-6, the dropout rate is
+0.3, and the epoch is 100.
+HFBSurv 7. The learning rate is set to 1e-3, the batch size is 16, λ = 3e-3, weight decay is 1e-6, and
+the epoch is 100.
+3https://github.com/CamDavidsonPilon/lifelines
+4https://github.com/czifan/DeepSurv.pytorch
+5https://github.com/mahmoodlab/PathomicFusion
+6https://github.com/isfj/GPDBN
+7https://github.com/Liruiqing-ustc/HFBSurv
+16
+
+0.8
+0.6
+C-Index
+0.4
+0.2
+0.0
+5
+10
+20
+30
+50
+3
+KFigure 6: Train and test loss for TCGA-KIRP under K = 3, 5, 10, 20, 50 for 5-fold cross-validation.
+17
+
+Fold 1
+Fold 2
+Fold 3
+Fold 4
+Fold 5
+Train
+K=3
+Test
+K=5
+K= 10
+K= 20
+1.06
+K=50
+ 0.65
+0.60
+0.0
+0.50C.3
+ADDITIONAL RESULTS
+Figure 7: Inspecting and interpreting PONET on TCGA-BLCA. Sankey diagram visualization of
+the inner layers of PONET shows the estimated relative importance of different nodes in each layer.
+Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes in
+each layer.
+Figure 8: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-BLCA.
+18
+
+Gene
+Pathways
+GNB1
+PI5P,PP2Aand IER3RegulatePI3K/AKTSignaling
+Toll Like Receptor 10 (TLR10) Cascade
+Cell-extracellular matrix interactions
+PPP2R5E
+SHC-related events triggered by IGF1R
+Cell death signalling via NRAGE, NRIF and NADE
+RNA Polymerase II Transcription Elongation
+KRAS
+MAP2K and MAPK activation
+Interferon gamma signaling
+rRNA processing in the mitochondrion
+Calmodulin induced events
+Mitotic Telophase/Cytokinesis
+Regulation of Hypoxia-inducible Factor (HIF) by oxygen
+PSMA7
+Activation of G protein gated Potassium channels
+FBXW7 Mutants and NOTCH1 in Cancer
+mRNA Splicing
+KPNA2
+outcome
+YWHAB
+Activation of NF-kappaB in B cells
+Golgi-to-ER retrograde transport
+TCR signaling
+GSK3B
+Gap junction degradation
+Signaling by FGFR1 in disease
+Mitotic Spindle Checkpoint
+HSP90AB1
+Phosphate bond hydrolysis by NUDT proteins
+Biosynthesis of DPA-derived SPMs
+ESR-mediated signaling
+TBK1
+NEP/NS2 Interacts with the Cellular Export Machinery
+TCF transactivating complex
+Fatty acid metabolism
+PIK3CA
+p53-IndependentDNADamageResponse
+Interleukin-17 signaling
+Effects of PIP2 hydrolysis
+residual
+residual
+residual
+residualTCGA-4Z-AA7Y
+PI5P, PP2A and IER3
+SHC-related events
+PI3K/AKT Signaling
+MAP2K and MAPK activation
+Calmodulin induced events
+Survival Month: 50
+triggered by IGF-1R
+High Attn
+TCGA-UY-A78N
+Survival Month: 86.76
+Low AttnFigure 9: Inspecting and interpreting PONET on TCGA-KIRC. Sankey diagram visualization of
+the inner layers of PONET shows the estimated relative importance of different nodes in each layer.
+Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes in
+each layer.
+Figure 10: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-KIRC.
+19
+
+TCGA-A3-3313
+Downregulation of
+MAP2K and MAPK activation
+Activation of the
+P53-Independent DNA
+ERBB2:ERBB3 signaling
+Survival Month: 24.15
+pre-replicative complex
+damage response
+High Attn
+TCGA-A3-3320
+Survival Month: 49.54
+Low AttnGene
+Pathways
+TFDP2
+Glucagon-type ligand receptors
+Class B/2 (Secretin family receptors)
+GPCR ligand binding
+MAPK3
+Downregulation of ERBB2:ERBB3 signaling
+G1/STransition
+G1/S DNA Damage Checkpoints
+PTPN11
+MAP2K and MAPK activation
+p53-lndependent G1/S DNA damage checkpoint
+Mitotic G1-G1/S phases
+ADCY5
+Activation of the pre-replicative complex
+RAF/MAP kinase cascade
+Defects in vitamin and cofactor metabolism
+PSMC2
+p53-Independent DNADamage Response
+CLEC7A (Dectin-1) signaling
+MAPK1/MAPK3 signaling
+MTRR
+outcome
+PLCB1
+Processing of DNA double-strand break ends
+Downregulation of ERBB2 signaling
+C-type lectin receptors (CLRs)
+PSMD11
+CLEC7A (Dectin-1) induces NFAT activation
+Defects in cobalamin (B12) metabolism
+HIV Infection
+PSMF1
+RHO GTPases Activate NADPH Oxidases
+HDR or Single Strand Annealing
+Signaling by ERBB2
+IL6ST
+SHC-related events triggered by IGF1R
+G2/M Transition
+Fatty acid metabolism
+Activation of NOXA and translocation to mitochondria
+Activation of BH3-only proteins
+Mitotic G2-G2/M phases
+residual
+residual
+residual
+residualFigure 11: Inspecting and interpreting PONET on TCGA-LUAD. Sankey diagram visualization
+of the inner layers of PONET shows the estimated relative importance of different nodes in each
+layer. Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes in
+each layer.
+Figure 12: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-LUAD.
+20
+
+Gene
+Pathways
+CCT3
+Processive synthesis on the lagging strand
+Lagging Strand Synthesis
+Class I MHC pathwiay
+PSEN1
+HDR through Homologous Recombination (HRR)
+Leading Strand Synthesis
+Signaling by EGFR in Cancer
+Phosphate bond hydrolysis by NUDT proteins
+Antigen processing-Cross presentation
+Intrinsic Pathway for Apoptosis
+EGFR
+Chk1/Chk2(Cds1) mediated inactivation of Cyclin B:Cdk1 complex
+RAF/MAP kinase cascade
+Nucleobase catabolism
+PSMD2
+Polymerase switching
+RHO GTPase Effectors
+Homology Directed Repait
+outcome
+CCT6A
+Purine catabolism
+G2/M Checkpoints
+ER-Phagosome pathway
+PTGES3
+HDR or Single Strand Annealing
+Signaling by Rho GTPases
+Golgi Cisternae Pericentriolar Stack Reorganization
+NUDT1
+G2/M DNA damage checkpoint
+MAPK1/MAPK3 signaling
+Downregulation of ERBB2:ERBB3 signaling
+YWHAZ
+Cap-dependent translation
+GPCR downstream signalling
+Regulation of RAS by GAPs
+RUNX1
+Signaling by Overexpressed Wild-Type EGFR in Cancer
+Glycosaminoglycan metabolism
+Inhibition of Signaling by Overexpressed EGFR
+AKT2 residual
+residual
+residual
+residualTCGA-55-8621
+Processive synthesis on
+HDR through
+Phosphaste bond hydrolysis
+Chk1/Chk2(Cds1) mediated
+Survival Month: 16.92
+the lagging strand
+homologous recombination
+by NUDT proteins
+inactivation of Cyclin B:Cdk1 complex
+High Attn
+TCGA-78-7153
+Survival Month: 119.42
+LowAttnFigure 13: Inspecting and interpreting PONET on TCGA-LUSC. Sankey diagram visualization
+of the inner layers of PONET shows the estimated relative importance of different nodes in each
+layer. Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes in
+each layer.
+Figure 14: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-LUSC.
+21
+
+Gene
+Pathways
+DLG1
+Calmodulin induced events
+TCF transactivating complex
+Cell-extracellular matrix interactions
+UBA52
+CD28 dependent PI3K/Akt signaling
+Pentose phosphate pathway disease
+Neurotransmitterclearance
+PPP2R5E
+PI5P, PP2A and IER3 Regulate PI3K/AKT Signaling
+Signaling by NTRK3 (TRKC)
+rRNA processing in the mitochondrion
+PSMC5
+NrCAM interactions
+Interferon gamma signaling
+TCR signaling
+RAC1
+Constitutive Signaling by NOTCH1
+Glutathione conjugation
+Hedgehog 'on' state
+outcome
+CREB1
+MAP2K and MAPK activation
+Toll Like Receptor 10 (TLR10) Cascade
+Base-Excision Repair, AP Site Formation
+ADAM17
+Negative regulation of MAPK pathway
+Defects in biotin (Btn) metabolism
+Hedgehog 'off state
+PAK2
+Cleavage of the damaged purine
+SUMO E3 ligases
+Regulation of Hypoxia-inducible Factor (HIF) by oxygen
+PSMC2
+AXIN missense mutants destabilize the destruction complex
+RAF-independent MAPK1/3 activation
+Signaling by NOTCH4
+NCOA1
+SHC-related events triggered by IGF1R
+Golgi-to-ER retrograde transport
+Nucleobase biosynthesis
+residual
+residual
+residual
+residualTCGA-18-3414
+CD28 dependent
+PI5P PP2A and IER3
+Calmodulin induced events
+ PI3K/Akt signaling
+regulate PI3K/AKT signaling
+NrCAM interactions
+Survival Month: 23.52
+High Attn
+TCGA-33-4538
+Survival Month: 97.86
+Low AttnFigure 15: Inspecting and interpreting PONET on TCGA-PAAD. Sankey diagram visualization
+of the inner layers of PONET shows the estimated relative importance of different nodes in each
+layer. Nodes in the first layer represent genes; the next layers represent pathways; and the final layer
+represents the model outcome. Different layers are linked by weights. Nodes with darker colors are
+more important, while transparent nodes represent the residual importance of undisplayed nodes in
+each layer.
+Figure 16: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-PAAD.
+22
+
+Gene
+Pathways
+PTGES3
+ SMAD4 MH2 Domain Mutants in Cancer
+Interleukin-6 family signaling
+Mitotic G1-G1/S phases
+C1QC
+SMAD2/3 MH2Domain Mutants in Cancer
+Signaling by FGFR3
+Class I MHC pathway
+PSMD3
+Synthesis of Prostaglandins and Thromboxanes
+AXIN mutants destabilize the destruction complex, activating WNT signalir
+Triglyceride metabolism
+CABIN1
+Regulation by c-FLIP
+Influenza Viral RNA Transcription and Replication 
+RIPK1-mediated regulated necrosis
+NRAS
+C1QB
+Formation of Senescence-Associated Heterochromatin Foci (SAHF)
+Thyroxine biosynthesis
+Reversal of alkylation damage by DNA dioxygenases
+outcome
+EIF3E
+Coenzyme A biosynthesis
+Fusion and Uncoating of the Infuenza Virion
+PIP3 activates AKT signaling
+MGAT4B
+PPCS
+RUNX3 regulates BCL2L11 (BIM) transcription
+EPH-Ephrin signaling
+Metabolism of cofactors
+YWHAG
+FGFR1 mutant receptor activation
+Signaling by NOTCH1 HD Domain Mutants in Cancer
+Platelet Aggregation (Plug Formation)
+MET activates RAP1 and RAC1
+Resolution of AP sites via the single-nucleotide replacement pathway
+GPCR downstream signalling
+Chk1/Chk2(Cds1) mediated inactivation of Cyclin B:Cdk1 complex
+Regulation of innate immune responses to cytosolic DNA
+RNA Polymerase I Promoter Clearance
+residual
+residual
+residual
+residualTCGA-2J-AABO
+Formation of senescence
+Survival Month: 14.45
+Regulation by c-FLIP
+associated heterochromatin foci
+Coenzyme A biosynthesis
+MET activates RAP1 and RAC1
+High Attn
+TCGA-3A-A9IH
+Survival Month: 33.54
+Low Attn
\ No newline at end of file
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new file mode 100644
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@@ -0,0 +1,1254 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf,len=1253
+page_content='DEEP BIOLOGICAL PATHWAY INFORMED PATHOLOGY-  GENOMIC MULTIMODAL SURVIVAL PREDICTION  Lin Qiu1, Aminollah Khormali2, Kai Liu1  1 Division of Research and Early Development, Genentech  {qiul13, liuk3}@gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='com  2 Department of Electrical and Computer Engineering, University of Central Florida  aminollah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='khormali@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='com  ABSTRACT  The integration of multi-modal data, such as pathological images and genomic data,  is essential for understanding cancer heterogeneity and complexity for personalized  treatments, as well as for enhancing survival predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Despite the progress made  in integrating pathology and genomic data, most existing methods cannot mine the  complex inter-modality relations thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Additionally, identifying explainable  features from these models that govern preclinical discovery and clinical prediction  is crucial for cancer diagnosis, prognosis, and therapeutic response studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We  propose PONET- a novel biological pathway informed pathology-genomic deep  model that integrates pathological images and genomic data not only to improve  survival prediction but also to identify genes and pathways that cause different  survival rates in patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Empirical results on six of The Cancer Genome Atlas  (TCGA) datasets show that our proposed method achieves superior predictive  performance and reveals meaningful biological interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The proposed  method establishes insight into how to train biologically informed deep networks on  multimodal biomedical data which will have general applicability for understanding  diseases and predicting response and resistance to treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  1  INTRODUCTION  Manual examination of hematoxylin and eosin (H&E)-stained slides of tumor tissue by pathologists  is currently the state-of-the-art for cancer diagnosis (Chan, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The recent advancements in deep  learning for digital pathology have enabled the use of whole-slide images (WSIs) for computational  image analysis tasks, such as cellular segmentation (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020), tissue  classification and characterization (Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Hekler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Iizuka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' While  H&E slides are important and sufficient to establish a profound diagnosis, genomics data can provide  a deep molecular characterization of the tumor, potentially offering the chance for prognostic and  predictive biomarker discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Cancer prognosis via survival outcome prediction is a standard method used for biomarker discovery,  stratification of patients into distinct treatment groups, and therapeutic response prediction (Cheng  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' WSIs exhibit enormous heterogeneity and most approaches adopt a  two-stage multiple instance learning-based (MIL) approach for the representation learning of WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Firstly, instance-level feature representations are extracted from image patches in the WSI, and then  global aggregation schemes are applied to the bag of instances to obtain a WSI-level representation  for subsequent modeling purpose (Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Courtiol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Wulczyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Lu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Therefore, multimodal survival prediction faces an additional challenge due to the  large data heterogeneity gap between WSIs and genomics, and many existing approaches use simple  multimodal fusion mechanisms for feature integration, which prevents mining important multimodal  interactions (Mobadersany et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  The incorporation of biological pathway databases in a model takes advantage of leveraging prior  biological knowledge so that potential prognostic factors of well-known biological functionality can  be identified (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Moreover, encoding biological pathway information into the neural  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='02383v1  [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='QM]  6 Jan 2023  Figure 1: Overview of PONET model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  networks achieved superior predictive performance compared with established models (Elmarakeby  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Based on the current challenges in multimodal fusion of pathology and genomics and the potential  prognostic interpretation to link pathways and clinical outcomes in pathway-based analysis, we  propose a novel biological pathway-informed pathology-genomic deep model, PONET, that uses H&E  WSIs and genomic profile features for survival prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The proposed method contains four major  contributions: 1) PONET formulates a biological pathway-informed deep hierarchical multimodal  integration framework for pathological images and genomic data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 2) PONET captures diverse and  comprehensive modality-specific and cross-modality relations among different data sources based  on the factorized bilinear model and graph fusion network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 3) PONET reveals meaningful model  interpretations on both genes and pathways for potential biomarker and therapeutic target discovery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  PONET also shows spatial visualization of the top genes/pathways which has enormous potential  for novel and prognostic morphological determinants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 4) We evaluate PONET on six public TCGA  datasets which showed superior survival prediction comparing to state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 1  shows our model framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  2  RELATED WORK  Multimodal Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Earlier works on multimodal fusion focus on early fusion and late fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Early  fusion approaches fuse features by simple concatenation which cannot fully explore intra-modality  dynamics (Wöllmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Poria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Zadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' In contrast, late fusion  fuses different modalities by weighted averaging which fails to model cross-modal interactions  (Nojavanasghari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Kampman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The exploitation of relations within each  modality has been successfully introduced in cancer prognosis via bilinear model (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021b)  and graph-based model (Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Adversarial Representation Graph Fusion (ARGF)  (Mai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020) interprets multimodal fusion as a hierarchical interaction learning procedure where  firstly bimodal interactions are generated based on unimodal dynamics, and then trimodal dynamics  are generated based on bimodal and unimodal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We propose a new hierarchical fusion  framework with modality-specific and cross-modality attentional factorized bilinear modules to  mine the comprehensive modality interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Our proposed hierarchical fusion framework is  different from ARGF in the following ways: 1) We take the sum of the weighted modality-specific  representation as the unimodal representation instead of calculating the weighted average of the  modality-specific representation in ARGF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 2) For higher level’s fusion, ARGF takes the original  embeddings of each modality as input while we use the weighted modality-specific representations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  3) We argue that ARGF takes redundant information during their trimodal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Multimodal Survival Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' There have been exciting attempts on multimodal fusion of pathol-  ogy and genomic data for cancer survival prediction (Mobadersany et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Cheerla & Gevaert,  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' However, these multimodal fusion based methods fail to model the interac-  tion between each subset of multiple modalities explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Kronecker product considers pairwise  interactions of two input feature vectors by producing a high-dimensional feature of quadratic ex-  pansion (Zadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017), and showed its superiority in cancer survival prediction (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=',  2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Despite promising results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' using Kronecker product in multimodal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Unimodal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Gene layer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Pathway layer Hidden layer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MFB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='>C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Atten  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='um  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='hm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='hm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='hm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='↑  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Bimodal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='himz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Gene  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='hg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Gene  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MFB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Multimodal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Atten  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='zuuy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Wm1m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Representation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='hm1m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='hm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Cox  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Pathway  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Pathology  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Trimodal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='↓  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='hp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='hmi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='fe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MFB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Atten  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Wm1m2m3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='hm1m2m3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='CNV + MUT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='hc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Spatial Pathway  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Data embedding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Hierarchical multimodal fusion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Interpretationfusion may introduce a large number of parameters that may lead to high computational cost and risk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='of overfitting (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021), thus limiting its applicability and improvement in  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' To overcome this drawback, hierarchical factorized bilinear fusion for cancer survival  prediction (HFBSurv) (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2022) uses factorized bilinear model to fuse genomic and image  features, dramatically reducing computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' PONET differs from HFBSurv in two  ways: 1) PONET’s multimodal framework has three levels of hierarchical fusion module includ-  ing unimodal, bimodal, and trimodal fusion while HFBSurv only considers within-modality and  cross-modality fusion which we argue it is not adequate for mining the comprehensive interactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  2) PONET leverages biological pathway informed network for better prediction and meaningful  interpretation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Pathway-associated Sparse Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The pathway-based analysis is an approach that a  number of studies have investigated to improve both predictive performance and biological inter-  pretability (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Cirillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Elmarakeby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Moreover,  pathway-based approaches have shown more reproducible analysis results than gene expression data  analysis alone (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Mallavarapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' These pathway-based deep neural networks  can only model genomic data which severely inhibits their applicability in current biomedical re-  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Additionally, the existing pathway-associated sparse neural network structures are limited  for disease mechanism investigation: there is only one pathway layer in PASNet (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2018)  which contains limited biological prior information to deep dive into the hierarchical pathway and  biological process relationships;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' P-NET (Elmarakeby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021) calculates the final prediction by  taking the average of all the gene and pathway layers’ outputs, and this will bias the learning process  because it will put more weights for some layers’ outputs while underestimating the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  3  METHODOLOGY  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='1  PROBLEM FORMULATION AND NOTATIONS  The model architecture of PONET is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 1, where three modalities are included as input:  gene expression g ∈ Rdg, pathological image p ∈ Rdp, and copy number (CNV) + mutation (MUT)  CNV + MUT ∈ Rdc, with dp being the dimensionality of p and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We define a hierarchical  factorized bilinear fusion model for PONET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We build a sparse biological pathway-informed embed-  ding network for gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' A fully connected (FC) embedding layer for both preprocessed  pathological image feature (fp) and the copy number + mutation (fc) to map feature into similar  embedding space for alleviating the statistical property differences between modalities, the three  network architecture details are in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We label the three modality embeddings as hm,  m ∈ {g, p, c}, the superscript/subscript u, b, and t represents unimodal fusion, biomodal fusion and  trimodal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' After that, the embeddings of each modality are first used as input for unimodal fusion  to generate the modality-specific representation hu  m = ωmˆhm, ωm represent the modality-specific im-  portance, the feature vector of the unimodal fusion is the sum of all modality-specific representations  hu = �  m hu  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' In the bimodal fusion, modality-specific representations from the output of unimodal  fusion are fused to yield cross-modality representations hb  m1m2 = ωm1m2ˆhm1m2, m1, m2 ∈ {p, c, g}  and m1 ̸= m2, ωm1m2 represents the corresponding cross-modality importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Similarly, the feature  vector of bimodal fusion is calculated as hb = �  m1,m2 hb  m1m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We propose to build a trimodal  fusion to take each cross-modality representation from the output of bimodal fusion to mine the  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Similarly to the bimodal fusion architecture, the trimodal fusion feature vector will  be ht = �  m1,m2,m3 ωm1m2m3ˆhm1m2m3, m1, m2, m3 ∈ {p, c, g} and m1 ̸= m2 ̸= m3, ωm1m2m3  represents the corresponding trimodal importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Finally, PONET concatenates hu, hb, ht to obtain  the final comprehensive multimodal representation and pass it to the Cox proportional hazards model  (Cox, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Cheerla & Gevaert, 2019) for survival prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' In the following sections we will  describe our hierarchical factorized bilinear fusion framework, l, o, s represents the dimensionality  of hm, zm, ˆhm1m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='2  SPARSE NETWORK  We design the sparse gene-pathway network consisting of one gene layer followed by three pathway  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' A patient sample of e gene expressions is formed as a column vector, which is denoted by  X = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', xe], each node represents one gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The gene layer is restricted to have connections  3  73,703 x 50,706 px  224 x 224 px, mpp: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='5  Whole Slide Image  WSI patching  Image Augmentation  𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' (#)  𝑞!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' (#)  𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' "(#)  𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' (#)  𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' "(#)  𝑦"  \'𝑦#  𝑧"  ̂𝑧#  𝑝"  𝐿(𝑝, 𝑧)  Visual representation learning using SSL ViT  Student Network  Teacher Network  Patch features  𝑣  𝑢  Figure 2: Overall framework of the visual representation extraction using pre-trained self-supervised  vision transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  reflecting the gene-pathway relationships curated by the Reactome pathway dataset (Fabregat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=',  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The connections are encoded by a binary matrix M ∈ Ra×e, where a is the number of  pathways and e is the number of genes, an element of M, mij, is set to one if gene j belongs to  pathway i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The connections that do not exist in the Reactome pathway dataset will be zero-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' For  the following pathway-pathway layers, a similar scheme is applied to control the connection between  consecutive layers to reflect the parent-child hierarchical relationships that exist in the Reactome  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The output of each layer is calculated as  y = f[(M ∗ W)T X + ϵ]  (1)  where f is the activation function, M represents the binary matrix, W is the weights matrix, X is the  input matrix, ϵ is the bias vector, and ∗ is the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We use tanh for the activation of  each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We allow the information flow from the biological prior informed network starting from  the first gene layer to the last pathway layer, and we label the last layer output embeddings of the  sparse network for gene expression as hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='3  UNIMODAL FUSION  Bilinear models (Tenenbaum & Freeman, 2000) provide richer representations than linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Given two feature vectors in different modalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', the visual features x ∈ Rm×1 for an image and  the genomic features y ∈ Rn×1 for a genomic profile, the bilinear model uses a quadratic expansion  of linear transformation considering every pair of features:  zi = xT Wiy  (2)  where Wi ∈ Rm×n is a projection matrix, zi ∈ R is the output of the bilinear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Bilinear  models introduce a large number of parameters which potentially lead to high computational cost  and overfitting risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' To address these issues, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' (2017) develop the Multi-modal Factorized  Bilinear pooling (MFB) method, which enjoys the dual benefits of compact output features and robust  expressive capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Inspired by the MFB (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017) and its application in pathology and genomic multimodal  learning (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2022), we propose unimodal fusion to capture modality-specific representations  and quantify their importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The unimodal fusion takes the embedding of each modality hm as  input and factorizes the projection matrix Wi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' (2) as two low-rank matrices:  zi  =  hT  mWihm =  k�  d=1  hT  mum,dvT  m,dhm  =  1T (U T  m,ihm ◦ V T  m,ihm), m ∈ {p, c, g}  (3)  we get the output feature zm:  zm = SumPooling  �  ˜U T  mhm◦ ˜V T  mhm, k  �  , m ∈ {p, c, g}  (4)  where k is the latent dimensionality of the factorized matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' SumPooling (x, k) function performs  sum pooling over x by using a 1-D non-overlapped window with the size k, ˜Um ∈ Rl×ko and  ˜Vm ∈ Rl×ko are 2-D matrices reshaped from Um and Vm, Um =[Um,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' , Um,h] ∈ Rl×k×o and  Vm = [Vm,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' , Vm,h] ∈ Rl×k×o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Each modality-specific representation ˆhm ∈ Rl+o is obtained  as:  ˆhm = hm©zm, m ∈ {p, c, g}  (5)  4  where © denotes vector concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We also introduce a modality attention network Atten ∈  Rl+o → R1 to determine the weight for each modality-specific representation to quantify its impor-  tance:  ωm = Atten(ˆhm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' ΘAtten), m ∈ {p, c, g}  (6)  where ωm is the weight of modality m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' In practice, Atten consists of a sigmoid activated dense layer  parameterized by ΘAtten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Therefore, the output of each modality in unimodal fusion, hu  m, is denoted  as ωmˆhm ∈ Rl+o, m ∈ {p, c, g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Accordingly, the output of unimodal fusion, hu, is the sum of each  weighted modality-specific representation ωmˆhm, m ∈ {p, c, g} which is different from ARGF (Mai  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020) that used the weighted average of different modalities as the unimodal fusion output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4  BIMODAL AND TRIMODAL FUSION  Bimodal fusion aims to fuse diverse information of different modalities and quantify different  importance for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' After receiving the modality-specific representations hu  m from the unimodal  fusion, we can generate the cross-modality representation ˆhm1m2 ∈ Rs similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' (4) :  ˆhm1,m2 = Sum Pooling  �  ˜U T  m1hu  m1◦ ˜V T  m2hu  m2, k  �  ,  m1, m2 ∈ {p, c, g}, m1 ̸= m2  (7)  where  ˜U T  m1 ∈ R(l+o)×ks and ˜V T  m2 ∈ R(l+o)×ks are 2-D matrices reshaped from Um1 and Vm2  and Um1 = [Um1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' , Um1,s] ∈ R(l+o)×k×s and Vm2 = [Vm2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' , Vm2,s] ∈ R(l+o)×k×s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We  leverage a bimodal attention network (Mai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020) to identify the importance of the cross-  modality representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The similarity Sm1m2 ∈ R1 of hu  m1 and hu  m2 is first estimated as follows:  Sm1,m2 =  l+o  �  i=1  �  eωm1 hu  m1,i  �l+o  j=1 eωm1hu  m1,j  � �  eωm2hu  m2,i  �l+o  j=1 eωm2hu  m2,j  �  (8)  where the computed similarity is in the range of 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Then, the cross-modality importance ωm1m2  is obtained by:  ωm1m2 =  eˆωmimj  �  mi̸=mj eˆωmimj , ˆωm1m2 = ωm1 + ωm2  Sm1m2 + S0  (9)  where S0 represents a pre-defined term controlling the relative contribution of similarity and modality-  specific importance, and here is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Therefore, the output of bimodal fusion, hb, is the sum of  each weighted cross-modality representation ωm1m2ˆhm1m2, m1, m2 ∈ {p, c, g} and m1 ̸= m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  In trimodal fusion, each bimodal fusion output is fused with the unimodal fusion output that does  not contribute to the formation of the bimodal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The output for each corresponding trimodal  representation is ˆhm1m2m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' In addition, trimodal attention was applied to identify the importance of  each trimodal representation, ωm1m2m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The output of the trimodal fusion, ht, is the sum of each  weighted trimodal representation ωm1m2m3ˆhm1m2m3, m1, m2, m3 ∈ {p, c, g} and m1 ̸= m2 ̸= m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='5  SURVIVAL LOSS FUNCTION  We train the model through the Cox partial likelihood loss (Cheerla & Gevaert, 2019) with l1  regularization for survival prediction, which is defined as:  ℓ(Θ) = −  �  i:Ei=1  �  �ˆhΘ (xi) − log  �  j:Ti>Tj  exp  �  ˆhΘ (xj)  �  �  � + λ (∥Θ∥1)  (10)  where the values Ei, Ti and xi for each patient represent the survival status, the survival time, and the  feature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Ei = 1 means event while Ei = 0 represents censor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' ˆhΘ is the neural network  model trained for predicting the risk of survival, Θ is the neural network model parameters, and λ is  a regularization hyperparameter to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  5  4  EXPERIMENTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='1  EXPERIMENTAL SETUP  Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' To validate our proposed method, we used six cancer datasets from The Cancer Genome  Atlas (TCGA), a public cancer data consortium that contains matched diagnostic WSIs and genomic  data with labeled survival times and censorship statuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The genomic profile features (mutation  status, copy number variation, RNA-Seq expression) are preprocessed by Porpoise 1 (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=',  2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' For this study, we used the following cancer types: Bladder Urothelial Carcinoma (BLCA)  (n = 437), Kidney Renal Clear Cell Carcinoma (KIRC) (n = 350), Kidney Renal Papillary Cell  Carcinoma (KIRP) (n = 284), Lung Adenocarcinoma (LUAD) (n = 515), Lung Squamous Cell  Carcinoma (LUSC) (n = 484), Pancreatic adenocarcinoma (PAAD) (n = 180).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We downloaded the  same diagnostic WSIs from the TCGA website 2 that were used in Porpoise study to match the  paired genomic features and survival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The feature alignment table for all the cancer types  is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' For each WSI, automated segmentation of tissue was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Following  segmentation, image patches of size 224 × 224 were extracted without overlap at the 20 X equivalent  pyramid level from all tissue regions identified while excluding the white background and selecting  only patches with at least 50% tissue regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Subsequently, a visual representation of those patches  is extracted with a vision transformer (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021a) pre-trained on the TCGA dataset through  a self-supervised constructive learning approach, such that each patch is represented as a 1 × 2048  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 2 shows the framework for the visual representation extraction by vision transformer  (VIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Survival outcome information is available at the patient level, we aggregated the patch-level  feature into slide level feature representations based on an attention-based method (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Ilse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Using the same 5-fold cross-validation splits for evaluating PONET, we implemented and  evaluated six state-of-the-art methods for survival outcome prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Additionally, we included  three variations of PONET: a) PONET-O represents only genomic data, and pathway architecture  for the gene expression are included in the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' b) PONET-OH represents only genomic and  pathological image data but without pathway architecture in the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' c) PONET is our full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  For all methods, we use the same VIT feature extraction pipeline for WSIs, as well as identical  training hyperparameters and loss functions for supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Training details and the parameters  tuning can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  CoxPH (Cox, 1972) represents the standard Cox proportional hazard models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  DeepSurv (Katzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2018) is the deep neural network version of the CoxPH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Pathomic Fusion (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2022a) as a pioneered deep learning-based framework for predicting  survival outcomes by fusing pathology and genomic multimodal data, in which Kronecker product is  taken to model pairwise feature interactions across modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  GPDBN (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021b) adopts Kronecker product to model inter-modality and intra-modality  relations between pathology and genomic data for cancer prognosis prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  HFBSurv (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2022) extended GPDBN using the factorized bilinear model to fuse genomic  and pathology features in a within-modality and cross-modalities hierarchical fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Porpoise (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2022b) applied the discrete survival model and Kronecker product to fuse  pathology and genomic data for survival prediction (Zadeh & Schmid, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' For each cancer dataset, we used the cross-validated concordance index (C-Index)  (Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='1) (Harrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 1982) to measure the predictive performance of correctly ranking the  predicted patient risk scores with respect to overall survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='2  RESULTS  Comparison with Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' In combing pathology image, genomics, and pathway network via  PONET, our approach outperforms CoxPH models, unimodal networks, and previous deep learning-  based approaches on pathology-genomic-based survival outcome prediction (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The results  show that deep learning-based approaches generally perform better than the CoxPH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' PONET  achieves superior C-index values in all six cancer types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' All versions of PONET outperform Pathomic  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='com/mahmoodlab/PORPOISE  2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='gov/about-nci/organization/ccg/research/structural-genomics/tcga  6  Table 1: C-Index (mean ± standard deviation) of PONET and ablation experiments in TCGA survival  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The top two performers are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Model  TCGA-BLCA  TCGA-KIRC  TCGA-KIRP  TCGA-LUAD  TCGA-LUSC  TCGA-PAAD  CoxPH (Age + Gender) (Cox, 1972)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='525 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='130  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='550 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='070  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='544 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='531 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='082  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='094  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='539 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='092  DeepSurv (Kampman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2018)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='076  GPDBN (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='527 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='664 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='761 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='093  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='623 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='062  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='538 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='598 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='027  PONET-OH (ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='625 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='063  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='695 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='776 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='123  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='618 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='049  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='553 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='049  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='591 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='050  PONET (ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='643 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='037  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='829 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='054  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='646 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='047  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='567 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='066  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='639 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='080  Table 2: Evaluation of PONET on different fusion methods and pathway designs by C-index (mean  ± standard deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The best performer is highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='652 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='049  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='554 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='065  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='525 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='066  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='568 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='075  Element-wise addition  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='592 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='047  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='655 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='055  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='587 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='065  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='522 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='046  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='588 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='055  Tensor fusion (Zadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='605 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='046  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='775 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='595 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='060  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='545 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='592 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='061  Hierarchical fusion  Unimodal  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='596 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='783 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='063  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='611 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='056  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='553 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='073  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='595 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='053  Bimodal  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='602 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='062  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='789 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='601 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='056  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='552 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='051  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='598 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='083  ARGF (Mai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='597 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='054  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='792 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='043  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='614 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='051  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='556 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='063  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='602 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='065  Unimodal + Bimodal  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='614 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='052  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='803 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='061  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='631 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='044  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='578 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='058  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='615 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='057  Pathway design  PASNet (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='793 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='051  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='621 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='061  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='069  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='625 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='057  P-NET (Elmarakeby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='622 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='047  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='802 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='071  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='625 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='562 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='054  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='627 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='073  PONET  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='643 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='037  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='829 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='054  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='641 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='046  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='567 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='066  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='639 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='070  Fusion by a big margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Pathomic Fusion uses Kronecker product to fuse the two modalities, and  that’s also the reason why other advanced fusion methods, like GPDBN and HFBSurv, achieve better  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Also, we argue that Pathomic Fusion extracts the region of interest of pathology image  for feature extraction might limit the understanding of the tumor microenvironment of the whole slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  HFBSurv shows better performance than GPDBN and Pathomic Fusion which is consistent with  their findings, and these results further demonstrate that the hierarchical factorized bilinear model  can better mine the rich complementary information among different modalities compared to the  Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Porpoise performs similarly with PONET on TCGA-KIRC and TCGA-KIRP and  outperformed HFBSurv in these two studies, this probably is due to Porpoise partitioned the survival  time into different non-overlapping bins and parameterized it as a discrete survival model (Zadeh  & Schmid, 2020) which works better for these two cancer types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' In other cases, Porpoise performs  similarly to HFBSurv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Note: the results of Porpoise are from their paper (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Additionally, we can see that PONET consistently outperforms PONET-O and PONET-OH indi-  cating the effectiveness of the biological pathway-informed neural network and the contribution of  pathological image for the overall survival prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Ablation Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' To assess whether the impact of hierarchical factorized bilinear fusion strategy  is indeed effective, we compare PONET with four single-fusion methods: 1) Simple concatenation:  concatenate each modality embeddings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 2) Element-wise addition: element-wise addition from each  modality embeddings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 3) Tensor fusion (Zadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017): Kronecker product from each modality  embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Table 2 shows the C-index values of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We can see that PONET  achieves the best performance and shows remarkable improvement over single-fusion methods on  different cancer type datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' For example, PONET outperforms the Simple concatenation by  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4% (TCGA-BLCA), 27% (TCGA-KIRP), 15% (TCGA-LUAD), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='0% (TCGA-LUSC), and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4%  (TCGA-PAAD), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Furthermore, we adopted five different configurations of PONET to evaluate each hierarchical  component of the proposed method: 1) Unimodal: unimodal fusion output as the final feature  representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 2) Bimodal: bimodal fusion output as the final feature representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 3) Unimodal  + Bimodal: hierarchical (include both unimodal and bimodal feature representation) fusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 4)  ARGF: ARGF (Mai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020) fusion strategy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 5) PONET: our proposed hierarchical strategy by  incorporating unimodal, bimodal, and trimodal fusion output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' As shown in Table 2, Unimodal +  Bimodal performs better than Unimodal and Bimodal which demonstrates that Unimodal + Bimodal  can capture the relations within each modality and across modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' ARGF performs worse than  Unimodal + Bimodal and far worse than PONET across all the cancer types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' PONET outperforms  7  Figure 3: Inspecting and interpreting PONET on TCGA-KIRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' a: Sankey diagram visualization  of inner layers of PONET shows the estimated relative importance of different nodes in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Nodes in the first layer represent genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' the next layers represent pathways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' and the final layer  represents the model outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Different layers are linked by weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Nodes with darker colors are  more important, while transparent nodes represent the residual importance of undisplayed nodes  in each layer, H1 presents the gene layer, and H2-H4 represent pathway layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' b: Co-attention  visualization of top 4 ranked pathways in one case of TCGA-KIRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Unimodal + Bimodal in 4 out of 5 cancer types indicating that three layers of hierarchical fusion can  mine the comprehensive interactions among different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  To evaluate our sparse gene-pathway network design, we compare PONET with PASNet (Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=',  2018) and P-NET (Elmarakeby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021) pathway architecture, PASNet performs the worst due to  the fact that it only has one pathway layer in the network, and thus limited prior information was used  to predict the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' PONET constantly outperforms P-NET across all the cancer types, which  demonstrates that averaging all the intermediate layers’ output for the final prediction cannot fully  capture the prior information flow among the hierarchical biological structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Model Interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We discuss the model interpretation results for cancer type TCGA-KIRP  here and the results for other cancer types are included in the Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' To understand the  interactions between different genes, pathways, and biological processes that contributed to the  predictive performance and to study the paths of impact from the input to the outcome, we visualized  the whole structure of PONET with the fully interpretable layers after training (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 3 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' To evaluate  the relative importance of specific genes contributing to the model prediction, we inspected the genes  layer and used the Integrated Gradients attribution (Sundararajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017) method to obtain  the total importance score of genes, and the modified ranking algorithm details are included in the  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Highly ranked genes included KRAS, PSMB6, RAC1, and CTNNB1 which are known  kidney cancer drivers previously (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Shan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Al-Obaidy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Guo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' GBN2, a member of the guanine nucleotide-binding proteins family, has been reported  that the decrease of its expression reduced tumor cell proliferation (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' A recent study  identified a strong dependency on BCL2L1, which encodes the BCL-XL anti-apoptotic protein, in a  subset of kidney cancer cells (Grubb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' This biological interpretability revealed established  and novel molecular features contributing to kidney cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' In addition, PONET selected a hierarchy  of pathways relevant to the model prediction, including downregulation of TGF-β receptor signaling,  regulation of PTEN stability and activity, the NLRP1 inflammasome, and noncanonical activation of  NOTCH3 by PSEN1, PSMB6, and BCL2L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' TGF-β signaling is increasingly recognized as a key  driver in cancer, and in progressive cancer tissues TGF-β promotes tumor formation, and its increased  expression often correlates with cancer malignancy (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Noncanonical activation of  NOTCH3 was reported to limit tumor angiogenesis and plays a vital role in kidney disease (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=',  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='H1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='H2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='H3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='H4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='KRAS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Downregulation of TGF-beta receptor signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='TGF-beta receptor signaling activates SMADs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Neurodegenerative Diseases  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PSMB6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Regulation of PTEN stability and activity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='NOTCH3 Activation and Transmission of Signal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Cellular Senescence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RAC1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Calmodulin induced events  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Semaphorin interactions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Signal amplification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='CTNNB1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='The NLRP1 inflammasome  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Downstream signaling events of B Cell Receptor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Signaling by FGFR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='BCL2L1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Noncanonical activation of NOTCH3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='M Phase  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='LeadingStrand Synthesis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='GNB2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Synthesis of PIPs in the nucleus  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Biosynthesis of the N-glycan precursor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='ER to Golgi Anterograde Transport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='outcome  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PSEN1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Regulation of PTEN gene transcription  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Biosynthesis of DHA-derived SPMs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Signaling by TGF-beta Receptor Complex in Cancer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MTMR4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RPA3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Viral mRNA Translation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Export of Viral Ribonucleoproteins from Nucleus  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Interferon Signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='HDAC3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Complex I biogenesis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='ZBP1(DAI) mediated induction of type I IFNs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Transportofvitaminsandnucleosides  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Formation of tubulin folding intermediates by CCT/TriC Amine Oxidase reactions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Transportofbilesalts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='organicacids,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='andmetalions  residual  residual  residual  b  TCGA-Q2-A5QZ  Downregulation of TGF-beta  Regulation of PTEN  Survival Month: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='06  receptor signaling  stability and activity  Calmodulin induced events  The NLRP1 inflammasome  High Attn  Low AttnFigure 4: Kaplan-Meier analysis of patient stratification of low and high risk patients via four  variations of PONET on TCGA-KIRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Low and high risks are defined by the median 50% percentile  of hazard predictions via each model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Log-rank test was used to test for statistical  significance in survival distributions between low and high risk patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  To further inspect the pathway spatial association with the WSI slide we adopted the co-attention  survival method MCAT (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021) between WSIs and genomic features on the top  pathways of the second layer, visualized as a WSI-level attention heatmap for each pathway  genomic embedding in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  3 b (algorithm details are included in the Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  We  used the gene list from the top 4 pathways as the genomic features and trained MCAT on the  TCGA-KIRP dataset for survival prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Overall, we observe that high attention in different  pathways showed different spatial pattern associations with the slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' This heatmap can reflect  genotype-phenotype relationships in cancer pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The high attention regions (red) of dif-  ferent pathways in the heatmap have positive associations with the predicted death risk while  the low attention regions (blue) have negative associations with the predicted risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' By further  checking the cell types in high attention patches we can gain insights of prognostic morpho-  logical determinants and have a better understanding of the complex tumor microenvironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Table 3: Comparison of model complexity  Methods  Number of Parameters  FLOPS  Pathomic Fusion  175M  168G  GPDBN  82M  91G  HFBSurv  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='3M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='5G  PONET  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='8M  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='1G  Patient Stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  In visualizing  the Kaplan-Meier survival curves of pre-  dicted high risk and low risk patient  populations, we plot four variations of  PONET in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' PONET-ARGF rep-  resents the model that we use the hier-  archical fusion strategy of ARGF in our  pathway-informed PONET model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' From  the results, PONET enables easy sepa-  ration of patients into low and high risk  groups with remarkably better stratifica-  tion (P-Value = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='60e-7) in comparison to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Complexity Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We compared PONET with Pathomic Fusion, GPDBN, and HFBSurv  since both Pathomic Fusion and GPDBN are based on Kronecker product to fuse different modalities  while GPDBN and HFBSurv modeled inter-modality and intra-modality relations which have similar  consideration to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' As illustrated in Table 3, PONET has 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='8M (M = Million) trainable  parameters, which is approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='6%, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4%, and 900% of the number of parameters of Pathomic  Fusion, GPDBN, and HFBSurv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' To assess the time complexity of PONET and the competitive  methods, we calculate each method’s floating-point operations per second (FLOPS) in testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The  results in Table 3 show that PONET needs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='1G during testing, compared with 168G, 91G, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='5G  in Pathomic Fusion, GPDBN, and HFBSurv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The main reason for fewer trainable parameters and the  number of FLOPS lies in that PONET and HFBSurv perform multimodal fusion using the factorized  bilinear model, and can significantly reduce the computational complexity and meanwhile obtain  more favorable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' PONET has one additional trimodal fusion which explains why it has  more trainable parameters than HFBSurv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  5  CONCLUSION  In this study, we pioneer propose a novel biological pathway-informed hierarchical multimodal  fusion model that integrates pathology image and genomic profile data for cancer prognosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' In  comparison to previous works, PONET deeply mines the interaction from multimodal data by  conducting unimodal, bimodal and trimodal fusion step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Empirically, PONET demonstrates  9  PONET-O  PONET-OH  PONET-ARGF  PONET  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='0 -  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='0  P-Value =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='90e-3  P-Value =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='49e-4  P-Value =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='27e-5  P-Value =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='60e-7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='7  Proportion s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='5  Low risk  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4  Low risk  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4  Low risk  Low risk  High risk  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4  High risk  High risk  High risk  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='3  0  25  50  75  100  125  0  25  50  75  100  125  0  25  50  75  100  125  0  25  50  75  100  125  Time (months)the effectiveness of the model architecture and the pathway-informed network for superior predictive  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Specifically, PONET provides insight on how to train biologically informed deep  networks on multimodal biomedical data for biological discovery in clinic genomic contexts which  will be useful for other problems in medicine that seek to combine heterogeneous data streams for  understanding diseases and predicting response and resistance to treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  REFERENCES  Khaleel I Al-Obaidy, John N Eble, Mehdi Nassiri, Liang Cheng, Mohammad K Eldomery, Sean R  Williamson, Wael A Sakr, Nilesh Gupta, Oudai Hassan, Muhammad T Idrees, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Recurrent kras  mutations in papillary renal neoplasm with reverse polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Modern Pathology, 33(6):1157–1164,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  John KC Chan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The wonderful colors of the hematoxylin-eosin stain in diagnostic surgical pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content=' Identification of potential diagnostic and prognostic biomarkers  for prostate cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Oncology Letters, 18(4):4237–4245, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  13  Table 4: TCGA Data Feature Alignment Summary  WSI  CNV  MUT  RNA  WSI+CNV+MUT  WSI+MUT+RNA  ALL  Cancer Type  BLCA  454  443  452  450  441  448  437  KIRC  517  509  357  514  352  355  350  KIRP  294  291  286  293  284  285  284  LUAD  528  522  523  522  519  519  515  LUSC  505  502  489  503  486  487  484  PAAD  208  201  187  195  187  180  180  A  DATA  Table 3 in Appendix A shows the number of patients with matched different data modalities: WSI  (Whole slide image),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' CNV (Copy number),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' MUT (Mutation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' RNA (RNA-Seq gene expression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' For  each TCGA dataset and each patient we have preprocessed data dimensions dg ∈ R1×2000 (RNA),  dc ∈ R1×227 (CNV + MUT), and dp ∈ R1×32 (WSI) which will be used for our multimodal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  B  METHODS  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='1  C-INDEX  We use concordance-index (C-index) (Harrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 1982) to measure the performance of survival  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' It evaluates the model by measuring the concordance of the ranking of predicted harzards  with the true survival time of patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The range of the C-index is [0, 1], and larger values indicate  better performance with a random guess leading to a C-index of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='2  WSI REPRESENTATION LEARNING  It has been shown that the WSI visual representations extracted by self-supervised learning methods  on histopathological images are more accurate and transferable than the supervised baseline models  on domain-irrelevant datasets such as ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' In this work, a pre-trained Vision Transformer (ViT)  model (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021a) that is trained on a large histopathological image dataset has been utilized  for tile feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The model is composed of two main neural networks that learn from each  other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', student and teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Parameters of the teacher model θt are updated using the  student network with parameter θs using the update rule represented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  θt ← τθt + (1 − τ)θs  (11)  Two different views of a given input H&E image x, uniformly selected from the training set I, are  generated using random augmentations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Then, student and teacher models generate two  different visual representations according to u and v as y1 = f θs (u) and ˆy2 = f θt (v), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Finally, the generated visual representations are transformed into latent space using linear projection as  p1 = gθs �  gθs (y1)  �  and ˆz2 = gθt (ˆy2) for student and teacher networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Similarly, feed-  ing v and u to student and teacher networks leads to y2 = f θs (v) , ˆy1 = f θt (u) , p2 = gθs �  gθs (y2)  �  and ˆz1 = gθt ( ˆy1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Finally, the symmetric objective function Lloss is optimized through minimizing  the ℓ2 − norm distance between student and teacher as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' (12)  Lloss = 1  2L (p1, ˆz2) + 1  2L (p2, ˆz1)  (12)  where L(p, z) = −  p  ∥p∥2 ·  z  ∥z∥2 and ∥ · ∥2 represents ℓ2 − norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  14  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='3  SPARSE NETWORK FEATURE INTERPRETATION  We use the Integrated Gradients attribution algorithm to rank the features in all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Inspired by  PNET (Elmarakeby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021), to reduce the bias introduced by over-annotation of certain nodes  (nodes that are members of too many pathways), we adjusted the Integrated Gradients scores using a  graph informed function f that considers the connectivity of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The importance score of  each node i, Cl  i is divided by the node degree dl  i if the node degree is larger than the mean of node  degrees plus 5σ where σ is the standard deviation of node degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  dl  i = fan − inl  i + fan − outl  i  adjusted Cl  i = f(x) =  � Cl  i  dl  i ,  dl  i > µ + 5σ  Cl  i,  otherwise  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4  CO-ATTENTION BASED PATHWAY VISUALIZATION  After we got the ranking of top genes and pathways, we adopted the co-attention survival model  (MCAT) (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=', 2021) to show the spatial visualization of genomic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We trained MACT  on all our TCGA datasets, and MACT learns how WSI patches attend to genes when predicting  patient survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We define each WSI patch representation and pathway genomic features as Hbag  and Gbag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The genomic features are the gene list values from the top pathways of each TCGA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  The model uses Gbag ∈ RN×dg to guide the feature aggregation of Hbag ∈ RN×dp into a clustered  set of gene-guided visual concepts �Hbag ∈ RN×dp , dg and dp represents the dimension for the  pathway (number of genes involved in the pathway) and patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Through the following mapping:  CoAttnG→H(G, H) = softmax  �  QK⊤  �  dp  �  = softmax  �  WqGH⊤W⊤  s  �  dp  �  WvH → Acoattn WvH → �H  where Wq, Ws, Wv ∈ Rdp×dp are trainable weight matrices multiplied to the queries Gbag and  key-value pair (Hbag , Hbag ), and Acoattn ∈ RN×M is the co-attention matrix for computing the  weighted average of Hbag .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Here, M represents the number of patches in one slide, and N represents  the number of pathways (We trained the top four pathways, so N = 4 in our study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  C  EXPERIMENTS  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='1  NETWORK ARCHITECTURE  Sparse network for gene: The final gene expression embedding is hg ∈ R1×50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Pathology network: The slide level image feature representation is passed through an image embed-  ding layer and encodes the embedding as hp ∈ R1×50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  CNV + MUT network: Similarly as the pathology network, the patient level CNV + MUT feature  representation is passed through an FC embedding layer and encodes the embedding as hc ∈ R1×50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='2  EXPERIMENTAL DETAILS  PONET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The latent dimensionality of the factorized matrices k is a very important tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  We tune k = [3, 5, 10, 20, 30, 50] based on the testing C-index value (Appendix Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 5) and the loss  of training and testing plot (Appendix Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 6) for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We choose k to maximize the C-index  value and also it should have stable convergence in both training and testing loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' For example, we  choose k = 10 in TCGA-KIRP for the optimized results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We can see that in Appendix Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' 5 the  testing loss is quite volatile when k is less than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Similarly, we choose k = [20, 10, 20, 20, 10] for  TCGA-BLCA, TCGA-KIRC, TCGA-LUAD, TCGA-LUSC, and TCGA-PAAD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  15  Figure 5: C-Index value under K = 3, 5, 10, 20, 30, 50 for TCGA-KIRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The mean value and standard  deviation for 5-fold cross-validation are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  The learning rate and the regularization hyperparameter λ for the Cox partial likelihood loss are  also tunable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The model is trained with Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' For each training/testing pair,  we first empirically preset the learning rate to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='2e-4 as a starting point for a grid search during  training, the optimal learning rate is determined through the 5-fold cross-validation on the training  set, C-index was used for the performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' After that, the model is trained on all the training  sets and evaluated on the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We use 2e-3 through the experiments for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The batch size is  set to 16, and the epoch is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' During the training process, we carefully observe the training and  testing loss for convergence (Figure 4 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The server used for experiments is NVIDIA  GeForce RTX 2080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  CoxPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We only include the age and gender for the survival prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Using CoxPHFitter from  lifelines 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  DeepSurv 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We concatenate preprocessed pathological image features, gene expression, and copy  number + mutant data in a vector to train the DeepSurv model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' L2 reg = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='0, dropout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4, hidden  layers sizes = [25, 25], learning rate = 1e-05, learning rate decay = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='001, momentum = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Pathomic Fusion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' We use the pathomicSurv model which takes our preprocessed image feature,  gene expression, and copy number + mutation as model input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' k = 20, Learning rate is 2e-3, weight  decay is 4e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The batch size is 16, and the epoch is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Drop out rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  GPDBN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The learning rate is 2e-3, the batch size is 16, the weight decay is 1e-6, the dropout rate is  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='3, and the epoch is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  HFBSurv 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' The learning rate is set to 1e-3, the batch size is 16, λ = 3e-3, weight decay is 1e-6, and  the epoch is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='com/CamDavidsonPilon/lifelines  4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='com/czifan/DeepSurv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='pytorch  5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='com/mahmoodlab/PathomicFusion  6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='com/isfj/GPDBN  7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='com/Liruiqing-ustc/HFBSurv  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='6  C-Index  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='0  5  10  20  30  50  3  KFigure 6: Train and test loss for TCGA-KIRP under K = 3, 5, 10, 20, 50 for 5-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  17  Fold 1  Fold 2  Fold 3  Fold 4  Fold 5  Train  K=3  Test  K=5  K= 10  K= 20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='06  K=50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='50C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='3  ADDITIONAL RESULTS  Figure 7: Inspecting and interpreting PONET on TCGA-BLCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Sankey diagram visualization of  the inner layers of PONET shows the estimated relative importance of different nodes in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Nodes in the first layer represent genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' the next layers represent pathways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' and the final layer  represents the model outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Different layers are linked by weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Nodes with darker colors are  more important, while transparent nodes represent the residual importance of undisplayed nodes in  each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Figure 8: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-BLCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  18  Gene  Pathways  GNB1  PI5P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PP2Aand IER3RegulatePI3K/AKTSignaling  Toll Like Receptor 10 (TLR10) Cascade  Cell-extracellular matrix interactions  PPP2R5E  SHC-related events triggered by IGF1R  Cell death signalling via NRAGE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' NRIF and NADE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RNA Polymerase II Transcription Elongation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='KRAS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MAP2K and MAPK activation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Interferon gamma signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='rRNA processing in the mitochondrion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Calmodulin induced events  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Mitotic Telophase/Cytokinesis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Regulation of Hypoxia-inducible Factor (HIF) by oxygen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PSMA7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Activation of G protein gated Potassium channels  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='FBXW7 Mutants and NOTCH1 in Cancer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='mRNA Splicing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='KPNA2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='outcome  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='YWHAB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Activation of NF-kappaB in B cells  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Golgi-to-ER retrograde transport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='TCR signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='GSK3B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Gap junction degradation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Signaling by FGFR1 in disease  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Mitotic Spindle Checkpoint  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='HSP90AB1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Phosphate bond hydrolysis by NUDT proteins  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Biosynthesis of DPA-derived SPMs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='ESR-mediated signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='TBK1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='NEP/NS2 Interacts with the Cellular Export Machinery  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='TCF transactivating complex  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Fatty acid metabolism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PIK3CA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='p53-IndependentDNADamageResponse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Interleukin-17 signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Effects of PIP2 hydrolysis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residualTCGA-4Z-AA7Y  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PI5P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' PP2A and IER3  SHC-related events  PI3K/AKT Signaling  MAP2K and MAPK activation  Calmodulin induced events  Survival Month: 50  triggered by IGF-1R  High Attn  TCGA-UY-A78N  Survival Month: 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='76  Low AttnFigure 9: Inspecting and interpreting PONET on TCGA-KIRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Sankey diagram visualization of  the inner layers of PONET shows the estimated relative importance of different nodes in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Nodes in the first layer represent genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' the next layers represent pathways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' and the final layer  represents the model outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Different layers are linked by weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Nodes with darker colors are  more important, while transparent nodes represent the residual importance of undisplayed nodes in  each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Figure 10: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-KIRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  19  TCGA-A3-3313  Downregulation of  MAP2K and MAPK activation  Activation of the  P53-Independent DNA  ERBB2:ERBB3 signaling  Survival Month: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='15  pre-replicative complex  damage response  High Attn  TCGA-A3-3320  Survival Month: 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='54  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Low AttnGene  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Pathways  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='TFDP2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Glucagon-type ligand receptors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Class B/2 (Secretin family receptors)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='GPCR ligand binding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MAPK3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Downregulation of ERBB2:ERBB3 signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='G1/STransition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='G1/S DNA Damage Checkpoints  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PTPN11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MAP2K and MAPK activation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='p53-lndependent G1/S DNA damage checkpoint  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Mitotic G1-G1/S phases  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='ADCY5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Activation of the pre-replicative complex  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RAF/MAP kinase cascade  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Defects in vitamin and cofactor metabolism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PSMC2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='p53-Independent DNADamage Response  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='CLEC7A (Dectin-1) signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MAPK1/MAPK3 signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MTRR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='outcome  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PLCB1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Processing of DNA double-strand break ends  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Downregulation of ERBB2 signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='C-type lectin receptors (CLRs)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PSMD11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='CLEC7A (Dectin-1) induces NFAT activation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Defects in cobalamin (B12) metabolism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='HIV Infection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PSMF1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RHO GTPases Activate NADPH Oxidases  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='HDR or Single Strand Annealing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Signaling by ERBB2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='IL6ST  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='SHC-related events triggered by IGF1R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='G2/M Transition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Fatty acid metabolism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Activation of NOXA and translocation to mitochondria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Activation of BH3-only proteins  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Mitotic G2-G2/M phases  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residualFigure 11: Inspecting and interpreting PONET on TCGA-LUAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Sankey diagram visualization  of the inner layers of PONET shows the estimated relative importance of different nodes in each  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Nodes in the first layer represent genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' the next layers represent pathways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' and the final layer  represents the model outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Different layers are linked by weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Nodes with darker colors are  more important, while transparent nodes represent the residual importance of undisplayed nodes in  each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Figure 12: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-LUAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Gene  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Pathways  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='CCT3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Processive synthesis on the lagging strand  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Lagging Strand Synthesis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Class I MHC pathwiay  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PSEN1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='HDR through Homologous Recombination (HRR)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Leading Strand Synthesis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Signaling by EGFR in Cancer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Phosphate bond hydrolysis by NUDT proteins  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Antigen processing-Cross presentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Intrinsic Pathway for Apoptosis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='EGFR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Chk1/Chk2(Cds1) mediated inactivation of Cyclin B:Cdk1 complex  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RAF/MAP kinase cascade  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Nucleobase catabolism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PSMD2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Polymerase switching  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RHO GTPase Effectors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Homology Directed Repait  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='outcome  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='CCT6A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Purine catabolism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='G2/M Checkpoints  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='ER-Phagosome pathway  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PTGES3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='HDR or Single Strand Annealing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Signaling by Rho GTPases  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Golgi Cisternae Pericentriolar Stack Reorganization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='NUDT1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='G2/M DNA damage checkpoint  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MAPK1/MAPK3 signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Downregulation of ERBB2:ERBB3 signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='YWHAZ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Cap-dependent translation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='GPCR downstream signalling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Regulation of RAS by GAPs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RUNX1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Signaling by Overexpressed Wild-Type EGFR in Cancer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Glycosaminoglycan metabolism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Inhibition of Signaling by Overexpressed EGFR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='AKT2 residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residualTCGA-55-8621  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Processive synthesis on  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='HDR through  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Phosphaste bond hydrolysis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Chk1/Chk2(Cds1) mediated  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Survival Month: 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='92  the lagging strand  homologous recombination  by NUDT proteins  inactivation of Cyclin B:Cdk1 complex  High Attn  TCGA-78-7153  Survival Month: 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='42  LowAttnFigure 13: Inspecting and interpreting PONET on TCGA-LUSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Sankey diagram visualization  of the inner layers of PONET shows the estimated relative importance of different nodes in each  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Nodes in the first layer represent genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' the next layers represent pathways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' and the final layer  represents the model outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Different layers are linked by weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Nodes with darker colors are  more important, while transparent nodes represent the residual importance of undisplayed nodes in  each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Figure 14: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-LUSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  21  Gene  Pathways  DLG1  Calmodulin induced events  TCF transactivating complex  Cell-extracellular matrix interactions  UBA52  CD28 dependent PI3K/Akt signaling  Pentose phosphate pathway disease  Neurotransmitterclearance  PPP2R5E  PI5P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=" PP2A and IER3 Regulate PI3K/AKT Signaling  Signaling by NTRK3 (TRKC)  rRNA processing in the mitochondrion  PSMC5  NrCAM interactions  Interferon gamma signaling  TCR signaling  RAC1  Constitutive Signaling by NOTCH1  Glutathione conjugation  Hedgehog 'on' state  outcome  CREB1  MAP2K and MAPK activation  Toll Like Receptor 10 (TLR10) Cascade  Base-Excision Repair," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' AP Site Formation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='ADAM17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Negative regulation of MAPK pathway  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Defects in biotin (Btn) metabolism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content="Hedgehog 'off state  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PAK2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Cleavage of the damaged purine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='SUMO E3 ligases  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Regulation of Hypoxia-inducible Factor (HIF) by oxygen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PSMC2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='AXIN missense mutants destabilize the destruction complex  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RAF-independent MAPK1/3 activation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Signaling by NOTCH4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='NCOA1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='SHC-related events triggered by IGF1R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Golgi-to-ER retrograde transport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Nucleobase biosynthesis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residualTCGA-18-3414  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='CD28 dependent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PI5P PP2A and IER3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Calmodulin induced events  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PI3K/Akt signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='regulate PI3K/AKT signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='NrCAM interactions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Survival Month: 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='52  High Attn  TCGA-33-4538  Survival Month: 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='86  Low AttnFigure 15: Inspecting and interpreting PONET on TCGA-PAAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Sankey diagram visualization  of the inner layers of PONET shows the estimated relative importance of different nodes in each  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Nodes in the first layer represent genes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' the next layers represent pathways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' and the final layer  represents the model outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Different layers are linked by weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' Nodes with darker colors are  more important, while transparent nodes represent the residual importance of undisplayed nodes in  each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  Figure 16: Co-attention visualization of top 4 ranked pathways in two cases of TCGA-PAAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='  22  Gene  Pathways  PTGES3  SMAD4 MH2 Domain Mutants in Cancer  Interleukin-6 family signaling  Mitotic G1-G1/S phases  C1QC  SMAD2/3 MH2Domain Mutants in Cancer  Signaling by FGFR3  Class I MHC pathway  PSMD3  Synthesis of Prostaglandins and Thromboxanes  AXIN mutants destabilize the destruction complex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content=' activating WNT signalir  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Triglyceride metabolism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='CABIN1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Regulation by c-FLIP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Influenza Viral RNA Transcription and Replication  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RIPK1-mediated regulated necrosis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='NRAS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='C1QB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Formation of Senescence-Associated Heterochromatin Foci (SAHF)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Thyroxine biosynthesis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Reversal of alkylation damage by DNA dioxygenases  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='outcome  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='EIF3E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Coenzyme A biosynthesis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Fusion and Uncoating of the Infuenza Virion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PIP3 activates AKT signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='MGAT4B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='PPCS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RUNX3 regulates BCL2L11 (BIM) transcription  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='EPH-Ephrin signaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Metabolism of cofactors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='YWHAG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='FGFR1 mutant receptor activation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Signaling by NOTCH1 HD Domain Mutants in Cancer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Platelet Aggregation (Plug Formation)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+page_content='GPCR downstream signalling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Chk1/Chk2(Cds1) mediated inactivation of Cyclin B:Cdk1 complex  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Regulation of innate immune responses to cytosolic DNA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='RNA Polymerase I Promoter Clearance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='residualTCGA-2J-AABO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Formation of senescence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='Survival Month: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='45  Regulation by c-FLIP  associated heterochromatin foci  Coenzyme A biosynthesis  MET activates RAP1 and RAC1  High Attn  TCGA-3A-A9IH  Survival Month: 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
+page_content='54  Low Attn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E0T4oBgHgl3EQfeAAf/content/2301.02383v1.pdf'}
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+The S-diagnostic—an a posteriori error
+assessment for single-reference coupled-cluster
+methods
+Fabian M. Faulstich,∗,† H˚akon E. Kristiansen,‡ Mihaly A. Csirik,‡ Simen Kvaal,‡
+Thomas Bondo Pedersen,‡ and Andre Laestadius¶,‡
+†Department of Mathematics, University of California, Berkeley
+‡Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University
+of Oslo, Norway
+¶Department of Computer Science, Oslo Metropolitan University, Norway
+E-mail: f.m.faulstich@berkeley.edu
+Abstract
+We propose a novel a posteriori error assessment for the single-reference coupled-
+cluster (SRCC) method called the S-diagnostic. We provide a derivation of the S-
+diagnostic that is rooted in the mathematical analysis of different SRCC variants. We
+numerically scrutinized the S-diagnostic, testing its performance for (1) geometry op-
+timizations, (2) electronic correlation simulations of systems with varying numerical
+difficulty, and (3) the square-planar copper complexes [CuCl4]2−, [Cu(NH3)4]2+, and
+[Cu(H2O)4]2+. Throughout the numerical investigations, the S-diagnostic is compared
+to other SRCC diagnostic procedures, that is, the T1, D1, and D2 diagnostics as well as
+different indices of multi-determinantal and multi-reference character in coupled-cluster
+theory. Our numerical investigations show that the S-diagnostic outperforms the T1,
+1
+arXiv:2301.11393v1  [physics.chem-ph]  26 Jan 2023
+
+D1, and D2 diagnostics and is comparable to the indices of multi-determinantal and
+multi-reference character in coupled-cluster theory in their individual fields of applica-
+bility. The experiments investigating the performance of the S-diagnostic for geometry
+optimizations using SRCC reveal that the S-diagnostic correlates well with different
+error measures at a high level of statistical relevance. The experiments investigating
+the performance of the S-diagnostic for electronic correlation simulations show that the
+S-diagnostic correctly predicts strong multi-reference regimes. The S-diagnostic more-
+over correctly detects the successful SRCC computations for [CuCl4]2−, [Cu(NH3)4]2+,
+and [Cu(H2O)4]2+, which have been known to be misdiagnosed by T1 and D1 diagnos-
+tics in the past. This shows that the S-diagnostic is a promising candidate for an a
+posteriori diagnostic for SRCC calculations.
+1
+Introduction
+While the underlying mathematical theory of the quantum many-body problem is, on a fun-
+damental level, well described, the governing equation, namely, the many-body Schr¨odinger
+equation, remains numerically intractable for a large number of particles. In fact, the many-
+body Schr¨odinger equation poses one of today’s hardest numerical challenges, mainly due
+to the exponential growth in computational complexity with the number of electrons. Over
+the past century, numerous numerical approximation techniques of various levels of cost
+and accuracy have been developed in order to overcome this curse of dimensionality. Ar-
+guably, the most successful approaches are based on coupled-cluster (CC) theory1, which
+defines a cost-efficient hierarchy of increasingly accurate methods, including the so-called gold
+standard of quantum chemistry—the coupled-cluster singles-and-doubles with perturbative
+triples (CCSD(T))2 model.
+Despite the great success of CC theory, its reliability is not yet fully quantifiable. More
+precisely, aside from a few heuristically derived results, there exists no universally reliable
+diagnostic that indicates if the computational result is to be trusted.
+This shortcoming
+2
+
+is most apparent in the regime of transition metal compounds and molecular bond break-
+ing/making processes, systems dominated by strong nondynamic electron-correlation effects,
+where several methods based on CC theory tend to fail along with all other numerically
+tractable approaches.
+Therefore, a posteriori error diagnostics are urgently needed in the field.
+Until very
+recently, the diagnostic approaches available were limited to the so-called T1 (also called
+τ1)3,4, D1, and D2 diagnostic5,6. Despite clear numerical evidence that diagnostics based
+on the single excitation amplitudes, such as the T1 and D1 diagnostics, do not provide
+reliable indicators7, they are commonly used due to the lack of alternatives.
+Recently,
+an alternative set of multi-reference indices was introduced which provided a number of a
+posteriori diagnostic tools8 christened the indices of multi-determinantal and multi-reference
+character in coupled-cluster theory. These tools are highly descriptive and able to determine
+different molecular scenarios in which CC theory may fail.
+We provide an alternative error diagnostic that is based on assumptions employed in the
+mathematical analysis CC theory. More precisely, our diagnostic is derived from the math-
+ematical analysis of CC theory that provides sufficient conditions for a locally unique and
+quasi-optimal solution to the CC working equations. Central to our derivation is the strong
+monotonicity property, as introduced by Schneider9, which is eponymous for our S-diagnostic.
+Compared to the recently suggested nine indices that describe the multi-determinantal and
+multi-reference character in coupled-cluster theory8, the S-diagnostic is a diagnostic tech-
+nique that can be applied to multi-determinantal and multi-reference scenarios alike. We
+complement our theoretical derivation of the S-diagnostic with numerical simulations scruti-
+nizing its validity for different geometry optimizations, and electronic correlation computa-
+tions for systems of varying numerical difficulty for single reference coupled-cluster methods.
+The rest of the article is structured as follows. We begin with a brief review of CC theory,
+followed by a short summary of the mathematical results derived in previous works which
+lay the mathematical foundation for the proposed S-diagnostics. Then, we derive the main
+3
+
+result, i.e., the S-diagnostic which is subsequently numerically scrutinized.
+2
+Theory
+2.1
+Brief overview of coupled-cluster theory
+In CC theory the wave function is parametrized by the exponential |ψ⟩ = e ˆT|φ0⟩. Here, |φ0⟩
+is the reference determinant defining the occupied spin orbitals, and ˆT = �
+µ tµ ˆXµ = �
+k ˆTk
+is a cluster operator, where ˆTk excites k = 1, . . . , N electrons—k is the excitation rank of a
+given ˆTk—from the occupied spin orbitals into the virtual spin-orbitals. All possible excited
+determinants can be expressed as |µ⟩ = ˆXµ|φ0⟩ for some multi-index µ labeling occupied and
+virtual spin-orbitals. The governing equations determining amplitudes (tµ), and therewith
+also the CC energy ECC(t), are given by fCC(t) = 0, where
+�
+�
+�
+�
+�
+ECC(t) = ⟨φ0|e− ˆT ˆHe
+ˆT|φ0⟩
+(fCC(t))µ = ⟨µ|e− ˆT ˆHe
+ˆT|φ0⟩.
+(1)
+More compactly, Eq. (1) can be expressed using the CC Lagrangian10,11
+L(t, z) = ECC(t) +
+�
+µ
+zµ(fCC(t))µ = ⟨φ0|(ˆI + ˆZ†)e− ˆT ˆHe
+ˆT|φ0⟩,
+(2)
+where (zµ) are the Lagrange multipliers which are the dual variables corresponding to (tµ). In
+the extended CC theory12–14 (ECC), which will be used to introduce additional information
+to our S-diagnostic, the Lagrangian is replaced with the more general energy expression
+EECC(t, λ) = ⟨φ0|e
+ˆΛ†e− ˆT ˆHe
+ˆT|φ0⟩.
+(3)
+4
+
+Consequently, through the substitution eˆΛ = ˆI + ˆZ, we have EECC(t, λ) = L(t, z).
+The
+stationarity condition can then be formulated as FECC = 0, where
+FECC = (∂ΛEECC, ∂TEECC)
+(4)
+is the so-called flipped gradient15. The partial derivatives with respect to the amplitudes in
+Eq. (4) are given by
+∂λµEECC = ⟨µ|e
+ˆΛ†e− ˆT ˆHe
+ˆT|φ0⟩,
+∂tµEECC = ⟨φ0|e
+ˆΛ†[e− ˆT ˆHe
+ˆT, ˆXµ]|φ0⟩.
+(5)
+Since the number of determinants, and therewith the size of the system’s governing
+equations, suffer in general from the curse of dimensionality (i.e., it grows exponentially fast
+with the number of electrons), restrictions are necessary to ensure the system’s numerical
+tractability. In practice this is achieved by restricting excitations to excited determinants
+that correspond to a preselected index set—this is referred to as truncation. Such excitation
+hierarchies are commonly denoted as singles (S), doubles (D), etc. We emphasize that the
+CC working equations, as a system of polynomial equations, typically have a large number
+of roots, and the corresponding landscape of said roots is highly non-trivial16. Consequently,
+different limit processes have to be considered separately and carefully studied. More pre-
+cisely, the convergence of the CC roots with respect to the basis set discretization, i.e.,
+convergence towards the complete basis set limit, is a fundamentally different limit process
+from the convergence with respect to the coupled-cluster truncations. Hence, it is important
+to note that the convergence of the numerical root finding procedure for the truncated stan-
+dard (or extended) CC equations does not by itself imply convergence of the roots to the
+corresponding exact roots. In other words, whether the discrete roots converge to the exact
+roots cannot simply be assumed to be true in general.
+Before proceeding further with the derivation of the S-diagnostic, we wish to provide
+the reader with a more precise description of the underlying mathematical conventions in
+5
+
+coupled-cluster theory. We first emphasize the distinction between the cluster amplitudes
+and the corresponding wave function. Although related, these objects live in different spaces
+which we shall elaborate on subsequently. First, the wave function object |ψ⟩ = e ˆT|φ0⟩ lives
+in the N-particle Hilbert space of square-integrable functions, i.e., L2 = {ψ :
+�
+|ψ|2 < +∞},
+with finite kinetic energy.1 We remind the reader of the notation for the L2-inner product
+⟨ψ′|ψ⟩, and its induced norm ∥ψ∥2
+L2 = ⟨ψ|ψ⟩.
+Second, operators that act on the wave
+function, e.g., the Hamiltonian or excitation operators. In this case, we can introduce a
+norm expression for the operator inherited from the function space it is defined on. For
+example, let O be an operator defined on L2 then we define the L2 operator norm
+∥O∥L2 = sup{∥OΨ∥L2 : ∥Ψ∥L2 = 1 and Ψ ∈ L2}.
+(6)
+Note that this reduces to the conventional matrix norm in the finite dimensional case. Third,
+the CC amplitudes (tµ) live in the Hilbert space of finite square summable sequences denoted
+the ℓ2-space. This space is equipped with the ℓ2-inner product17, i.e., let x = (xµ) and
+y = (yµ) be two finite sequences, the ℓ2-inner product is defined as
+⟨x, y⟩ℓ2 =
+�
+µ
+xµyµ,
+which induces the norm ∥x∥2
+ℓ2 = ⟨x, x⟩ℓ2. Henceforth, we shall denote the full amplitude space
+by V, and the truncated amplitude space, e.g., the space only containing single and double
+amplitudes, by V(d); note that we use “d” in this section to distinguish objects that are subject
+to imposed truncations. We moreover follow the mathematically convenient convention that
+1Mathematically, assuming finite kinetic energy is important for the well-posedness of the Schr¨odinger
+equation. In a “weak” formulation this is given by (here for simplicity leaving out spin degrees of freedom)
+�
+R3N |∇ψ(r1, . . . , rN)|2dr1 . . . drN < +∞.
+In the mathematical literature this can be summarized by ψ ∈ H1 (Sobolev space)17. This extra constraint
+of finite kinetic energy is moreover important for the “continuous” (i.e., infinite dimensional) formulation of
+coupled-cluster18.
+6
+
+uses a generic constant C, independent of the main variables under consideration, for the
+different estimations performed subsequently.
+Having laid down the basic definitions, we now recall a result that gives insight into the
+root convergence of CC theory which can be established using a basic existence result of
+nonlinear analysis9,15,18–20. To state this result, we need two more definitions.
+First, local strong monotonicity. Let t, t′, t∗ be cluster amplitudes with ˆT, ˆT ′ and ˆT∗
+denoting the corresponding cluster operators. Set
+∆(t, t′) = ⟨fCC(t) − fCC(t′), t − t′⟩ℓ2,
+(7)
+and furthermore ∆ ˆT = ˆT − ˆT ′. Then the CC function fCC is said to be locally strongly
+monotone at t∗ if for some r > 0, γ > 0 and all t, t′ within the distance r of t∗
+∆(t, t′) ≥ γ∥t − t′∥2
+ℓ2.
+(8)
+Second, local Lipschitz continuity. The function fCC is said to be locally Lipschitz con-
+tinuous at t∗ with Lipschitz constant L > 0 if
+∥fCC(t) − fCC(t′)∥ℓ2 ≤ L∥t − t′∥ℓ2
+(9)
+for any t, t′ in a ball around t∗. Note that in the finite-dimensional case, fCC is indeed locally
+Lipschitz since it is continuously differentiable.
+With these definitions at hand, we can recall the following result9,19:
+Let fCC(t∗) = 0 and assume that fCC is locally strongly monotone with constant γ > 0 at
+t∗. Furthermore, let V(d) ⊂ V be a truncated amplitude space with Pd being the orthogonal
+projector onto V(d) and fd a discretization of fCC, i.e., fd = PdfCC. Then, the following
+holds:
+1. t∗ is locally unique, i.e., |ψ∗⟩ = eT∗|φ0⟩ is the only solution within a sufficiently small
+7
+
+ball.
+2. There exists a sufficiently large d0, such that for any d > d0, there exists t(d)
+∗
+∈ V(d)
+such that fd(t(d)
+∗ ) = 0. This root is unique in a ball centered at t∗ (for some radius r)
+and we have quasi-optimality of the discrete solution t(d)
+∗
+i.e.
+∥t(d)
+∗
+− t∗∥ℓ2 ≤ L
+γ dist(t∗, V(d)),
+(10)
+where dist(v, V(d)) is the distance from v to V(d) measured using the norm of V, and L
+is the Lipschitz constant of fCC at t∗.
+3. For d > d0, the discrete equations fd(t(d)
+∗ ) = 0 have locally unique solutions, and in
+addition to the error estimate (10), we have the quadratic energy error bound
+|ECC(t(d)
+∗ ) − E0| ≤ C1∥t∗ − t(d)
+∗ ∥2
+ℓ2 + C2∥t∗ − t(d)
+∗ ∥ℓ2∥z∗ − z(d)
+∗ ∥ℓ2,
+(11)
+where E0 is the ground state energy and z∗ and z(d)
+∗
+are the Lagrange multiplier of the
+exact and truncated equations, respectively. The constants C1, C2 > 0 arise in general
+from particular continuity considerations18,19 which shall not be further characterized
+here.
+We emphasize that the result in Ref. 18 is more elaborate since it is concerned with an
+infinite dimensional amplitude space. Here, we implicitly assume a finite-dimensional am-
+plitude space which allows us to present the result in the simpler but equivalent ℓ2-topology.
+This result ensures that the CC method is convergent as the truncated cluster amplitude
+space V(d) approaches the untruncated limit and that the energy converges quadratically.
+Note also that the above results hold for conventional single-reference CC theory but can be
+formulated for the extended CC theory as well with some slight modifications (see Ref. 15).
+8
+
+2.2
+Strong Monotonicity Property
+The local strong monotonicity at a root of the CC equations is the mathematical basis of what
+we deem as a reliable solution obtained from a truncated CC calculation since this implies
+a unique solution of fd = 0 for sufficiently good approximate V(d) as well as a quadratic
+convergence in the energy. Moreover, it follows that the Jacobian of both fCC and fd are
+non-degenerate at such a solution. In order to derive the S-diagnostic, we start with a brief
+review of the proof presented in the literature15,18,20 while making some slight improvements.
+We subsequently establish Eq. (8) up to second order in ∥t−t′∥ℓ2 under certain assumptions.
+To that end, we define
+∆2(t∗; t, t′) = ⟨∆ ˆTφ0|e− ˆT∗( ˆH − E0)e
+ˆT∗|∆ ˆTφ0⟩.
+(12)
+Now, suppose that fCC(t∗) = 0, then by Taylor expansion we find
+∆(t, t′) = ∆2(t∗; t, t′) + O((∆t)3).
+(13)
+For the proof, we refer the reader to Ref. 19. We emphasize that the core idea of the proof
+is a Taylor expansion of e ˆT and e ˆT ′ around ˆT∗, which does not require t∗ itself to be small,
+rather, the assumption is that we are within a certain neighborhood of t∗.
+By Eq. (13), if ∆2(t∗; t, t′) ≥ γ′∥t − t′∥2
+ℓ2 with γ′ > 0 for t, t′ within distance r′ from t∗,
+then it is possible to find r > 0 such that Eq. (8) is true for γ ∈ (0, γ′] for t, t′ at distance at
+most r ≤ r′) from t∗. Consequently, we wish to establish
+∆2(t∗; t, t′) ≥ γ′∥t − t′∥2
+ℓ2
+(14)
+for some γ′ = γ′(t∗) > 0.
+We subsequently assume that the ground state of ˆH exists and is non-degenerate, and
+that ˆH admits a spectral gap γ∗ > 0 between the ground-state energy E0 and the rest of the
+9
+
+spectrum of ˆH, i.e.,
+γ∗ = inf
+�
+⟨ψ| ˆH − E0|ψ⟩
+⟨ψ|ψ⟩
+: |ψ⟩ ⊥ |ψ∗⟩
+�
+> 0.
+(15)
+Moreover, we assume that the reference |φ0⟩ is such that it is not orthogonal to the ground-
+state wave function.
+With these assumptions, we can establish an improved version of
+Lemma 11 in Ref. 15 and Lemma 3.5 in Ref. 19: If t∗ solves fCC(t∗) = 0 then for |ψ⟩ ⊥ |φ0⟩
+⟨ψ| ˆH − E0|ψ⟩ ≥ γeff
+∗ ∥ψ∥2
+L2,
+(16)
+where
+γeff
+∗
+=
+γ∗
+∥eT∗φ0∥2
+L2
+.
+(17)
+For the sake of clarity, we here display the used L2-norm. Equation 16 can be obtained as
+follows: Let P∗ be the projection onto the solution |ψ∗⟩, then
+⟨ψ|( ˆH − E0)ψ⟩ = ⟨ψ − P∗(ψ)| ˆH − E0|ψ − P∗(ψ)⟩
+≥ γ∗∥ψ − P∗(ψ)∥2
+L2
+= ∥ψ∥2
+L2 − 2Re⟨ψ|P∗(ψ)⟩ + ∥P∗(ψ)∥2
+L2
+= ∥ψ∥2
+L2 − |⟨ψ|ψ∗⟩|2
+∥ψ∗∥2
+L2
+= ∥ψ∥2
+L2 − |⟨ψ|(eT∗ − I)φ0⟩|2
+∥ψ∗∥2
+L2
+.
+(18)
+We next note that
+|⟨ψ|(eT∗ − I)φ0⟩|2
+∥ψ∗∥2
+L2
+≤ ∥ψ∥2
+L2 ∥(eT∗ − I)φ0∥2
+L2
+∥ψ∗∥2
+L2
+= ∥ψ∥2
+L2
+�
+1 −
+1
+∥ψ∗∥2
+L2
+�
+,
+which inserted in Eq. (18) yields the desired result.
+10
+
+With the inequality (16) at hand, we can establish the inequality
+∆2(t∗; t, t′) = ⟨∆ ˆTφ0|e− ˆT∗( ˆH − E0)e
+ˆT∗|∆ ˆTφ0⟩
+≥ γeff
+∗ ∥∆ ˆTφ0∥2
+L2 − CGCC(T∗)∥∆ ˆTφ0∥2
+H1,
+(19)
+where C is a constant that depends on the Hamiltonian ˆH and
+GCC(T∗) = ∥e
+ˆT∗ − I∥L2 + ∥e− ˆT †
+∗ − I∥L2∥e
+ˆT∗∥L2.
+(20)
+Equation (19) follows from the definition of ∆2 and that
+∆2 = ⟨∆ ˆTφ0| ˆH − E0|∆ ˆTφ0⟩ + ⟨∆ ˆTφ0| ˆH − E0|(e
+ˆT∗ − I)∆ ˆTφ0⟩
++ ⟨(e− ˆT †
+∗ − I)∆ ˆTφ0| ˆH − E0|e
+ˆT∗∆ ˆTφ0⟩,
+then, using that ˆH is a bounded operator in the energy norm and the estimate in Eq. (16),
+we obtain the desired result in Eq. (19).
+3
+The S-Diagnostic
+Given the reformulation of the strong monotonicity property in Eq. (19), we consider a
+computation to be successful if the results fulfill Eq. (19). In order to derive an a posterioi
+diagnostic, we reformulate this inequality in a way that yields a function that indicates
+a reliable computation. To ensure the tractability of the said function we introduce the
+following approximations, which will yield diagnostic functions of different flavors, later
+referred to as S1, S2, and S3, respectively.
+11
+
+Approximation (i)
+A first-order Taylor approximation of e ˆT∗ and the trivial operator
+norm inequality 2 yields
+∥e
+ˆT∗φ0∥2
+L2 ≈ 1 + ∥ ˆT∗∥2
+L2.
+(21)
+Approximation (ii)
+For GCC we use (i) and make the approximation (linearization)
+GCC(T) ≈ 2∥ ˆT∥L2.
+(22)
+Approximation (iii)
+As outlined in Ref. 20, we can moreover estimate
+(1 + ∥ ˆZ∗∥2
+L2)1/2 ≈ (1 + ∥ ˆT∗∥2
+L2)−1/2.
+(23)
+This approximation follows by equating the bra and ket wave functions (in the bivariational
+formulation) e− ˆT †
+∗(ˆI + ˆZ∗)|φ0⟩ = ∥e ˆT∗φ0∥−2
+L2 e ˆT∗|φ0⟩ with eˆΛ∗ = ˆI + ˆZ∗ and approximating
+e− ˆT †
+∗(ˆI + ˆZ∗)|φ0⟩ ≈ (ˆI + ˆZ∗)|φ0⟩.
+(24)
+With these approximations at hand, we can derive three variants of the S-diagnostic that
+we shall investigate subsequently.
+3.1
+The S1-diagnostic
+Starting from Eq. (19), we first note that we are considering the finite-dimensional case, and
+therefore there exists a constant C > 0 such that
+∆2(t∗; t, t′) ≥
+�
+γeff
+∗ − CGCC( ˆT∗)
+�
+∥∆ ˆTφ0∥2
+L2
+(25)
+2
+∥ ˆT∗φ0∥L2 ≤ ∥ ˆT∗∥L2∥φ0∥L2 = ∥ ˆT∗∥L2
+12
+
+holds. Next, we employ Approximation (ii) in the definition of GCC( ˆT∗), and combine Ap-
+proximation (i) with the definition of γeff
+∗
+in Eq. (17), i.e.,
+γeff
+∗
+≈
+γ∗
+1 + ∥ ˆT∗∥2
+L2
+.
+(26)
+This yields
+γeff
+∗ − CGCC( ˆT∗) ≈
+γ∗
+1 + ∥ ˆT∗∥2
+L2
+− 2C∥ ˆT∗∥L2.
+(27)
+Requiring that this expression is positive, we obtain the success condition
+1
+2 > C
+γ∗
+(1 + ∥ ˆT∗∥2
+L2)∥ ˆT∗∥L2.
+(28)
+3.2
+The S2-diagnostic
+By applying Approximation (iii) to Eq. (28), we obtain a success condition that involves the
+Lagrange multipliers, namely,
+1
+2 > C
+γ∗
+∥ ˆT∗∥2
+L2
+(1 + ∥ ˆZ∗∥2
+L2)
+.
+(29)
+3.3
+The S3-diagnostic
+To obtain a diagnostic that includes the Lagrangian multipliers without making use of Ap-
+proximation (iii), we shall follow the argument on strong monotonicity of the extended CC
+function FECC defined above. Note that although we use the extended CC formalism in this
+section (i.e., where the Lagrange multipliers are treated as a second set of cluster amplitudes),
+the derived diagnostic is for the conventional single reference CC method. Subsequently, we
+assume that truncations of ˆT and ˆΛ are at the same rank, i.e., the truncated scheme follows
+as described above for V(d) but takes the double form V(d) × V(d) and with Pd being the
+orthogonal projector onto Vd × Vd. Note that this aligns with practical implementations of
+the CC Lagrangian. For brevity, let ˆU = ( ˆT, ˆΛ), ˆU∗ = ( ˆT∗, ˆΛ∗) and ˆU (d)
+∗
+= ( ˆT (d)
+∗ , ˆΛ(d)
+∗ ) and
+furthermore, set Fd to be the Galerkin discretization of FECC, i.e., Fd( ˆU (d)) = PdFECC( ˆU (d)).
+13
+
+In Ref. 15 strong monotonicity of FECC was established under certain assumptions, and
+recently generalized to a class of extended CC theories21. We, therefore, refer the reader
+to these references for the full proof, here we shall only address those parts relevant to our
+diagnostics.
+Similarly to the CC case, local strong monotonicity of FECC holds if
+∆ECC := ⟨FECC(u) − FECC(u′), u − u′⟩ ≥ γ∥u − u′∥2
+(30)
+for some positive constant γ. Note that we here extended the notation such that u carries
+both the primal-, and dual variables. Furthermore, we let ∆ECC up to second order in ∥u−u′∥
+be denoted ∆ECC
+2
+and similarly to Eq. (19) we have
+∆ECC
+2
+(u∗; u, u′) ≥ γeff
+∗ ∥∆ ˆUφ0∥2
+L2 − CGECC( ˆU∗)∥∆ ˆUφ0∥2
+H1,
+(31)
+where
+GECC( ˆU) = GECC( ˆT, ˆΛ)
+= ∥e− ˆT †e
+ˆΛ∥L2∥e
+ˆT − I∥L2 + ∥e− ˆT †e
+ˆΛ − I∥L2 + K∥φ0∥H1∥e− ˆT †∥L2∥e
+ˆT∥L2∥e
+ˆΛ − I∥L2.
+for some positive constant K
+Starting from Eq. (31), we note again that since we are considering finite-dimensional
+Hilbert spaces, there exists a constant C > 0 such that
+∆ECC
+2
+(u∗; u, u′) ≥
+�
+γeff
+∗ − CGECC( ˆU∗)
+�
+∥∆ ˆUφ0∥2
+L2.
+(32)
+We next employ a variation of Approximation (iii): For GECC we make the substitution
+eˆΛ = ˆI + ˆZ and approximate with a low-order Taylor expansion
+˜GECC( ˆT, ˆZ) := GECC( ˆT, ˆΛ( ˆZ)) ≈ C(∥ ˆT∥L2 + ∥ ˆZ∥L2).
+(33)
+14
+
+Hence, we arrive at the approximation (and we remind the reader that C is used as a generic
+constant)
+γeff
+∗ − CGECC( ˆU∗) ≈
+γ∗
+1 + ∥ ˆT∗∥2
+L2
+− C(∥ ˆT∗∥L2 + ∥ ˆZ∗∥L2).
+(34)
+Requiring that this expression is positive, we find the condition
+1 > C
+γ∗
+�
+(1 + ∥ ˆT∗∥2
+L2)(∥ ˆT∗∥L2 + ∥ ˆZ∗∥L2)
+�
+≈ C
+γ∗
+�
+(1 + ∥ ˆT∗∥2
+L2)∥ ˆT∗∥L2 +
+∥ ˆZ∗∥L2
+1 + ∥ ˆZ∗∥L2
+�
+. (35)
+3.4
+Approximation of operator norms using singular values
+The above-derived success conditions Eqs. (28), (29) and (35) can be directly implemented,
+however, the quantities involved will depend on the system size. This can be illustrated
+by simply placing copies of a molecular system at a distance such that they are at least
+numerically non-interacting. In that case, the reliability of the overall CC calculation is
+determined by the CC calculations of a single copy, yet, the operator norm of the cluster
+operator ∥ ˆT∥L2 will scale with the system’s size.
+To remedy this serious difficulty, we consider an alternative interpretation of the clus-
+ter operators22: The CCSD method yields a set of single amplitudes (ta
+i ) forming a ma-
+trix in Rnocc×nvirt and a set of double amplitudes (tab
+ij ) forming a fourth-order tensor in
+Rnocc×nocc×nvirt×nvirt. As outlined in Ref. 22, in order to capture the pair correlation we re-
+shape the fourth-order tensor that describes the double amplitudes as a matrix in Rn2
+occ×n2
+virt,
+an operation that is also known as “matricization”. In order to include pair correlations
+captured by the single amplitudes, we can moreover extend (tab
+ij ) to also include products of
+single amplitudes which yields MT ∈ Rn2
+occ×n2
+virt with matrix elements
+[MT]ij,ab = tab
+ij + (ta
+i tb
+j − tb
+ita
+j).
+(36)
+15
+
+The singular value decomposition then yields
+MT = UTΣTV ⊤
+T ,
+(37)
+where UT, VT are real orthogonal matrix and ΣT is diagonal. We will subsequently use the
+spectral norm, i.e., the largest singular value, here denoted as σ(MT) to approximate the
+operator norm, i.e.,
+∥ ˆT∥L2 ≈ σ(MT) =: σ(t)
+(38)
+and similarly for the dual variable z. Incorporating this into the success conditions Eqs. (28),
+(29) and (35) yields the S-diagnostic functions used in this article
+S1(t) := 1
+γ∗
+(1 + σ(t)2)σ(t),
+(39a)
+S2(t, z) := 1
+γ∗
+σ(t)
+1 + σ(z)2,
+(39b)
+S3(t, z) := 1
+γ∗
+�
+(1 + σ(t)2)σ(t) +
+σ(z)
+1 + σ(z)2
+�
+.
+(39c)
+For computed cluster amplitudes (t) and Lagrange multipliers (z), the above functions
+will yield an S-diagnostic value.
+In the following numerical investigations, we will first
+investigate the statistical correlation between the computed S-diagnostic value and different
+measures of error. Second, we will investigate a quantitative bound for the S-diagnostic value
+beyond which the computations may not be reliable and further benchmark computations
+with more profound error classifications are advised.
+4
+Numerical simulations
+In this section, we numerically scrutinize the proposed S-diagnostic procedures derived in
+the previous sections. All simulations are performed using the Python-based Simulations
+of Chemistry Framework (PySCF)23–25.
+First, we perform geometry optimizations on a
+16
+
+medium-sized set of molecules comprising all molecules that were investigated in Refs. 3,5,6
+to test the T1, D1, and D2 diagnostic, respectively. With this data at hand, we can propose
+an initial set of values, beyond which our diagnostic suggests interpreting the computational
+results with caution and if possible benchmarking with additional methods that allow for
+a more profound error classification. Second, we target small model systems whose multi-
+reference character can be controlled by simple geometric changes. Third, we numerically
+investigate transition metal complexes that have been shown to be misdiagnosed by the T1
+and D1 diagnostics7.
+4.1
+Correlation in Geometry Optimization
+In order to quantify the correlation between the S-diagnostics and the error of the CC
+method, we numerically investigate the Spearman correlation26 between the error of in sil-
+ico geometry optimizations and the corresponding value of the S-diagnostics. We perform
+geometry optimizations for 34 small to medium-sized molecules that were previously studied
+in relation to CC error classifications3,5,6, see Table 1.
+Table 1: Molecules which are used in the geometry optimization presented here.
+H2N2
+HOF
+C2H2
+ClOH
+H2S
+O3
+FNO
+ClNO
+C2
+C3
+CO
+HNO
+HNC
+HOF
+Cl2O
+P2
+N2H2
+HCN
+CH2NH
+N2
+C2H4
+F2
+HOCl
+Cl2
+HF
+CH4
+H2O
+SiH4
+NH3
+HCl
+CO2
+BeO
+H2CO
+CH2
+The calculations are performed using the CC method with singles and doubles (CCSD)
+using the cc-pVDZ basis set provided by PySCF; the geometry optimization is performed
+using the interface to PyBerny27. The numerically obtained results are compared with exper-
+imentally measured geometries of the considered systems in their gas phases extracted from
+the Computational Chemistry Comparison and Benchmark Data Base (CCCBDB)28. Since
+the computed atomic positions cannot be directly compared, we introduce the bond-length
+matrix that describes the pairwise distance between the atoms in the molecular compound.
+17
+
+This bond-length matrix can be directly compared with the bond-length matrix provided by
+CCCBDB if we label and order the atoms of the corresponding system accordingly. We in-
+vestigate the correlation between the S-diagnostics and three possible error characterizations
+obtained from the absolute difference of the bond-length matrices denoted D(diff):
+i) The maximal absolute error (∆r(max)
+abs
+): the maximal absolute deviation of the numeri-
+cally obtained bond-length matrix to the experimentally obtained bond-length matrix,
+i.e.,
+∆r(max)
+abs
+= max
+i,j D(diff)
+ij
+ii) The averaged absolute error (∆r(ave)
+abs ): the averaged absolute deviation of the numeri-
+cally obtained bond-length matrix to the experimentally obtained bond-length matrix,
+i.e.,
+∆r(ave)
+abs
+=
+�
+i,j D(diff)
+i,j
+Natoms
+iii) The averaged relative error (∆r(ave)
+rel
+): the averaged relative deviation of the numerically
+obtained bond-length matrix to the experimentally obtained bond-length matrix, i.e.,
+∆r(ave)
+rel
+=
+�
+i,j D(diff)
+i,j
+Natoms maxi,j D(diff)
+ij
+Computing the Spearman correlation between the errors listed above and the proposed S-
+diagnostics, we find that all suggested S-diagnostics correlate well with all the error measures
+suggested, i.e., we consistently find correlations of rsp > 0.5 with p < 0.0008, see Table 2. The
+largest correlation is observed between the maximal absolute error (∆r(max)
+abs
+) and S2 and S3
+where we find a correlation of rsp = 0.58476 with p = 0.00018. For comparison, we compute
+the Spearman correlation for the previously suggested T1, D1, and D2 diagnostic in Table 2.
+We find that T1, and D1, are uncorrelated to all the errors that we investigate here, i.e.,
+rsp < 0.3 with p > 0.1. The D2 diagnostic6 shows a correlation with the averaged absolute
+error (∆r(ave)
+abs ) and the averaged relative error (∆r(ave)
+rel
+), where we find a correlation of rsp =
+18
+
+0.36886 with p = 0.026847 and rsp = 0.35496 with p = 0.033646, respectively. We moreover
+compare the S-diagnostics with the recently suggested indices of multi-determinantal and
+multi-reference character in CC theory8. We find that similar to the S-diagnostics, the EEN
+index8 correlates well with the maximal absolute error (∆r(max)
+abs
+); we observe a correlation
+of rsp = 0.53572 with p = 0.000759.
+Directly comparing the Spearman correlation of the S-diagnostics with the T1, D1, and
+D2 diagnostic, we see that the S-diagnostics have a significantly higher correlation than the
+heuristically motivated diagnostics T1, D1 and D2 diagnostics while exhibiting a higher level
+of stochastic significance. Comparing the Spearman correlation of the S-diagnostics with the
+indices of multi-determinantal and multi-reference character in CC theory, we find that the
+S-diagnostic and EEN show similar correlation with the maximal absolute error (∆r(max)
+abs
+)
+with a comparable level of stochastic significance.
+Table 2: Spearman correlation between the S-diagnostic computed form CCSD amplitudes
+and different errors in geometry optimization. The pair-entries show the rank correlation
+and the corresponding p-value, i.e., (rsp, p).
+∆r(max)
+abs
+∆r(ave)
+abs
+∆r(ave)
+rel
+S1
+(0.57910, 0.000215)
+(0.57761, 0.000225)
+(0.53668, 0.000740)
+S2
+(0.58476, 0.000180)
+(0.58584, 0.000174)
+(0.54543, 0.000581)
+S3
+(0.58476, 0.000180)
+(0.58584, 0.000174)
+(0.54543, 0.000581)
+T1
+(0.03025, 0.863034)
+(0.00489, 0.977416)
+(0.02265, 0.895674)
+D1
+(0.27675, 0.107522)
+(-0.00541, 0.975040)
+(-0.02034, 0.906294)
+D2
+(0.16974, 0.329625)
+(0.36886, 0.026847)
+(0.35496, 0.033646)
+EEN
+(0.53572, 0.000759)
+(0.42059, 0.010643)
+(0.33694, 0.044488)
+In order to obtain an approximate trusted region suggested by the S-diagnostics, we
+require a descriptive function that maps the value obtained from the S-diagnostic to the
+error in geometry. Since the Spearman correlation describes a monotone relation between
+the quantities, we may not assume that this relation is linear. Unfortunately, the Spearman
+correlation does not indicate the type of relation that connects the two measured quanti-
+ties. We, therefore, perform a piecewise linear fit to the data obtained in this simulation,
+see Fig. 1. We here allow for four segments which are optimized to reach the best approx-
+19
+
+imation by means of a piecewise linear and monotone function. We emphasize that larger
+numbers of segments yield similar approximations, see Fig. 1b. Performing this piecewise
+linear fit, we observe that the function is constant on some segments. Based on the data dis-
+tribution, we conclude that this constant behavior is artificial and caused by the test set not
+being sufficiently versatile. In particular, no quantitative conclusions can be drawn from the
+piecewise linear fit function for values S3 > 1. Therefore, from the geometry optimizations
+performed here, we can merely conjecture to raise a concern about the validity of CC calcu-
+lations performed for values of the S-diagnostics v(3)
+crit ≥ 1. Based on the piecewise linear fit,
+S3 = 1 corresponds to an error larger than 0.035 a0. A larger statistical investigation with
+a larger variety of molecules and basis set discretizations is delegated to future works. We
+emphasize that this first estimation of vcrit is particularly pessimistic since the data set is not
+versatile enough to give a precise estimation of vcrit. Indeed, in the subsequently performed
+simulations, we show a more refined estimation of vcrit that reveals v(2)
+crit = 1.9 and v(3)
+crit = 1.8,
+for S2, and S3, respectively.
+(a)
+(b)
+Figure 1:
+The maximal error in geometry optimization as a function of the S2 value. (a)
+The orange line corresponds to a piecewise linear fit to the data using four segments for
+the piecewise linear function. (b) Piecewise linear fits to the data with a varying number of
+segments.
+Aside from CC-based simulations, we can also perform MP2 simulations, and use the
+obtained doubles amplitudes to compute the S-diagnostics.
+We find that the proposed
+20
+
+0.150
+4-seg.
+0.125
+0.100
+Max. diff.
+0.075
+0.050
+0.025
+0.000
+0.5
+1.0
+1.5
+2.0
+S3 value0.150
+3-seg.
+0.125
+4-seg.
+5-seg.
+0.100
+Max. diff.
+6-seg.
+0.075
+0.050
+0.025
+0.000
+0.5
+1.0
+1.5
+2.0
+S3 valueS-diagnostics correlate similarly well with MP2 based calculations as it does for CCSD,
+see Table 3
+Table 3: Spearman correlation between S-diagnostics computed from MP2 doubles ampli-
+tudes and different errors in geometry optimization.
+∆r(max)
+abs
+∆r(ave)
+abs
+∆r(ave)
+rel
+S1
+(0.55992, 0.000384)
+(0.54569, 0.000577)
+(0.49781, 0.002006)
+S2
+(0.56687, 0.000313)
+(0.54801, 0.000541)
+(0.49858, 0.001968)
+S3
+(0.55992, 0.000384)
+(0.54569, 0.000577)
+(0.49781, 0.002006)
+4.2
+Model Systems
+In this section we investigate the use of the proposed S-diagnostics for four model systems
+whose multi-reference character can be controlled by simple geometric change: (1) twisting
+ethylene, (2) the C2v insertion pathway for BeH2 (Be · · · H2)29, (3) the H4 model (transition
+from square to linear geometry)30 (4) the H4 model (symmetrically disturbed on a circle);
+the computations are performed in cc-pVTZ basis.
+4.2.1
+Twisting ethylene
+We begin by numerically investigating the proposed S-diagnostics for ethylene twisted around
+the carbon–carbon bond, see Fig. 2.
+Θ
+H
+C
+H
+H
+C
+H
+H
+C
+H
+C
+H
+H
+Figure 2: Depiction of the ethylene (C2H4) model with twist angle Θ.
+At a twist angle of 90°, this system shows a strong multi-reference character. This can
+be seen as follows: At the equilibrium geometry, i.e., in a planar geometry, the two carbon
+p orbitals are perpendicular to the molecular plane form bonding π and anti-bonding π∗
+orbitals. In this geometry, the ground state doubly occupies the π-orbital. As we twist around
+21
+
+the carbon–carbon bond, the overlap between the two p orbitals decreases and becomes zero
+at 90°. Therefore, at 90° the π and π∗ orbitals become degenerate and the π-bond is broken.
+This (quasi) degeneracy can also be observed numerically by computing the HOMO-LUMO
+gap as a function of the twist angle, see Fig. 3a. Computing the corresponding ground state
+energy as a function of the twist angle, we observe the characteristic energy cusp at exactly
+90°, see Fig. 3b.
+(a)
+(b)
+Figure 3:
+(a) HOMO-LUMO gap of C2H4 as a function of the twist angle (b) RHF and
+RCCSD energies of C2H4 as a function of the twist angle
+Due to the quasi degeneracy around 90°, we compare the S-diagnostic with the MRI
+index suggested in Ref. 8. We clearly see the indication of the quasi degeneracy in the MRI
+index, see Fig. 4b. The S-diagnostic also indicates the problematic region around 90°. By
+numerical comparison, we find that a cut-off value of v(2)
+crit = 1.9 and v(3)
+crit = 1.8 for S2 and
+S3, respectively, indicates the same region of quasi degeneracy as the MRI index.
+4.2.2
+C2v insertion pathway for BeH2
+Next we shall investigate the C2v insertion pathway for BeH2 (Be · · · H2)29. The model
+represents an insertion of the Be atom into the H2 molecule. The transformation coordinate
+connects the non-interacting subsystems (Be + H2) with the linear equilibrium state (H-Be-
+H), see Fig. 5
+22
+
+HOMO-LUMO gap
+0.50
+0.45
+0.40
+0.35
+0.30
+0
+1
+2
+3
+Twist angle/ RadianGround state energy
+-78.0
+RHF
+-78.2
+CCSD
+-78.4
+0
+1
+2
+3
+Twist angle/ Radian(a)
+(b)
+Figure 4:
+(a) The proposed S-diagnostics of C2H4 as a function of the twist angle, the
+dotted green and red horizontal lines correspond to v(2)
+crit = 1.9 and v(3)
+crit = 1.8, respectively.
+(b) The previously suggested MRI of C2H4 as a function of the twist angle
+H
+Be
+H
+H
+Be
+H
+Figure 5: Depiction of the C2v insertion pathway for BeH2.
+23
+
+3.0
+S1
+S2
+2.5
+S3
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Twist angle/ Radian1.0
+0.5
+0.0
+-0.5
+MRI
+-1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Twist angle/ RadianWe here follow the insertion pathway outlined in Ref. 29 and denote the position of
+the beryllium atom by X-position, where X-position equal to zero corresponds to the linear
+equilibrium state and X-position equal to five corresponds to the non-interacting subsystems.
+The transition state of this chemical transformation has a pronounced multi-reference char-
+acter. Another distinguishing feature of this model system is a change in the character of the
+dominating determinant in the wave function along the potential energy surface. There are
+two leading determinants in the wave function, each of which dominates in a certain region
+of the potential energy surface while both are quasi-degenerate around the transition-state
+geometry. This leads yields to discontinuities as can be seen in Figs. 6a and 6b
+(a)
+(b)
+Figure 6:
+(a) HOMO-LUMO gap as a function of the X-position (b) RHF and RCCSD
+energies as a function of the X-position.
+Due to the quasi-degeneracy that appears along the transition path, we again compare
+the proposed S-diagnostics with the MRI index suggested in Ref. 8. We clearly see the
+indication of the quasi degeneracy in the MRI index, see Fig. 7b. The region indicated by
+MRI< −0.99 corresponds to x ∈ [2.6, 3.05]. The S-diagnostic also indicates a region where
+the CC computations are potentially unreliable. It is worth mentioning that choosing the
+critical values similar to the previous example, i.e., v(2)
+crit = 1.9 and v(3)
+crit = 1.8, the predicted
+region corresponds to x ∈ [2.5, 4.5] and x ∈ [2.5, 4.25], respectively. In order to reproduce
+the same region of quasi-degeneracy as indicated by the MRI index, the critical values have
+24
+
+Ground state energy
+-15.0
+RHF
+Hartree
+CCSD
+-15.2
+一15.4
+Energyl
+-15.6
+-15.8
+0
+1
+2
+3
+4
+5
+X-position/ aoHOMO-LUMO gap
+0.5
+0.4
+0.3
+0.2
+0
+2
+3
+4
+5
+1
+X-position/ aoto be adjusted to v(2)
+crit = 3.8 and v(3)
+crit = 3.5, respectively.
+(a)
+(b)
+Figure 7: (a) shows the S-diagnostics, the dotted green, and red horizontal lines correspond
+to v(2)
+crit = 1.9 and v(3)
+crit = 1.8, respectively. (b) shows the previously suggested MRI
+4.2.3
+H4 model (transition from square to linear geometry)
+Next, we shall investigate the proposed S-diagnostics applied to the H4 model. The H4
+model is a standard transition model that allows steering the quasi-degeneracy using a single
+parameter, namely, the transition angle α where α = 0 corresponds to a square geometry
+and α = π/2 corresponds to a linear geometry. Following Ref.30, we set a = 2.0 (a.u.),
+see Fig. 8.
+a
+a
+a
+a
+a
+α
+α
+a
+a
+a
+a
+Figure 8: Depiction of the H4 model undergoing the transition from a square geometry to
+linear geometry model by the angle α.
+We see that as the transition angle α tends to zero, the HOMO-LUMO gap closes and
+the system shows signs of (quasi-) degeneracy, see Fig. 9a
+25
+
+So
+S
+S2
+101
+100
+0
+2
+3
+4
+5
+X-position/ ao1.0
+0.5
+0.0
+-0.5
+MRI
+-1.0
+0
+1
+2
+3
+4
+5
+X-position/ ao(a)
+(b)
+Figure 9: (a) HOMO-LUMO gap of H4 as a function of the transition angle (b) RHF, CCSD
+and FCI energies of H4 as a function of the transition angle
+Due to the quasi degeneracy near α = 0, we again compare the proposed S-diagnostics
+with the MRI index. We clearly see the indication of the quasi degeneracy in the MRI index,
+see Fig. 10b. The S-diagnostic also indicates the problematic region near zero transition
+angle.
+A cut-off value of v(2)
+crit = 1.9 and v(3)
+crit = 1.8 results in S2 and S3, respectively,
+indicating the same region of quasi degeneracy as the MRI index.
+(a)
+(b)
+Figure 10:
+(a) The S-diagnostics of H4 as a function of the transition angle, the dotted
+green, and red horizontal lines correspond to v(2)
+crit = 1.9 and v(3)
+crit = 1.8, respectively. (b) The
+previously suggested MRI of H4 as a function of the transition angle.
+For this small model Hamiltonian, it is moreover feasible to perform computations at the
+26
+
+HOMO-LUMO gap
+0.50
+0.45
+0.40
+0.35
+0.30
+0.0
+0.5
+1.0
+1.5
+Angle/ radianGround state energy
+RHF
+-2.0
+CCSD
+FCI
+2.1
+-2.2
+0.0
+0.5
+1.0
+Angle/ radian6
+So
+S1
+5
+S2
+4
+3
+2
+1
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Angle/ radian1.0
+0.5
+0.0
+-0.5
+MRI
+-1.0
+0.00
+0.25
+0.50 0.75
+¥1.00
+1.25
+1.50
+Angle/ radianFCI level of theory, see Fig. 12. This comparison yields a quantitative comparison of error
+and S-diagnostic.
+Figure 11
+Figure 12: The energy error of CCSD compared to the FCI reference energy using semi-log
+scales. The area left of the vertical solid (black), dashed (green), and dotted-dashed (red)
+lines correspond to the regions where the MRI, S2, and S3 diagnostic indicate a potential
+failure of CCSD, respectively.
+4.2.4
+H4 model (symmetrically disturbed on a circle)
+Another variant of the H4 model that is commonly employed to evaluate CC methods consists
+of four hydrogen atoms symmetrically distributed on a circle of radius R = 1.738 ˚A31.
+For small or large angles, the system resembles two H2 molecules that are reasonably well
+separated, but as the angle passes through 90, the four atoms form a square yielding a
+degenerate ground state. The exact energy is smooth as a function of the angle, but at the
+RHF level, we observe a cusp at 90, similar to the rotation of the carbon-carbon bond in
+ethylene. We follow the system’s geometry configuration outlined in Ref.32, see Fig. 13.
+We see that as the transition angle Θ tends to π/2 radians (90°), the HOMO-LUMO gap
+closes and the system shows signs of (quasi) degeneracy, see Fig. 14a
+Due to the quasi degeneracy near Θ = π/2 (90°), we again compare the proposed S-
+diagnostics with the MRI index. We clearly see the indication of the quasi degeneracy in
+27
+
+Ground state energy error
+EcCSD - FFCI
+X
+10-3
+10-3
+3
+ X
+2
+× 10-3
+0.0
+0.5
+1.0
+Angle/ radianΘ
+Θ
+Figure 13: Depiction of the H4 model undergoing a symmetric disturbance on a circle modeled
+by the angle Θ.
+(a)
+(b)
+Figure 14:
+(a) HOMO-LUMO gap of H4 as a function of the transition angle (b) RHF,
+RCCSD energies of H4 as a function of the transition angle.
+28
+
+HOMO-LUMO gap
+0.5
+0.4
+0.3
+0.2
+1.0
+1.5
+2.0
+Angle/ RadianGround state energy
+RHF
+.8
+CCSD
+FCI
+-1.9
+-2.0
+-2.1
+-2.2
+1.0
+1.5
+2.0
+Angle/ Radianthe MRI index, see Fig. 15b. The S-diagnostic also indicates the problematic region near
+zero transition angle. A cut-off value of v(2)
+crit = 1.9 and v(3)
+crit = 1.8 results in S2 and S3,
+respectively, indicating the same region of quasi degeneracy as the MRI index.
+(a)
+(b)
+Figure 15:
+(a) The S-diagnostics of H4 as a function of the transition angle, the dotted
+green, and red horizontal lines correspond to v(2)
+crit = 1.9 and v(3)
+crit = 1.8, respectively. (b) The
+previously suggested MRI of H4 as a function of the transition angle.
+For this small model Hamiltonian, it is moreover feasible to perform computations at
+the FCI level of theory, see Fig. 16. This comparison reveals the variational collapse of the
+CCSD energy, see Fig. 16a, and moreover yields a quantitative comparison of error and S-
+diagnostic. The trusted region suggested by the S-diagnostic corresponds to a CCSD energy
+error smaller than 2 · 10−4 a.u. which is below the chemical accuracy threshold.
+Since the simulations performed in the previous section suggest that the previously used
+T1, D1, and D2 diagnostics are uncorrelated, or merely weakly correlated, we do not report
+their performance here. The computations showing the performance of the T1, D1, and D2
+diagnostics can be found in the Appendix, see Figs. 26 to 29
+4.3
+Transition metal complexes
+In this section we investigate three square-planar copper complexes [CuCl4]2−, [Cu(NH3)4]2+,
+and [Cu(H2O)4]2+. Transition metal complexes are in general considered to be strongly corre-
+29
+
+S1
+15
+S2
+S3
+10
+5
+0
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+Angle/ RadianMRI
+0.5
+0.0
+-0.5
+-1.0
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+Angle/ Radian(a)
+(b)
+Figure 16: (a) The energy error of CCSD compared to the FCI reference energy. Note that
+in the region of 1.3-1.8 radians the CCSD energy is lower than the FCI reference energy,
+which indicates the variational collapse of the CCSD energy in this region. (b) The absolute
+value of the energy error of CCSD compared to the FCI reference energy using semi-log
+scales. The area between the vertical solid (black), dashed (green), and dotted-dashed (red)
+lines correspond to the regions where the MRI, S2, and S3 diagnostic indicate a potential
+failure of CCSD, respectively.
+lated systems and complete active space self-consistent field (CASSCF) theory is commonly
+applied, with multi-reference perturbation or truncated CI corrections for dynamic correla-
+tion. However, as shown in Ref. 7, the single reference CC method performs very well despite
+the large D1 diagnostic value. We use these systems to scrutinize the proposed S-diagnostics
+for larger systems that are known to be misleadingly diagnosed by the D1 diagnostics.
+Similar to Ref. 7, we perform the simulation of [CuCl4]2−, [Cu(NH3)4]2+, and [Cu(H2O)4]2+
+in 6-31G basis using UHF and ROHF as reference states. Also, He, Ne, and Ar cores were
+frozen in the nitrogen, chlorine, and copper atoms, respectively, resulting in 41 electrons
+in 50, 66, and 74 orbitals for the [CuCl4]2−, [Cu(H2O)4]2+, and [Cu(NH3)4]2+ molecules,
+respectively. We list the ground state energies obtained at the mean-field level of theory and
+the corresponding CCSD results in Table 4; we moreover list the HOMO-LUMO gap which
+enters in the S-diagnostics.
+The results in Table 4 show that UHF and ROHF calculations predict similar energy
+values. Moreover, using the UHF, or ROHF reference state results in similar CCSD energy
+30
+
+0.000
+Iartree
+-0.002
+-0.004
+Energyl
+-0.006
+-0.008
+EcOSD -EFCI
+1.0
+1.5
+2.0
+Angle/ Radian10-2.
+10
+IEcCSD －EFCll
+10-5
+1.0
+1.5
+2.0
+Angle/ RadianTable 4: Energies values and HOMO-LUMO gap obtained with UHF, ROHF, and UCCSD
+calculations given the reference state from UHF and ROHF, respectively.
+UHF
+γUHF
+UCCSD
+RHOF
+γROHF
+UCCSD
+[CuCl4]2−
+-3476.764
+0.453
+-3477.119
+-3476.763
+0.146
+-3477.119
+[Cu(NH3)4]2+
+-1862.977
+0.564
+-1863.663
+-1862.976
+0.351
+-1863.663
+[Cu(H2O)4]2+
+-1942.225
+0.677
+-1942.914
+-1942.224
+0.340
+-1942.914
+values. It is worth noticing that ROHF yields a generally smaller HOMO-LUMO gap. Since
+the performed CCSD calculations differ in their reference, we can compute the S-diagnostics
+for both sets of calculations. The results obtained from a UHF and ROHF reference are
+listed in Table 5 and in Table 6, respectively.
+Table 5: S-diagnostics obtained for the three square-planar copper complexes [CuCl4]2−,
+[Cu(NH3)4]2+, and [Cu(H2O)4]2+ in spin unrestricted formulation with UHF reference.
+S1
+S2
+S3
+T1
+D1
+D2
+[CuCl4]2−
+0.208
+0.409
+0.406
+0.019
+0.158
+0.110
+[Cu(NH3)4]2+
+0.203
+0.403
+0.398
+0.014
+0.130
+0.121
+[Cu(H2O)4]2+
+0.155
+0.308
+0.305
+0.011
+0.072
+0.116
+We see that all S-diagnostic variants suggest that the CCSD calculations were successful,
+and do not require additional numerical confirmation. This is opposed to the D1 diagnostics,
+which aligns with the results reported in Ref. 7.
+Table 6: S-diagnostics obtained for the three square-planar copper complexes [CuCl4]2−,
+[Cu(NH3)4]2+, and [Cu(H2O)4]2+ in spin unrestricted formulation with ROHF reference.
+S0
+S1
+S2
+T1
+D1
+D2
+[CuCl4]2−
+0.645
+1.285
+1.27
+0.020
+0.167
+0.110
+[Cu(NH3)4]2+
+0.326
+0.646
+0.638
+0.015
+0.139
+0.121
+[Cu(H2O)4]2+
+0.309
+0.614
+0.607
+0.011
+0.077
+0.116
+Similar to the results in Table 5, we see that all variants of the S-diagnostic suggest that
+the CCSD calculations were successful. However, it is worth noticing that the S-diagnostic
+values have increased compared to the values reported in Table 5.
+31
+
+5
+Conclusion
+In this article, we proposed three a posteriori diagnostics for single-reference CC calcula-
+tions which we called S-diagnostics, due to their origin in the strong monotonicity analysis.
+Contrary to previously suggested CC diagnostics, the S-diagnostics are motivated by math-
+ematical principles that have been used to analyze CC methods of different flavors in the
+past9,15,18,19,33.
+We performed a set of geometry optimizations for small to medium-sized molecules in
+order to reveal the correlation between the S-diagnostics and the error in geometry from
+CCSD calculations. The test set comprised all molecules that were used in previous articles
+concerning CC diagnostics3–6. Our investigations revealed that the S-diagnostics correlate
+well and with large statistical relevance with different errors in geometry. This yields a first
+estimate of the critical values for the S-diagnostics beyond which the computational results
+should be confirmed using further and more careful numerical investigations. The observed
+correlation between the S-diagnostics and the different errors in geometry are comparable
+to the recently suggested EEN index8. A heuristic test revealed that the S-diagnostics also
+correlate well and with large statistical relevance with the error in geometry at the MP2 level
+of theory. This suggests that the S-diagnostics can also be used as an a posteriori diagnostic
+for MP2 calculations. Our numerical simulations moreover showed that diagnostics based on
+single excitation cluster amplitudes, i.e., D1 and T1, are uncorrelated to errors in geometry
+optimization.
+Following we investigated the S-diagnostics for transition state models that undergo a
+transition from a region in which CC calculations are reliable to a regime where the CC cal-
+culations require further numerical investigations—in this case, due to (quasi-) degeneracy of
+the ground state. The S-diagnostic detects the corresponding regions of (quasi-) degeneracy
+well. In fact, its performance is comparable to the recently suggested MRI indicator—an a
+posteriori indicator for multi-reference character8.
+The last set of numerical simulations targeted transition metal complexes which have
+32
+
+recently been carefully benchmarked7. The previously performed benchmark calculations
+revealed that diagnostics based on single excitation amplitudes severely misdiagnose the
+performance of CCSD for these transition metal complexes. Our computations confirm this,
+and moreover, show that the S-diagnostic correctly confirms the accuracy of the CCSD
+results outlined in Ref. 7.
+These carefully performed numerical investigations suggest that the S-diagnostic is a
+promising candidate for an a posteriori diagnostic for single-reference CC and MP2 calcu-
+lations. To further confirm this, benchmarks on a larger set of molecules will be performed
+in the future. Moreover, since the mathematical analysis of the single-reference CC method
+generalizes to periodic systems as well, we believe that the S-diagnostic can moreover be
+applied to simulations of solids at the CC and MP2 level of theory.
+Throughout our numerical investigations, we observe a subpar performance of the T1 and
+D1 diagnostics. This suggests that those diagnostics should once and for all be removed as
+a posteriori diagnostic tools for single-reference CC calculations.
+Acknowledgement
+This work was partially supported by the Air Force Office of Scientific Research under the
+award number FA9550-18-1-0095 and by the Simons Targeted Grants in Mathematics and
+Physical Sciences on Moir´e Materials Magic (F.M.F.), by the Peder Sather Grant Program
+(A.L., M.A.C., F.M.F.,), and by the Research Council of Norway (A.L., M.A.C.) through
+Project No.
+287906 (CCerror) and its Centres of Excellence scheme (Hylleraas Centre)
+Project No. 262695.
+Some of the calculations were performed on resources provided by
+Sigma2 - the National Infrastructure for High Performance Computing and Data Storage in
+Norway (Project No. NN4654K). We also want to thank Prof. Lin Lin, Prof. Trygve Helgaker,
+Prof. Anna Krylov, Dr. Pavel Pokhilko, Dr. Tanner P. Culpitt, Dr. Laurens Peters, and Dr.
+Tilmann Bodenstein for fruitful discussions.
+33
+
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+37
+
+6
+Appendix
+6.1
+Correlation in Geometry Optimization
+Since we can correlate three S-diagnostic variants with three error measures, we can in
+principle perform the piecewise linear fit that is presented in Section on Correlation in
+Geometry Optimization for nine different scenarios. We here present the piecewise linear fits
+which were not addressed in the above article.
+6.1.1
+S1-diagnostic Correlations
+(a)
+(b)
+Figure 17: The averaged relative error in geometry optimization as a function of the S1 value.
+(a) The orange line corresponds to a piecewise linear fit to the data using four segments for
+the piecewise linear function. (b) Piecewise linear fits to the data with a varying number of
+segments.
+38
+
+4-seg.
+0.03
+Ave. rel. diff.
+0.02
+0.01
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value0.03
+Ave. rel. diff.
+0.02
+3-seg.
+4-seg.
+0.01
+5-seg.
+6-seg.
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value(a)
+(b)
+Figure 18:
+The maximal absolute error in geometry optimization as a function of the S1
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+(a)
+(b)
+Figure 19:
+The averaged absolute error in geometry optimization as a function of the S1
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+39
+
+0.150
+4-seg.
+0.125
+0.100
+Max. diff.
+0.075
+0.050
+.
+0.025
+★+
+0.000
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value0.150
+3-seg.
+0.125
+4-seg.
+5-seg.
+0.100
+Max. diff.
+6-seg.
+0.075
+0.050
+.
+0.025
+★★
+0.000
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+0.02
+0.01
+4-seg.
+0.00
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+3-seg.
+0.02
+4-seg.
+5-seg.
+0.01
+6-seg.
+0.00
+0.2
+0.4
+0.6
+0.8
+1.0
+Si value6.1.2
+S2-diagnostic Correlations
+(a)
+(b)
+Figure 20: The averaged relative error in geometry optimization as a function of the S2 value.
+(a) The orange line corresponds to a piecewise linear fit to the data using four segments for
+the piecewise linear function. (b) Piecewise linear fits to the data with a varying number of
+segments.
+40
+
+4-seg.
+0.03
+Ave. rel. diff.
+0.02
+0.01
+0.5
+1.0
+1.5
+2.0
+S2 value0.03
+Ave. rel. diff.
+0.02
+3-seg.
+4-seg.
+0.01
+5-seg.
+6-seg.
+0.5
+1.0
+1.5
+2.0
+S2 value(a)
+(b)
+Figure 21:
+The maximal absolute error in geometry optimization as a function of the S2
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+(a)
+(b)
+Figure 22:
+The averaged absolute error in geometry optimization as a function of the S2
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+41
+
+0.150
+4-seg.
+0.125
+0.100
+Max. diff.
+0.075
+0.050
+0.025
+0.000
+0.5
+1.0
+1.5
+2.0
+S2 value0.150
+3-seg.
+0.125
+4-seg.
+5-seg.
+0.100
+Max. diff.
+6-seg.
+0.075
+0.050
+0.025
+☆
+0.000
+0.5
+1.0
+1.5
+2.0
+S2 value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+0.02
+0.01
+4-seg.
+0.00
+0.5
+1.0
+1.5
+2.0
+S2 value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+3-seg.
+0.02
+4-seg.
+5-seg.
+0.01
+6-seg.
+0.00
+0.5
+1.0
+1.5
+2.0
+S2 value6.1.3
+S2-diagnostic Correlations
+(a)
+(b)
+Figure 23: The averaged relative error in geometry optimization as a function of the S3 value.
+(a) The orange line corresponds to a piecewise linear fit to the data using four segments for
+the piecewise linear function. (b) Piecewise linear fits to the data with a varying number of
+segments.
+42
+
+4-seg.
+0.03
+Ave. rel. diff.
+0.02
+0.01
+0.5
+1.0
+1.5
+2.0
+S3 value7
+0.03
+Ave. rel. diff.
+0.02
+3-seg.
+4-seg.
+0.01
+5-seg.
+6-seg.
+0.5
+1.0
+1.5
+2.0
+S3 value(a)
+(b)
+Figure 24:
+The maximal absolute error in geometry optimization as a function of the S3
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+(a)
+(b)
+Figure 25:
+The averaged absolute error in geometry optimization as a function of the S3
+value. (a) The orange line corresponds to a piecewise linear fit to the data using four segments
+for the piecewise linear function. (b) Piecewise linear fits to the data with a varying number
+of segments.
+43
+
+0.150
+4-seg.
+0.125
+0.100
+Max. diff.
+0.075
+0.050
+0.025
+0.000
+0.5
+1.0
+1.5
+2.0
+S3 value0.150
+3-seg.
+0.125
+4-seg.
+5-seg.
+0.100
+Max. diff.
+6-seg.
+0.075
+0.050
+0.025
+0.000
+0.5
+1.0
+1.5
+2.0
+S3 value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+0.02
+0.01
+4-seg.
+0.00
+0.5
+1.0
+1.5
+2.0
+S3 value0.06
+★
+0.05
+Ave. abs. diff.
+0.04
+0.03
+3-seg.
+0.02
+4-seg.
+5-seg.
+0.01
+6-seg.
+0.00
+0.5
+1.0
+1.5
+2.0
+S3 value6.2
+Transition State Models
+Here we shall compare the performance of the S-diagnostics and the previously used T1, D1,
+and D2 diagnostics.
+(a)
+(b)
+Figure 26:
+(a) shows the S-diagnostics (b) shows the previously suggested T1, D1 and D2
+diagnostics
+(a)
+(b)
+Figure 27:
+(a) shows the S-diagnostics (b) shows the previously suggested T1, D1 and D2
+diagnostics
+44
+
+So
+S
+S2
+101
+100
+0
+2
+3
+4
+5
+X-position/ ao10-1
+T1
+D1
+D2
+10-2
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Twist angle/ Radian100
+T1
+D1
+D2
+10-1
+10-2
+0
+2
+3
+4
+5
+1
+X-position/ ao3.0
+S1
+S2
+2.5
+S3
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Twist angle/ Radian(a)
+(b)
+Figure 28:
+(a) shows the S-diagnostics (b) shows the previously suggested T1, D1 and D2
+diagnostics
+(a)
+(b)
+Figure 29:
+(a) shows the S-diagnostics (b) shows the previously suggested T1, D1 and D2
+diagnostics
+45
+
+6
+So
+S1
+5
+S2
+4
+3
+2
+1
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Angle/ radianS1
+15
+S2
+S3
+10
+5
+0
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+Angle/ RadianT1
+D1
+D:
+10-
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Angle/ radian100
+T1
+D1
+D2
+10-1
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+Angle/ RadianGraphical TOC Entry
+Running CCSD ...
+S-Diagnostic
+46
+
diff --git a/39FIT4oBgHgl3EQf6SuL/content/tmp_files/load_file.txt b/39FIT4oBgHgl3EQf6SuL/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..075055c3a44d83d9cad7480f118fa33ebb215a5b
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@@ -0,0 +1,1506 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf,len=1505
+page_content='The S-diagnostic—an a posteriori error  assessment for single-reference coupled-cluster  methods  Fabian M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Faulstich,∗,† H˚akon E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Kristiansen,‡ Mihaly A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Csirik,‡ Simen Kvaal,‡  Thomas Bondo Pedersen,‡ and Andre Laestadius¶,‡  †Department of Mathematics, University of California, Berkeley  ‡Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University  of Oslo, Norway  ¶Department of Computer Science, Oslo Metropolitan University, Norway  E-mail: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='faulstich@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='edu  Abstract  We propose a novel a posteriori error assessment for the single-reference coupled-  cluster (SRCC) method called the S-diagnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We provide a derivation of the S-  diagnostic that is rooted in the mathematical analysis of different SRCC variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We  numerically scrutinized the S-diagnostic, testing its performance for (1) geometry op-  timizations, (2) electronic correlation simulations of systems with varying numerical  difficulty, and (3) the square-planar copper complexes [CuCl4]2−, [Cu(NH3)4]2+, and  [Cu(H2O)4]2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Throughout the numerical investigations, the S-diagnostic is compared  to other SRCC diagnostic procedures, that is, the T1, D1, and D2 diagnostics as well as  different indices of multi-determinantal and multi-reference character in coupled-cluster  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Our numerical investigations show that the S-diagnostic outperforms the T1,  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='11393v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='chem-ph]  26 Jan 2023  D1, and D2 diagnostics and is comparable to the indices of multi-determinantal and  multi-reference character in coupled-cluster theory in their individual fields of applica-  bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The experiments investigating the performance of the S-diagnostic for geometry  optimizations using SRCC reveal that the S-diagnostic correlates well with different  error measures at a high level of statistical relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The experiments investigating  the performance of the S-diagnostic for electronic correlation simulations show that the  S-diagnostic correctly predicts strong multi-reference regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The S-diagnostic more-  over correctly detects the successful SRCC computations for [CuCl4]2−, [Cu(NH3)4]2+,  and [Cu(H2O)4]2+, which have been known to be misdiagnosed by T1 and D1 diagnos-  tics in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This shows that the S-diagnostic is a promising candidate for an a  posteriori diagnostic for SRCC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  1  Introduction  While the underlying mathematical theory of the quantum many-body problem is, on a fun-  damental level, well described, the governing equation, namely, the many-body Schr¨odinger  equation, remains numerically intractable for a large number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In fact, the many-  body Schr¨odinger equation poses one of today’s hardest numerical challenges, mainly due  to the exponential growth in computational complexity with the number of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Over  the past century, numerous numerical approximation techniques of various levels of cost  and accuracy have been developed in order to overcome this curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Ar-  guably, the most successful approaches are based on coupled-cluster (CC) theory1, which  defines a cost-efficient hierarchy of increasingly accurate methods, including the so-called gold  standard of quantum chemistry—the coupled-cluster singles-and-doubles with perturbative  triples (CCSD(T))2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Despite the great success of CC theory, its reliability is not yet fully quantifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' More  precisely, aside from a few heuristically derived results, there exists no universally reliable  diagnostic that indicates if the computational result is to be trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  This shortcoming  2  is most apparent in the regime of transition metal compounds and molecular bond break-  ing/making processes, systems dominated by strong nondynamic electron-correlation effects,  where several methods based on CC theory tend to fail along with all other numerically  tractable approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Therefore, a posteriori error diagnostics are urgently needed in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Until very  recently, the diagnostic approaches available were limited to the so-called T1 (also called  τ1)3,4, D1, and D2 diagnostic5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Despite clear numerical evidence that diagnostics based  on the single excitation amplitudes, such as the T1 and D1 diagnostics, do not provide  reliable indicators7, they are commonly used due to the lack of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Recently,  an alternative set of multi-reference indices was introduced which provided a number of a  posteriori diagnostic tools8 christened the indices of multi-determinantal and multi-reference  character in coupled-cluster theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' These tools are highly descriptive and able to determine  different molecular scenarios in which CC theory may fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  We provide an alternative error diagnostic that is based on assumptions employed in the  mathematical analysis CC theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' More precisely, our diagnostic is derived from the math-  ematical analysis of CC theory that provides sufficient conditions for a locally unique and  quasi-optimal solution to the CC working equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Central to our derivation is the strong  monotonicity property, as introduced by Schneider9, which is eponymous for our S-diagnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Compared to the recently suggested nine indices that describe the multi-determinantal and  multi-reference character in coupled-cluster theory8, the S-diagnostic is a diagnostic tech-  nique that can be applied to multi-determinantal and multi-reference scenarios alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We  complement our theoretical derivation of the S-diagnostic with numerical simulations scruti-  nizing its validity for different geometry optimizations, and electronic correlation computa-  tions for systems of varying numerical difficulty for single reference coupled-cluster methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  The rest of the article is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We begin with a brief review of CC theory,  followed by a short summary of the mathematical results derived in previous works which  lay the mathematical foundation for the proposed S-diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Then, we derive the main  3  result, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', the S-diagnostic which is subsequently numerically scrutinized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  2  Theory  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='1  Brief overview of coupled-cluster theory  In CC theory the wave function is parametrized by the exponential |ψ⟩ = e ˆT|φ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Here, |φ0⟩  is the reference determinant defining the occupied spin orbitals, and ˆT = �  µ tµ ˆXµ = �  k ˆTk  is a cluster operator, where ˆTk excites k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' , N electrons—k is the excitation rank of a  given ˆTk—from the occupied spin orbitals into the virtual spin-orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' All possible excited  determinants can be expressed as |µ⟩ = ˆXµ|φ0⟩ for some multi-index µ labeling occupied and  virtual spin-orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The governing equations determining amplitudes (tµ), and therewith  also the CC energy ECC(t), are given by fCC(t) = 0, where  �  �  �  �  �  ECC(t) = ⟨φ0|e− ˆT ˆHe  ˆT|φ0⟩  (fCC(t))µ = ⟨µ|e− ˆT ˆHe  ˆT|φ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (1)  More compactly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (1) can be expressed using the CC Lagrangian10,11  L(t, z) = ECC(t) +  �  µ  zµ(fCC(t))µ = ⟨φ0|(ˆI + ˆZ†)e− ˆT ˆHe  ˆT|φ0⟩,  (2)  where (zµ) are the Lagrange multipliers which are the dual variables corresponding to (tµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In  the extended CC theory12–14 (ECC), which will be used to introduce additional information  to our S-diagnostic, the Lagrangian is replaced with the more general energy expression  EECC(t, λ) = ⟨φ0|e  ˆΛ†e− ˆT ˆHe  ˆT|φ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (3)  4  Consequently, through the substitution eˆΛ = ˆI + ˆZ, we have EECC(t, λ) = L(t, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  The  stationarity condition can then be formulated as FECC = 0, where  FECC = (∂ΛEECC, ∂TEECC)  (4)  is the so-called flipped gradient15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The partial derivatives with respect to the amplitudes in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (4) are given by  ∂λµEECC = ⟨µ|e  ˆΛ†e− ˆT ˆHe  ˆT|φ0⟩,  ∂tµEECC = ⟨φ0|e  ˆΛ†[e− ˆT ˆHe  ˆT, ˆXµ]|φ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (5)  Since the number of determinants, and therewith the size of the system’s governing  equations, suffer in general from the curse of dimensionality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', it grows exponentially fast  with the number of electrons), restrictions are necessary to ensure the system’s numerical  tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In practice this is achieved by restricting excitations to excited determinants  that correspond to a preselected index set—this is referred to as truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Such excitation  hierarchies are commonly denoted as singles (S), doubles (D), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We emphasize that the  CC working equations, as a system of polynomial equations, typically have a large number  of roots, and the corresponding landscape of said roots is highly non-trivial16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Consequently,  different limit processes have to be considered separately and carefully studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' More pre-  cisely, the convergence of the CC roots with respect to the basis set discretization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=',  convergence towards the complete basis set limit, is a fundamentally different limit process  from the convergence with respect to the coupled-cluster truncations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Hence, it is important  to note that the convergence of the numerical root finding procedure for the truncated stan-  dard (or extended) CC equations does not by itself imply convergence of the roots to the  corresponding exact roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In other words, whether the discrete roots converge to the exact  roots cannot simply be assumed to be true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Before proceeding further with the derivation of the S-diagnostic, we wish to provide  the reader with a more precise description of the underlying mathematical conventions in  5  coupled-cluster theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We first emphasize the distinction between the cluster amplitudes  and the corresponding wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Although related, these objects live in different spaces  which we shall elaborate on subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' First, the wave function object |ψ⟩ = e ˆT|φ0⟩ lives  in the N-particle Hilbert space of square-integrable functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', L2 = {ψ :  �  |ψ|2 < +∞},  with finite kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='1 We remind the reader of the notation for the L2-inner product  ⟨ψ′|ψ⟩, and its induced norm ∥ψ∥2  L2 = ⟨ψ|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Second, operators that act on the wave  function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', the Hamiltonian or excitation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In this case, we can introduce a  norm expression for the operator inherited from the function space it is defined on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' For  example, let O be an operator defined on L2 then we define the L2 operator norm  ∥O∥L2 = sup{∥OΨ∥L2 : ∥Ψ∥L2 = 1 and Ψ ∈ L2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (6)  Note that this reduces to the conventional matrix norm in the finite dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Third,  the CC amplitudes (tµ) live in the Hilbert space of finite square summable sequences denoted  the ℓ2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This space is equipped with the ℓ2-inner product17, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', let x = (xµ) and  y = (yµ) be two finite sequences, the ℓ2-inner product is defined as  ⟨x, y⟩ℓ2 =  �  µ  xµyµ,  which induces the norm ∥x∥2  ℓ2 = ⟨x, x⟩ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Henceforth, we shall denote the full amplitude space  by V, and the truncated amplitude space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', the space only containing single and double  amplitudes, by V(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' note that we use “d” in this section to distinguish objects that are subject  to imposed truncations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We moreover follow the mathematically convenient convention that  1Mathematically, assuming finite kinetic energy is important for the well-posedness of the Schr¨odinger  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In a “weak” formulation this is given by (here for simplicity leaving out spin degrees of freedom)  �  R3N |∇ψ(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' , rN)|2dr1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' drN < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  In the mathematical literature this can be summarized by ψ ∈ H1 (Sobolev space)17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This extra constraint  of finite kinetic energy is moreover important for the “continuous” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', infinite dimensional) formulation of  coupled-cluster18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  6  uses a generic constant C, independent of the main variables under consideration, for the  different estimations performed subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Having laid down the basic definitions, we now recall a result that gives insight into the  root convergence of CC theory which can be established using a basic existence result of  nonlinear analysis9,15,18–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' To state this result, we need two more definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  First, local strong monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Let t, t′, t∗ be cluster amplitudes with ˆT, ˆT ′ and ˆT∗  denoting the corresponding cluster operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Set  ∆(t, t′) = ⟨fCC(t) − fCC(t′), t − t′⟩ℓ2,  (7)  and furthermore ∆ ˆT = ˆT − ˆT ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Then the CC function fCC is said to be locally strongly  monotone at t∗ if for some r > 0, γ > 0 and all t, t′ within the distance r of t∗  ∆(t, t′) ≥ γ∥t − t′∥2  ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (8)  Second, local Lipschitz continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The function fCC is said to be locally Lipschitz con-  tinuous at t∗ with Lipschitz constant L > 0 if  ∥fCC(t) − fCC(t′)∥ℓ2 ≤ L∥t − t′∥ℓ2  (9)  for any t, t′ in a ball around t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Note that in the finite-dimensional case, fCC is indeed locally  Lipschitz since it is continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  With these definitions at hand, we can recall the following result9,19:  Let fCC(t∗) = 0 and assume that fCC is locally strongly monotone with constant γ > 0 at  t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Furthermore, let V(d) ⊂ V be a truncated amplitude space with Pd being the orthogonal  projector onto V(d) and fd a discretization of fCC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', fd = PdfCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Then, the following  holds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' t∗ is locally unique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', |ψ∗⟩ = eT∗|φ0⟩ is the only solution within a sufficiently small  7  ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' There exists a sufficiently large d0, such that for any d > d0, there exists t(d)  ∗  ∈ V(d)  such that fd(t(d)  ∗ ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This root is unique in a ball centered at t∗ (for some radius r)  and we have quasi-optimality of the discrete solution t(d)  ∗  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  ∥t(d)  ∗  − t∗∥ℓ2 ≤ L  γ dist(t∗, V(d)),  (10)  where dist(v, V(d)) is the distance from v to V(d) measured using the norm of V, and L  is the Lipschitz constant of fCC at t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' For d > d0, the discrete equations fd(t(d)  ∗ ) = 0 have locally unique solutions, and in  addition to the error estimate (10), we have the quadratic energy error bound  |ECC(t(d)  ∗ ) − E0| ≤ C1∥t∗ − t(d)  ∗ ∥2  ℓ2 + C2∥t∗ − t(d)  ∗ ∥ℓ2∥z∗ − z(d)  ∗ ∥ℓ2,  (11)  where E0 is the ground state energy and z∗ and z(d)  ∗  are the Lagrange multiplier of the  exact and truncated equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The constants C1, C2 > 0 arise in general  from particular continuity considerations18,19 which shall not be further characterized  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  We emphasize that the result in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 18 is more elaborate since it is concerned with an  infinite dimensional amplitude space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Here, we implicitly assume a finite-dimensional am-  plitude space which allows us to present the result in the simpler but equivalent ℓ2-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  This result ensures that the CC method is convergent as the truncated cluster amplitude  space V(d) approaches the untruncated limit and that the energy converges quadratically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Note also that the above results hold for conventional single-reference CC theory but can be  formulated for the extended CC theory as well with some slight modifications (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2  Strong Monotonicity Property  The local strong monotonicity at a root of the CC equations is the mathematical basis of what  we deem as a reliable solution obtained from a truncated CC calculation since this implies  a unique solution of fd = 0 for sufficiently good approximate V(d) as well as a quadratic  convergence in the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Moreover, it follows that the Jacobian of both fCC and fd are  non-degenerate at such a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In order to derive the S-diagnostic, we start with a brief  review of the proof presented in the literature15,18,20 while making some slight improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  We subsequently establish Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (8) up to second order in ∥t−t′∥ℓ2 under certain assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  To that end, we define  ∆2(t∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' t, t′) = ⟨∆ ˆTφ0|e− ˆT∗( ˆH − E0)e  ˆT∗|∆ ˆTφ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (12)  Now, suppose that fCC(t∗) = 0, then by Taylor expansion we find  ∆(t, t′) = ∆2(t∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' t, t′) + O((∆t)3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (13)  For the proof, we refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We emphasize that the core idea of the proof  is a Taylor expansion of e ˆT and e ˆT ′ around ˆT∗, which does not require t∗ itself to be small,  rather, the assumption is that we are within a certain neighborhood of t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (13), if ∆2(t∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' t, t′) ≥ γ′∥t − t′∥2  ℓ2 with γ′ > 0 for t, t′ within distance r′ from t∗,  then it is possible to find r > 0 such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (8) is true for γ ∈ (0, γ′] for t, t′ at distance at  most r ≤ r′) from t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Consequently, we wish to establish  ∆2(t∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' t, t′) ≥ γ′∥t − t′∥2  ℓ2  (14)  for some γ′ = γ′(t∗) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  We subsequently assume that the ground state of ˆH exists and is non-degenerate, and  that ˆH admits a spectral gap γ∗ > 0 between the ground-state energy E0 and the rest of the  9  spectrum of ˆH, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=',  γ∗ = inf  �  ⟨ψ| ˆH − E0|ψ⟩  ⟨ψ|ψ⟩  : |ψ⟩ ⊥ |ψ∗⟩  �  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (15)  Moreover, we assume that the reference |φ0⟩ is such that it is not orthogonal to the ground-  state wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  With these assumptions, we can establish an improved version of  Lemma 11 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 15 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 19: If t∗ solves fCC(t∗) = 0 then for |ψ⟩ ⊥ |φ0⟩  ⟨ψ| ˆH − E0|ψ⟩ ≥ γeff  ∗ ∥ψ∥2  L2,  (16)  where  γeff  ∗  =  γ∗  ∥eT∗φ0∥2  L2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (17)  For the sake of clarity, we here display the used L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Equation 16 can be obtained as  follows: Let P∗ be the projection onto the solution |ψ∗⟩, then  ⟨ψ|( ˆH − E0)ψ⟩ = ⟨ψ − P∗(ψ)| ˆH − E0|ψ − P∗(ψ)⟩  ≥ γ∗∥ψ − P∗(ψ)∥2  L2  = ∥ψ∥2  L2 − 2Re⟨ψ|P∗(ψ)⟩ + ∥P∗(ψ)∥2  L2  = ∥ψ∥2  L2 − |⟨ψ|ψ∗⟩|2  ∥ψ∗∥2  L2  = ∥ψ∥2  L2 − |⟨ψ|(eT∗ − I)φ0⟩|2  ∥ψ∗∥2  L2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (18)  We next note that  |⟨ψ|(eT∗ − I)φ0⟩|2  ∥ψ∗∥2  L2  ≤ ∥ψ∥2  L2 ∥(eT∗ − I)φ0∥2  L2  ∥ψ∗∥2  L2  = ∥ψ∥2  L2  �  1 −  1  ∥ψ∗∥2  L2  �  ,  which inserted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (18) yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  10  With the inequality (16) at hand, we can establish the inequality  ∆2(t∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' t, t′) = ⟨∆ ˆTφ0|e− ˆT∗( ˆH − E0)e  ˆT∗|∆ ˆTφ0⟩  ≥ γeff  ∗ ∥∆ ˆTφ0∥2  L2 − CGCC(T∗)∥∆ ˆTφ0∥2  H1,  (19)  where C is a constant that depends on the Hamiltonian ˆH and  GCC(T∗) = ∥e  ˆT∗ − I∥L2 + ∥e− ˆT †  ∗ − I∥L2∥e  ˆT∗∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (20)  Equation (19) follows from the definition of ∆2 and that  ∆2 = ⟨∆ ˆTφ0| ˆH − E0|∆ ˆTφ0⟩ + ⟨∆ ˆTφ0| ˆH − E0|(e  ˆT∗ − I)∆ ˆTφ0⟩  + ⟨(e− ˆT †  ∗ − I)∆ ˆTφ0| ˆH − E0|e  ˆT∗∆ ˆTφ0⟩,  then, using that ˆH is a bounded operator in the energy norm and the estimate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (16),  we obtain the desired result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  3  The S-Diagnostic  Given the reformulation of the strong monotonicity property in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (19), we consider a  computation to be successful if the results fulfill Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In order to derive an a posterioi  diagnostic, we reformulate this inequality in a way that yields a function that indicates  a reliable computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' To ensure the tractability of the said function we introduce the  following approximations, which will yield diagnostic functions of different flavors, later  referred to as S1, S2, and S3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  11  Approximation (i)  A first-order Taylor approximation of e ˆT∗ and the trivial operator  norm inequality 2 yields  ∥e  ˆT∗φ0∥2  L2 ≈ 1 + ∥ ˆT∗∥2  L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (21)  Approximation (ii)  For GCC we use (i) and make the approximation (linearization)  GCC(T) ≈ 2∥ ˆT∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (22)  Approximation (iii)  As outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 20, we can moreover estimate  (1 + ∥ ˆZ∗∥2  L2)1/2 ≈ (1 + ∥ ˆT∗∥2  L2)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (23)  This approximation follows by equating the bra and ket wave functions (in the bivariational  formulation) e− ˆT †  ∗(ˆI + ˆZ∗)|φ0⟩ = ∥e ˆT∗φ0∥−2  L2 e ˆT∗|φ0⟩ with eˆΛ∗ = ˆI + ˆZ∗ and approximating  e− ˆT †  ∗(ˆI + ˆZ∗)|φ0⟩ ≈ (ˆI + ˆZ∗)|φ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (24)  With these approximations at hand, we can derive three variants of the S-diagnostic that  we shall investigate subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='1  The S1-diagnostic  Starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (19), we first note that we are considering the finite-dimensional case, and  therefore there exists a constant C > 0 such that  ∆2(t∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' t, t′) ≥  �  γeff  ∗ − CGCC( ˆT∗)  �  ∥∆ ˆTφ0∥2  L2  (25)  2  ∥ ˆT∗φ0∥L2 ≤ ∥ ˆT∗∥L2∥φ0∥L2 = ∥ ˆT∗∥L2  12  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Next, we employ Approximation (ii) in the definition of GCC( ˆT∗), and combine Ap-  proximation (i) with the definition of γeff  ∗  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (17), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=',  γeff  ∗  ≈  γ∗  1 + ∥ ˆT∗∥2  L2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (26)  This yields  γeff  ∗ − CGCC( ˆT∗) ≈  γ∗  1 + ∥ ˆT∗∥2  L2  − 2C∥ ˆT∗∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (27)  Requiring that this expression is positive, we obtain the success condition  1  2 > C  γ∗  (1 + ∥ ˆT∗∥2  L2)∥ ˆT∗∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (28)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2  The S2-diagnostic  By applying Approximation (iii) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (28), we obtain a success condition that involves the  Lagrange multipliers, namely,  1  2 > C  γ∗  ∥ ˆT∗∥2  L2  (1 + ∥ ˆZ∗∥2  L2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (29)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='3  The S3-diagnostic  To obtain a diagnostic that includes the Lagrangian multipliers without making use of Ap-  proximation (iii), we shall follow the argument on strong monotonicity of the extended CC  function FECC defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Note that although we use the extended CC formalism in this  section (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', where the Lagrange multipliers are treated as a second set of cluster amplitudes),  the derived diagnostic is for the conventional single reference CC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Subsequently, we  assume that truncations of ˆT and ˆΛ are at the same rank, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', the truncated scheme follows  as described above for V(d) but takes the double form V(d) × V(d) and with Pd being the  orthogonal projector onto Vd × Vd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Note that this aligns with practical implementations of  the CC Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' For brevity, let ˆU = ( ˆT, ˆΛ), ˆU∗ = ( ˆT∗, ˆΛ∗) and ˆU (d)  ∗  = ( ˆT (d)  ∗ , ˆΛ(d)  ∗ ) and  furthermore, set Fd to be the Galerkin discretization of FECC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', Fd( ˆU (d)) = PdFECC( ˆU (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  13  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 15 strong monotonicity of FECC was established under certain assumptions, and  recently generalized to a class of extended CC theories21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We, therefore, refer the reader  to these references for the full proof, here we shall only address those parts relevant to our  diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Similarly to the CC case, local strong monotonicity of FECC holds if  ∆ECC := ⟨FECC(u) − FECC(u′), u − u′⟩ ≥ γ∥u − u′∥2  (30)  for some positive constant γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Note that we here extended the notation such that u carries  both the primal-, and dual variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Furthermore, we let ∆ECC up to second order in ∥u−u′∥  be denoted ∆ECC  2  and similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (19) we have  ∆ECC  2  (u∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' u, u′) ≥ γeff  ∗ ∥∆ ˆUφ0∥2  L2 − CGECC( ˆU∗)∥∆ ˆUφ0∥2  H1,  (31)  where  GECC( ˆU) = GECC( ˆT, ˆΛ)  = ∥e− ˆT †e  ˆΛ∥L2∥e  ˆT − I∥L2 + ∥e− ˆT †e  ˆΛ − I∥L2 + K∥φ0∥H1∥e− ˆT †∥L2∥e  ˆT∥L2∥e  ˆΛ − I∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  for some positive constant K  Starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (31), we note again that since we are considering finite-dimensional  Hilbert spaces, there exists a constant C > 0 such that  ∆ECC  2  (u∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' u, u′) ≥  �  γeff  ∗ − CGECC( ˆU∗)  �  ∥∆ ˆUφ0∥2  L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (32)  We next employ a variation of Approximation (iii): For GECC we make the substitution  eˆΛ = ˆI + ˆZ and approximate with a low-order Taylor expansion  ˜GECC( ˆT, ˆZ) := GECC( ˆT, ˆΛ( ˆZ)) ≈ C(∥ ˆT∥L2 + ∥ ˆZ∥L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (33)  14  Hence, we arrive at the approximation (and we remind the reader that C is used as a generic  constant)  γeff  ∗ − CGECC( ˆU∗) ≈  γ∗  1 + ∥ ˆT∗∥2  L2  − C(∥ ˆT∗∥L2 + ∥ ˆZ∗∥L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (34)  Requiring that this expression is positive, we find the condition  1 > C  γ∗  �  (1 + ∥ ˆT∗∥2  L2)(∥ ˆT∗∥L2 + ∥ ˆZ∗∥L2)  �  ≈ C  γ∗  �  (1 + ∥ ˆT∗∥2  L2)∥ ˆT∗∥L2 +  ∥ ˆZ∗∥L2  1 + ∥ ˆZ∗∥L2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (35)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='4  Approximation of operator norms using singular values  The above-derived success conditions Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (28), (29) and (35) can be directly implemented,  however, the quantities involved will depend on the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This can be illustrated  by simply placing copies of a molecular system at a distance such that they are at least  numerically non-interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In that case, the reliability of the overall CC calculation is  determined by the CC calculations of a single copy, yet, the operator norm of the cluster  operator ∥ ˆT∥L2 will scale with the system’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  To remedy this serious difficulty, we consider an alternative interpretation of the clus-  ter operators22: The CCSD method yields a set of single amplitudes (ta  i ) forming a ma-  trix in Rnocc×nvirt and a set of double amplitudes (tab  ij ) forming a fourth-order tensor in  Rnocc×nocc×nvirt×nvirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' As outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 22, in order to capture the pair correlation we re-  shape the fourth-order tensor that describes the double amplitudes as a matrix in Rn2  occ×n2  virt,  an operation that is also known as “matricization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In order to include pair correlations  captured by the single amplitudes, we can moreover extend (tab  ij ) to also include products of  single amplitudes which yields MT ∈ Rn2  occ×n2  virt with matrix elements  [MT]ij,ab = tab  ij + (ta  i tb  j − tb  ita  j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (36)  15  The singular value decomposition then yields  MT = UTΣTV ⊤  T ,  (37)  where UT, VT are real orthogonal matrix and ΣT is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We will subsequently use the  spectral norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', the largest singular value, here denoted as σ(MT) to approximate the  operator norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=',  ∥ ˆT∥L2 ≈ σ(MT) =: σ(t)  (38)  and similarly for the dual variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Incorporating this into the success conditions Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (28),  (29) and (35) yields the S-diagnostic functions used in this article  S1(t) := 1  γ∗  (1 + σ(t)2)σ(t),  (39a)  S2(t, z) := 1  γ∗  σ(t)  1 + σ(z)2,  (39b)  S3(t, z) := 1  γ∗  �  (1 + σ(t)2)σ(t) +  σ(z)  1 + σ(z)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (39c)  For computed cluster amplitudes (t) and Lagrange multipliers (z), the above functions  will yield an S-diagnostic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  In the following numerical investigations, we will first  investigate the statistical correlation between the computed S-diagnostic value and different  measures of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Second, we will investigate a quantitative bound for the S-diagnostic value  beyond which the computations may not be reliable and further benchmark computations  with more profound error classifications are advised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  4  Numerical simulations  In this section, we numerically scrutinize the proposed S-diagnostic procedures derived in  the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' All simulations are performed using the Python-based Simulations  of Chemistry Framework (PySCF)23–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  First, we perform geometry optimizations on a  16  medium-sized set of molecules comprising all molecules that were investigated in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 3,5,6  to test the T1, D1, and D2 diagnostic, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' With this data at hand, we can propose  an initial set of values, beyond which our diagnostic suggests interpreting the computational  results with caution and if possible benchmarking with additional methods that allow for  a more profound error classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Second, we target small model systems whose multi-  reference character can be controlled by simple geometric changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Third, we numerically  investigate transition metal complexes that have been shown to be misdiagnosed by the T1  and D1 diagnostics7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='1  Correlation in Geometry Optimization  In order to quantify the correlation between the S-diagnostics and the error of the CC  method, we numerically investigate the Spearman correlation26 between the error of in sil-  ico geometry optimizations and the corresponding value of the S-diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We perform  geometry optimizations for 34 small to medium-sized molecules that were previously studied  in relation to CC error classifications3,5,6, see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Table 1: Molecules which are used in the geometry optimization presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  H2N2  HOF  C2H2  ClOH  H2S  O3  FNO  ClNO  C2  C3  CO  HNO  HNC  HOF  Cl2O  P2  N2H2  HCN  CH2NH  N2  C2H4  F2  HOCl  Cl2  HF  CH4  H2O  SiH4  NH3  HCl  CO2  BeO  H2CO  CH2  The calculations are performed using the CC method with singles and doubles (CCSD)  using the cc-pVDZ basis set provided by PySCF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' the geometry optimization is performed  using the interface to PyBerny27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The numerically obtained results are compared with exper-  imentally measured geometries of the considered systems in their gas phases extracted from  the Computational Chemistry Comparison and Benchmark Data Base (CCCBDB)28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Since  the computed atomic positions cannot be directly compared, we introduce the bond-length  matrix that describes the pairwise distance between the atoms in the molecular compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  17  This bond-length matrix can be directly compared with the bond-length matrix provided by  CCCBDB if we label and order the atoms of the corresponding system accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We in-  vestigate the correlation between the S-diagnostics and three possible error characterizations  obtained from the absolute difference of the bond-length matrices denoted D(diff):  i) The maximal absolute error (∆r(max)  abs  ): the maximal absolute deviation of the numeri-  cally obtained bond-length matrix to the experimentally obtained bond-length matrix,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=',  ∆r(max)  abs  = max  i,j D(diff)  ij  ii) The averaged absolute error (∆r(ave)  abs ): the averaged absolute deviation of the numeri-  cally obtained bond-length matrix to the experimentally obtained bond-length matrix,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=',  ∆r(ave)  abs  =  �  i,j D(diff)  i,j  Natoms  iii) The averaged relative error (∆r(ave)  rel  ): the averaged relative deviation of the numerically  obtained bond-length matrix to the experimentally obtained bond-length matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=',  ∆r(ave)  rel  =  �  i,j D(diff)  i,j  Natoms maxi,j D(diff)  ij  Computing the Spearman correlation between the errors listed above and the proposed S-  diagnostics, we find that all suggested S-diagnostics correlate well with all the error measures  suggested, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', we consistently find correlations of rsp > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5 with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0008, see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The  largest correlation is observed between the maximal absolute error (∆r(max)  abs  ) and S2 and S3  where we find a correlation of rsp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='58476 with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='00018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' For comparison, we compute  the Spearman correlation for the previously suggested T1, D1, and D2 diagnostic in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  We find that T1, and D1, are uncorrelated to all the errors that we investigate here, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=',  rsp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='3 with p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The D2 diagnostic6 shows a correlation with the averaged absolute  error (∆r(ave)  abs ) and the averaged relative error (∆r(ave)  rel  ), where we find a correlation of rsp =  18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='36886 with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='026847 and rsp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='35496 with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='033646, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We moreover  compare the S-diagnostics with the recently suggested indices of multi-determinantal and  multi-reference character in CC theory8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We find that similar to the S-diagnostics, the EEN  index8 correlates well with the maximal absolute error (∆r(max)  abs  );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' we observe a correlation  of rsp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='53572 with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Directly comparing the Spearman correlation of the S-diagnostics with the T1, D1, and  D2 diagnostic, we see that the S-diagnostics have a significantly higher correlation than the  heuristically motivated diagnostics T1, D1 and D2 diagnostics while exhibiting a higher level  of stochastic significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Comparing the Spearman correlation of the S-diagnostics with the  indices of multi-determinantal and multi-reference character in CC theory, we find that the  S-diagnostic and EEN show similar correlation with the maximal absolute error (∆r(max)  abs  )  with a comparable level of stochastic significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Table 2: Spearman correlation between the S-diagnostic computed form CCSD amplitudes  and different errors in geometry optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The pair-entries show the rank correlation  and the corresponding p-value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', (rsp, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  ∆r(max)  abs  ∆r(ave)  abs  ∆r(ave)  rel  S1  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='57910, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000215)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content='000581)  T1  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='03025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='863034)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content='895674)  D1  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='27675, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='107522)  (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='00541, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='975040)  (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='02034, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='906294)  D2  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='16974, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='329625)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='36886, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='026847)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='35496, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='033646)  EEN  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='53572, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000759)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='42059, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='010643)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='33694, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='044488)  In order to obtain an approximate trusted region suggested by the S-diagnostics, we  require a descriptive function that maps the value obtained from the S-diagnostic to the  error in geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Since the Spearman correlation describes a monotone relation between  the quantities, we may not assume that this relation is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Unfortunately, the Spearman  correlation does not indicate the type of relation that connects the two measured quanti-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We, therefore, perform a piecewise linear fit to the data obtained in this simulation,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We here allow for four segments which are optimized to reach the best approx-  19  imation by means of a piecewise linear and monotone function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We emphasize that larger  numbers of segments yield similar approximations, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Performing this piecewise  linear fit, we observe that the function is constant on some segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Based on the data dis-  tribution, we conclude that this constant behavior is artificial and caused by the test set not  being sufficiently versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In particular, no quantitative conclusions can be drawn from the  piecewise linear fit function for values S3 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Therefore, from the geometry optimizations  performed here, we can merely conjecture to raise a concern about the validity of CC calcu-  lations performed for values of the S-diagnostics v(3)  crit ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Based on the piecewise linear fit,  S3 = 1 corresponds to an error larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='035 a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' A larger statistical investigation with  a larger variety of molecules and basis set discretizations is delegated to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We  emphasize that this first estimation of vcrit is particularly pessimistic since the data set is not  versatile enough to give a precise estimation of vcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Indeed, in the subsequently performed  simulations, we show a more refined estimation of vcrit that reveals v(2)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='9 and v(3)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8,  for S2, and S3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (a)  (b)  Figure 1:  The maximal error in geometry optimization as a function of the S2 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (a)  The orange line corresponds to a piecewise linear fit to the data using four segments for  the piecewise linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (b) Piecewise linear fits to the data with a varying number of  segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Aside from CC-based simulations, we can also perform MP2 simulations, and use the  obtained doubles amplitudes to compute the S-diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  We find that the proposed  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='150  4-seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='100  Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  S3 value0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='150  3-seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='125  4-seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  5-seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='100  Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  6-seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  S3 valueS-diagnostics correlate similarly well with MP2 based calculations as it does for CCSD,  see Table 3  Table 3: Spearman correlation between S-diagnostics computed from MP2 doubles ampli-  tudes and different errors in geometry optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  ∆r(max)  abs  ∆r(ave)  abs  ∆r(ave)  rel  S1  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='55992, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000384)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='54569, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000577)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='49781, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='002006)  S2  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='56687, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000313)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='54801, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000541)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='49858, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='001968)  S3  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='55992, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000384)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='54569, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000577)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='49781, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='002006)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2  Model Systems  In this section we investigate the use of the proposed S-diagnostics for four model systems  whose multi-reference character can be controlled by simple geometric change: (1) twisting  ethylene, (2) the C2v insertion pathway for BeH2 (Be · · · H2)29, (3) the H4 model (transition  from square to linear geometry)30 (4) the H4 model (symmetrically disturbed on a circle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  the computations are performed in cc-pVTZ basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='1  Twisting ethylene  We begin by numerically investigating the proposed S-diagnostics for ethylene twisted around  the carbon–carbon bond, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Θ  H  C  H  H  C  H  H  C  H  C  H  H  Figure 2: Depiction of the ethylene (C2H4) model with twist angle Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  At a twist angle of 90°, this system shows a strong multi-reference character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This can  be seen as follows: At the equilibrium geometry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', in a planar geometry, the two carbon  p orbitals are perpendicular to the molecular plane form bonding π and anti-bonding π∗  orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In this geometry, the ground state doubly occupies the π-orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' As we twist around  21  the carbon–carbon bond, the overlap between the two p orbitals decreases and becomes zero  at 90°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Therefore, at 90° the π and π∗ orbitals become degenerate and the π-bond is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  This (quasi) degeneracy can also be observed numerically by computing the HOMO-LUMO  gap as a function of the twist angle, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Computing the corresponding ground state  energy as a function of the twist angle, we observe the characteristic energy cusp at exactly  90°, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (a)  (b)  Figure 3:  (a) HOMO-LUMO gap of C2H4 as a function of the twist angle (b) RHF and  RCCSD energies of C2H4 as a function of the twist angle  Due to the quasi degeneracy around 90°, we compare the S-diagnostic with the MRI  index suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We clearly see the indication of the quasi degeneracy in the MRI  index, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The S-diagnostic also indicates the problematic region around 90°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' By  numerical comparison, we find that a cut-off value of v(2)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='9 and v(3)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8 for S2 and  S3, respectively, indicates the same region of quasi degeneracy as the MRI index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2  C2v insertion pathway for BeH2  Next we shall investigate the C2v insertion pathway for BeH2 (Be · · · H2)29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The model  represents an insertion of the Be atom into the H2 molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The transformation coordinate  connects the non-interacting subsystems (Be + H2) with the linear equilibrium state (H-Be-  H), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 5  22  HOMO-LUMO gap  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='30  0  1  2  3  Twist angle/ RadianGround state energy  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  RHF  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2  CCSD  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='4  0  1  2  3  Twist angle/ Radian(a)  (b)  Figure 4:  (a) The proposed S-diagnostics of C2H4 as a function of the twist angle, the  dotted green and red horizontal lines correspond to v(2)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='9 and v(3)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (b) The previously suggested MRI of C2H4 as a function of the twist angle  H  Be  H  H  Be  H  Figure 5: Depiction of the C2v insertion pathway for BeH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  23  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  S1  S2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  S3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  Twist angle/ Radian1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  MRI  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  Twist angle/ RadianWe here follow the insertion pathway outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 29 and denote the position of  the beryllium atom by X-position, where X-position equal to zero corresponds to the linear  equilibrium state and X-position equal to five corresponds to the non-interacting subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  The transition state of this chemical transformation has a pronounced multi-reference char-  acter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Another distinguishing feature of this model system is a change in the character of the  dominating determinant in the wave function along the potential energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' There are  two leading determinants in the wave function, each of which dominates in a certain region  of the potential energy surface while both are quasi-degenerate around the transition-state  geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This leads yields to discontinuities as can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 6a and 6b  (a)  (b)  Figure 6:  (a) HOMO-LUMO gap as a function of the X-position (b) RHF and RCCSD  energies as a function of the X-position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Due to the quasi-degeneracy that appears along the transition path, we again compare  the proposed S-diagnostics with the MRI index suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We clearly see the  indication of the quasi degeneracy in the MRI index, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The region indicated by  MRI< −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='99 corresponds to x ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The S-diagnostic also indicates a region where  the CC computations are potentially unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' It is worth mentioning that choosing the  critical values similar to the previous example, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', v(2)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='9 and v(3)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8, the predicted  region corresponds to x ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5] and x ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='25], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In order to reproduce  the same region of quasi-degeneracy as indicated by the MRI index, the critical values have  24  Ground state energy  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  RHF  Hartree  CCSD  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2  一15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='4  Energyl  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='6  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8  0  1  2  3  4  5  X-position/ aoHOMO-LUMO gap  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2  0  2  3  4  5  1  X-position/ aoto be adjusted to v(2)  crit = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8 and v(3)  crit = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (a)  (b)  Figure 7: (a) shows the S-diagnostics, the dotted green, and red horizontal lines correspond  to v(2)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='9 and v(3)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (b) shows the previously suggested MRI  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='3  H4 model (transition from square to linear geometry)  Next, we shall investigate the proposed S-diagnostics applied to the H4 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The H4  model is a standard transition model that allows steering the quasi-degeneracy using a single  parameter, namely, the transition angle α where α = 0 corresponds to a square geometry  and α = π/2 corresponds to a linear geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='30, we set a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' ),  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  a  a  a  a  a  α  α  a  a  a  a  Figure 8: Depiction of the H4 model undergoing the transition from a square geometry to  linear geometry model by the angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  We see that as the transition angle α tends to zero, the HOMO-LUMO gap closes and  the system shows signs of (quasi-) degeneracy, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 9a  25  So  S  S2  101  100  0  2  3  4  5  X-position/ ao1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  MRI  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0  1  2  3  4  5  X-position/ ao(a)  (b)  Figure 9: (a) HOMO-LUMO gap of H4 as a function of the transition angle (b) RHF, CCSD  and FCI energies of H4 as a function of the transition angle  Due to the quasi degeneracy near α = 0, we again compare the proposed S-diagnostics  with the MRI index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We clearly see the indication of the quasi degeneracy in the MRI index,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 10b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The S-diagnostic also indicates the problematic region near zero transition  angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  A cut-off value of v(2)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='9 and v(3)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8 results in S2 and S3, respectively,  indicating the same region of quasi degeneracy as the MRI index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (a)  (b)  Figure 10:  (a) The S-diagnostics of H4 as a function of the transition angle, the dotted  green, and red horizontal lines correspond to v(2)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='9 and v(3)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (b) The  previously suggested MRI of H4 as a function of the transition angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  For this small model Hamiltonian, it is moreover feasible to perform computations at the  26  HOMO-LUMO gap  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  Angle/ radianGround state energy  RHF  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  CCSD  FCI  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  Angle/ radian6  So  S1  5  S2  4  3  2  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='50  Angle/ radian1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  MRI  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='75  ¥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='50  Angle/ radianFCI level of theory, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This comparison yields a quantitative comparison of error  and S-diagnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Figure 11  Figure 12: The energy error of CCSD compared to the FCI reference energy using semi-log  scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The area left of the vertical solid (black), dashed (green), and dotted-dashed (red)  lines correspond to the regions where the MRI, S2, and S3 diagnostic indicate a potential  failure of CCSD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='4  H4 model (symmetrically disturbed on a circle)  Another variant of the H4 model that is commonly employed to evaluate CC methods consists  of four hydrogen atoms symmetrically distributed on a circle of radius R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='738 ˚A31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  For small or large angles, the system resembles two H2 molecules that are reasonably well  separated, but as the angle passes through 90, the four atoms form a square yielding a  degenerate ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The exact energy is smooth as a function of the angle, but at the  RHF level, we observe a cusp at 90, similar to the rotation of the carbon-carbon bond in  ethylene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We follow the system’s geometry configuration outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='32, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  We see that as the transition angle Θ tends to π/2 radians (90°), the HOMO-LUMO gap  closes and the system shows signs of (quasi) degeneracy, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 14a  Due to the quasi degeneracy near Θ = π/2 (90°), we again compare the proposed S-  diagnostics with the MRI index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We clearly see the indication of the quasi degeneracy in  27  Ground state energy error  EcCSD - FFCI  X  10-3  10-3  3  X  2  × 10-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  Angle/ radianΘ  Θ  Figure 13: Depiction of the H4 model undergoing a symmetric disturbance on a circle modeled  by the angle Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (a)  (b)  Figure 14:  (a) HOMO-LUMO gap of H4 as a function of the transition angle (b) RHF,  RCCSD energies of H4 as a function of the transition angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  28  HOMO-LUMO gap  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  Angle/ RadianGround state energy  RHF  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8  CCSD  FCI  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  Angle/ Radianthe MRI index, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 15b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The S-diagnostic also indicates the problematic region near  zero transition angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' A cut-off value of v(2)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='9 and v(3)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8 results in S2 and S3,  respectively, indicating the same region of quasi degeneracy as the MRI index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  (a)  (b)  Figure 15:  (a) The S-diagnostics of H4 as a function of the transition angle, the dotted  green, and red horizontal lines correspond to v(2)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='9 and v(3)  crit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (b) The  previously suggested MRI of H4 as a function of the transition angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  For this small model Hamiltonian, it is moreover feasible to perform computations at  the FCI level of theory, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This comparison reveals the variational collapse of the  CCSD energy, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 16a, and moreover yields a quantitative comparison of error and S-  diagnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The trusted region suggested by the S-diagnostic corresponds to a CCSD energy  error smaller than 2 · 10−4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' which is below the chemical accuracy threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Since the simulations performed in the previous section suggest that the previously used  T1, D1, and D2 diagnostics are uncorrelated, or merely weakly correlated, we do not report  their performance here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The computations showing the performance of the T1, D1, and D2  diagnostics can be found in the Appendix, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 26 to 29  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='3  Transition metal complexes  In this section we investigate three square-planar copper complexes [CuCl4]2−, [Cu(NH3)4]2+,  and [Cu(H2O)4]2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Transition metal complexes are in general considered to be strongly corre-  29  S1  15  S2  S3  10  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='25  Angle/ RadianMRI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='25  Angle/ Radian(a)  (b)  Figure 16: (a) The energy error of CCSD compared to the FCI reference energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Note that  in the region of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='3-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='8 radians the CCSD energy is lower than the FCI reference energy,  which indicates the variational collapse of the CCSD energy in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' (b) The absolute  value of the energy error of CCSD compared to the FCI reference energy using semi-log  scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The area between the vertical solid (black), dashed (green), and dotted-dashed (red)  lines correspond to the regions where the MRI, S2, and S3 diagnostic indicate a potential  failure of CCSD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  lated systems and complete active space self-consistent field (CASSCF) theory is commonly  applied, with multi-reference perturbation or truncated CI corrections for dynamic correla-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' However, as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 7, the single reference CC method performs very well despite  the large D1 diagnostic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We use these systems to scrutinize the proposed S-diagnostics  for larger systems that are known to be misleadingly diagnosed by the D1 diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Similar to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 7, we perform the simulation of [CuCl4]2−, [Cu(NH3)4]2+, and [Cu(H2O)4]2+  in 6-31G basis using UHF and ROHF as reference states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Also, He, Ne, and Ar cores were  frozen in the nitrogen, chlorine, and copper atoms, respectively, resulting in 41 electrons  in 50, 66, and 74 orbitals for the [CuCl4]2−, [Cu(H2O)4]2+, and [Cu(NH3)4]2+ molecules,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We list the ground state energies obtained at the mean-field level of theory and  the corresponding CCSD results in Table 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' we moreover list the HOMO-LUMO gap which  enters in the S-diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  The results in Table 4 show that UHF and ROHF calculations predict similar energy  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Moreover, using the UHF, or ROHF reference state results in similar CCSD energy  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='000  Iartree  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='004  Energyl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='008  EcOSD -EFCI  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  Angle/ Radian10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  10  IEcCSD －EFCll  10-5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='0  Angle/ RadianTable 4: Energies values and HOMO-LUMO gap obtained with UHF, ROHF, and UCCSD  calculations given the reference state from UHF and ROHF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  UHF  γUHF  UCCSD  RHOF  γROHF  UCCSD  [CuCl4]2−  3476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='764  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='453  3477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='119  3476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='763  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='146  3477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='119  [Cu(NH3)4]2+  1862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='977  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='564  1863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='663  1862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='976  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='351  1863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='663  [Cu(H2O)4]2+  1942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='225  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='677  1942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='914  1942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='224  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='340  1942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='914  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' It is worth noticing that ROHF yields a generally smaller HOMO-LUMO gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Since  the performed CCSD calculations differ in their reference, we can compute the S-diagnostics  for both sets of calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The results obtained from a UHF and ROHF reference are  listed in Table 5 and in Table 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Table 5: S-diagnostics obtained for the three square-planar copper complexes [CuCl4]2−,  [Cu(NH3)4]2+, and [Cu(H2O)4]2+ in spin unrestricted formulation with UHF reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  S1  S2  S3  T1  D1  D2  [CuCl4]2−  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='208  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='409  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='406  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='158  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='110  [Cu(NH3)4]2+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='203  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='403  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='398  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='130  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='121  [Cu(H2O)4]2+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='155  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='308  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='305  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='072  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='116  We see that all S-diagnostic variants suggest that the CCSD calculations were successful,  and do not require additional numerical confirmation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This is opposed to the D1 diagnostics,  which aligns with the results reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Table 6: S-diagnostics obtained for the three square-planar copper complexes [CuCl4]2−,  [Cu(NH3)4]2+, and [Cu(H2O)4]2+ in spin unrestricted formulation with ROHF reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  S0  S1  S2  T1  D1  D2  [CuCl4]2−  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='645  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='285  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='167  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='110  [Cu(NH3)4]2+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='326  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='646  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='638  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='139  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='121  [Cu(H2O)4]2+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='309  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='614  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='607  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='077  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='116  Similar to the results in Table 5, we see that all variants of the S-diagnostic suggest that  the CCSD calculations were successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' However, it is worth noticing that the S-diagnostic  values have increased compared to the values reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  31  5  Conclusion  In this article, we proposed three a posteriori diagnostics for single-reference CC calcula-  tions which we called S-diagnostics, due to their origin in the strong monotonicity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Contrary to previously suggested CC diagnostics, the S-diagnostics are motivated by math-  ematical principles that have been used to analyze CC methods of different flavors in the  past9,15,18,19,33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  We performed a set of geometry optimizations for small to medium-sized molecules in  order to reveal the correlation between the S-diagnostics and the error in geometry from  CCSD calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The test set comprised all molecules that were used in previous articles  concerning CC diagnostics3–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Our investigations revealed that the S-diagnostics correlate  well and with large statistical relevance with different errors in geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This yields a first  estimate of the critical values for the S-diagnostics beyond which the computational results  should be confirmed using further and more careful numerical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The observed  correlation between the S-diagnostics and the different errors in geometry are comparable  to the recently suggested EEN index8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' A heuristic test revealed that the S-diagnostics also  correlate well and with large statistical relevance with the error in geometry at the MP2 level  of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This suggests that the S-diagnostics can also be used as an a posteriori diagnostic  for MP2 calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Our numerical simulations moreover showed that diagnostics based on  single excitation cluster amplitudes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=', D1 and T1, are uncorrelated to errors in geometry  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Following we investigated the S-diagnostics for transition state models that undergo a  transition from a region in which CC calculations are reliable to a regime where the CC cal-  culations require further numerical investigations—in this case, due to (quasi-) degeneracy of  the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The S-diagnostic detects the corresponding regions of (quasi-) degeneracy  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' In fact, its performance is comparable to the recently suggested MRI indicator—an a  posteriori indicator for multi-reference character8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  The last set of numerical simulations targeted transition metal complexes which have  32  recently been carefully benchmarked7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' The previously performed benchmark calculations  revealed that diagnostics based on single excitation amplitudes severely misdiagnose the  performance of CCSD for these transition metal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Our computations confirm this,  and moreover, show that the S-diagnostic correctly confirms the accuracy of the CCSD  results outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  These carefully performed numerical investigations suggest that the S-diagnostic is a  promising candidate for an a posteriori diagnostic for single-reference CC and MP2 calcu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' To further confirm this, benchmarks on a larger set of molecules will be performed  in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Moreover, since the mathematical analysis of the single-reference CC method  generalizes to periodic systems as well, we believe that the S-diagnostic can moreover be  applied to simulations of solids at the CC and MP2 level of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Throughout our numerical investigations, we observe a subpar performance of the T1 and  D1 diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' This suggests that those diagnostics should once and for all be removed as  a posteriori diagnostic tools for single-reference CC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Acknowledgement  This work was partially supported by the Air Force Office of Scientific Research under the  award number FA9550-18-1-0095 and by the Simons Targeted Grants in Mathematics and  Physical Sciences on Moir´e Materials Magic (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' 262695.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Lin Lin, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Trygve Helgaker,  Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Laurens Peters, and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Tilmann Bodenstein for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  33  References  (1) Bartlett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Musial, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Coupled-cluster theory in quantum chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Schaefer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Janssen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Alavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Interplay between Electronic Correlation  and Metal–Ligand Delocalization in the Spectroscopy of Transition Metal Compounds:  Case Study on a Series of Planar Cu2+ Complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content='  (8) Bartlett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Bauman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Melnichuk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Ranasinghe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Ravi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='  Perera, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Index of multi-determinantal and multi-reference character in coupled-cluster  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content='  McClain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Sayfutyarova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Laestadius, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Legeza, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content=' Schneider, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Kvaal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Analysis of the  tailored coupled-cluster method in quantum chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+page_content='  37  6  Appendix  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content='1  Correlation in Geometry Optimization  Since we can correlate three S-diagnostic variants with three error measures, we can in  principle perform the piecewise linear fit that is presented in Section on Correlation in  Geometry Optimization for nine different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
+page_content=' We here present the piecewise linear fits  which were not addressed in the above article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FIT4oBgHgl3EQf6SuL/content/2301.11393v1.pdf'}
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+Strongly correlated physics in organic open-shell quantum systems
+G. Gandus,1, ∗ D. Passerone,2 R. Stadler,3 M. Luisier,1 and A. Valli3, 4, †
+1Integrated Systems Laboratory, ETH Z¨urich, Gloriastrasse 35, 8092 Z¨urich, Switzerland
+2Empa, Swiss Federal Laboratories for Materials Science and Technology,
+¨Uberlandstrasse 129, CH-8600, D¨ubendorf, Switzerland
+3Institute for Theoretical Physics, Vienna University of Technology,
+Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
+4Department of Theoretical Physics, Institute of Physics,
+Budapest University of Technology and Economics, M¨uegyetem rkp. 3., H-1111 Budapest, Hungary
+Strongly correlated physics arises due to electron-electron scattering within partially-filled or-
+bitals, and in this perspective, organic molecules in open-shell configuration are good candidates to
+exhibit many-body effects. With a focus on neutral organic radicals with a molecular orbital host-
+ing a single unpaired electron (SOMO) we investigate many-body effects on electron transport in a
+single-molecule junction setup. Within a combination of density functional theory and many-body
+techniques, we perform numerical simulations for an effective model for which all the parameters,
+including the Coulomb tensor, are derived ab-initio. We demonstrate that the SOMO resonance is
+prone towards splitting, and identify a giant electronic scattering rate as the driving many-body
+mechanism, akin to a Mott metal-to-insulator transition. The nature of the splitting, and thus of
+the resulting gap, as well as the spatial distribution of the SOMO and its coupling to the electrodes,
+have dramatic effects on the transport properties of the junction. We argue that the phenomenon
+and the underlying microscopic mechanism are general, and apply to a wide family of open-shell
+molecular systems.
+I.
+INTRODUCTION
+Strongly correlated electronic physics arises in par-
+tially occupied orbitals in the presence of competing en-
+ergy scales.
+Due to the Coulomb repulsion, electrons
+display a collective behavior, leading to the breakdown
+of the single-particle picture and the emergence of com-
+plex quantum phenomena.
+Electronic correlations are
+also enhanced due to spatial confinement effects in low-
+dimensional and nanoscopic systems. While in solid-state
+physics the concept of a “strongly-correlated metal” is
+well-established, its analog for molecules is not obvious.
+In chemistry, the majority of stable organic molecules
+have closed-shell electronic configurations, and electrons
+are paired in delocalized molecular orbitals (MOs) that
+are either completely filled or empty. The energy differ-
+ence between the frontier MOs, i.e., the highest occupied
+(HOMO) and the lowest unoccupied (LUMO) orbitals
+defines the spectral gap. In particular, π-conjugated sys-
+tems display a wide HOMO-LUMO gap (∆ ∼ eV) which
+is controlled by the overlap of neighboring pz orbitals. A
+molecular system in an open-shell configuration (radical)
+is characterized by unpaired valence electrons residing
+in non-bonding singly-occupied MOs (SOMOs) found at
+intermediate energies between HOMO and LUMO. Rad-
+icals can form by breaking bonds or by adding/removing
+electrons (e.g., in photoinduced processes) and are inter-
+mediate products of chemical reactions.
+While open-shell configurations are typically associ-
+ated with high chemical reactivity, there exist also species
+∗ gandusgui@gmail.com
+† valli.angelo@ttk.bme.hu
+of relatively stable radicals, which possess interesting
+electronic, magnetic, and optical functionalities that are
+relevant to technological applications ranging from next-
+generation spintronics to quantum information [1–3].
+Tremendous advances in the synthesis and character-
+ization of organic radicals triggered recent experimen-
+tal studies with organic species that are stable enough
+to be trapped in break-junctions [4, 5] or investigated
+with scanning tunneling spectroscopy [6–9], which fu-
+eled a revival of interest in the molecular Kondo ef-
+fect [4, 6–12].
+There is a growing experimental and
+theoretical effort to unravel how many-body effects can
+dramatically influence electronic and transport proper-
+ties in light of technological applications.
+In the con-
+text of molecular electronics, noteworthy organic radicals
+include triphenylmethyl [4, 5, 12], Blatter radical [13],
+polyacetylene [14, 15], benzyl [16, 17], together with the
+whole family of polycyclic hydrocarbons with non-Kekul`e
+structure [7, 18–20]. Molecular organic frameworks with
+transition-metal centers (e.g., iron-porphyrin) are also
+typically open-shell, and have been recently suggested
+as molecular transistors [21, 22].
+From the theoretical point of view, in wide-gap semi-
+conductors, the electron-electron scattering rate is low
+due to the lack of electronic states at the Fermi energy.
+The accuracy of ab-inito prediction of the gap is a long-
+standing issue [23], and numerical simulations for insula-
+tors [24, 25] and molecules [25–32] predict a many-body
+renormalization of the spectral gap. However, these ef-
+fects do not change qualitatively the transport proper-
+ties. In open-shell configurations instead, it can be ex-
+pected that electron-electron scattering within the par-
+tially filled SOMO and many-body effects have a promi-
+nent role.
+arXiv:2301.00282v1  [cond-mat.str-el]  31 Dec 2022
+
+2
+In computational quantum chemistry,
+it is well-
+established that open-shell molecular configurations re-
+quire careful treatment (see, e.g., [33] for an overview)
+but the accuracy of quantum chemical methods comes
+at a high numerical cost. Hence, we recently witnessed
+significant advances in developing alternative simulation
+schemes, that are suitable to describe complex devices
+relevant to molecular electronics [11, 34, 35]. In the en-
+deavor to achieve predictive power and allow for a quan-
+titative comparison with experiments, a suitable method
+should be high-throughput — i.e., scalable and automa-
+tized as much as possible, and able to describe a real-
+istic chemical environment and many-body correlations
+within an ab-initio framework. This would allow a coop-
+erative effort between theory and experiments, and pave
+the path to future breakthroughs for next-generation
+quantum technologies.
+II.
+SCOPE OF THIS WORK
+The scope of this work is to investigate the emergence
+of strongly correlated electron physics in the electronic
+and transport properties of single-molecule junctions.
+To this end, we have developed a comprehensive nu-
+merical workflow that combines density functional theory
+(DFT) with quantum field theoretical methods, and it is
+able to address the complexity of a realistic chemical en-
+vironment as well as electronic correlation effects beyond
+the single-particle picture within an ab-initio framework.
+With both aspects taken into account, we are able to un-
+ravel the origin of many-body transport effects in single-
+molecule junctions.
+The art of combining ab-initio and many-body com-
+putational schemes lies in a transformation from non-
+orthogonal atomic orbitals (AOs) to recently introduced
+local orbitals (LOs) [36]. The LOs are by construction
+orthogonal within the same atom and localized in space.
+They take over the symmetries of the original AOs, while
+inheriting the information of the environment. This al-
+lows to represent the electronic wavefunction in a region
+of the spectrum close to the Fermi energy with a mini-
+mal set of orbitals, making them an ideal basis for many-
+body calculations. So far, LOs have been employed in the
+context of DFT [36]. In what follows, we also evaluate
+the Coulomb integrals that describe the electron-electron
+repulsion in the LO basis, and thus map to the origi-
+nal Hamiltonian onto an effective many-body problem,
+which we can feasibly solve with appropriate numerical
+methods. This recipe is particularly suitable to address
+strong correlation effects in the transport properties of
+molecular junctions.
+In terms of applications, we focus on molecular break-
+junctions in which the central molecule bridging the elec-
+trodes is in an open-shell configuration, which are strong
+candidates to manifest many-body effects. Specifically,
+we select a linear and a cyclic molecular bridge, i.e., a
+polyene radical, and a benzene molecule substituted with
+a methylene (CH2) radical group. While both molecules
+are π-radicals with one electron in the SOMO, we show
+that many-body effects bring out profound differences.
+We identify the fingerprint of strong electronic correla-
+tions in the splitting of the SOMO resonance. The details
+of the splitting and the spatial distribution of the SOMO
+on the molecular backbone have dramatic consequences
+on the transport properties of the junction.
+Finally, we demonstrate that such a splitting cannot
+be obtained with less sophisticated techniques, such as
+many-body perturbation theory. We argue that this phe-
+nomenon and the underlying microscopic mechanism are
+general, and apply to a wide family of open-shell molec-
+ular systems.
+III.
+METHODS
+A.
+Local orbitals and low-energy models
+The LOs method [36] is a transformation-based ap-
+proach that aims at retrieving hydrogen-like orbitals for
+atoms in molecules and solids. By construction, LOs are
+locally orthogonal on each atom. The starting point is a
+DFT calculation in an AOs basis set. The Hilbert space
+H is then spanned by a finite set of non-orthogonal or-
+bitals {|i⟩}, i.e., with a overlap matrix ⟨i|j⟩ = (S)ij ̸= δij
+for |i⟩ , |j⟩ ∈ H. A set of LOs {|m⟩} ∈ M ⊆ H can be
+obtained for any atom α in subspace M by a subdiago-
+nalization of the corresponding Hamiltonian sub-block
+Hα |m⟩ = ϵmSα |m⟩
+(1)
+The LOs are then linear combinations of AOs and are
+by definition orthogonal on each atom. This allows for a
+more natural physical interpretation of the LOs as atomic
+orbitals [36].
+In order to obtain an ab-initio effective
+model, we formally separate the Hilbert space into an ac-
+tive space (A) and an environment (E). The active space
+consists of a subset of LOs {|a⟩} = A ⊆ M which are ex-
+pected to describe the relevant physics close to the Fermi
+energy, and at the same time can be efficiently treated
+within quantum many-body techniques.
+Insytead, the
+environment consists of all the remaining LOs and AOs,
+i.e., {|e⟩} ∈ E ≡ H \A. Embedding the active space into
+the environment ensures that the effective model pre-
+serves all information of the original single-particle DFT
+Hamiltonian [36]. Finally, it is convenient to perform a
+L¨owdin orthogonalization [37] of the LO {|a⟩} states and
+redefine the A subspace in terms of this new orthonormal
+basis set with elements
+��a⊥�
+=
+�
+a
+(S−1/2)aa⊥ |a⟩ .
+(2)
+Since the overlap between LOs on different atoms is typi-
+cally low, i.e., (S)ij ≪ 1, the L¨owdin orthonormalization
+of the active space results only in a weak deformation of
+the original LOs, which preserves their atomic-like sym-
+metry.
+
+3
+In practice, the LO low-energy model is constructed
+embedding the active subspace into the environment
+through a downfolding procedure [38, 39]. Taking into
+account the non-orthogonality between the A and E sub-
+spaces [34], we write the Green’s function projected onto
+the A subspace as
+GA(z) = S−1
+A SAHGH(z)SHAS−1
+A ,
+(3)
+where z = E + iη is a complex energy with an infinites-
+imal shift η → 0+. GH denotes the Green’s function of
+the full Hilbert space, and SAH the overlap matrix be-
+tween orbitals
+��a⊥�
+∈ A and orbitals |i⟩ ∈ H, while the
+overlap SA between the
+��a⊥�
+states is, by construction,
+the identity matrix and will be omitted in what follows
+for notational simplicity. The effect of the environment
+on the A subspace is described by the hybridization func-
+tion
+∆A(z) = g−1
+A (z) − GA(z)−1,
+(4)
+where
+gA =
+�
+z − HA
+�−1
+(5)
+is Green’s function of the isolated A subspace. Rewriting
+GA in terms of ∆A and using the definition of gA yields
+GA(z) =
+�
+z − HA − ∆A(z)
+�−1.
+(6)
+Then, GA can be seen as the resolvent of an effective
+A subspace renormalized by the environment through a
+dynamical hybridization. The Green’s function describes
+the physics of the whole system, projected onto a sub-
+space.
+For a single-particle Hamiltonian, the partition above
+is arbitrary, and the procedure remains valid indepen-
+dently of the subset of LOs included in the active space.
+In the context of π-conjugated organic molecules, the
+projection onto a single pz LO per C atom (and pos-
+sibly other species such as N or S) is usually sufficient
+to achieve a faithful representation of the frontier MOs,
+and hence suitable to describe the physics close to the
+Fermi energy [36]. The possibility of considering a re-
+stricted subset of LOs in the effective model is of pivotal
+importance in view of performing computationally-heavy
+many-body simulations.
+B.
+cRPA and ab-initio Coulomb parameters
+In order to derive the electronic interaction parameters
+in the A subspace beyond the semi-local density approx-
+imations, we employ the constrained Random Phase Ap-
+proximation (cRPA) [34, 40, 41]. Within the cRPA, we
+select a region R ⊃ A where the formation of electron-
+hole pairs is expected to screen the Coulomb interaction
+between the A electrons. Because of the strong local na-
+ture of the LOs, it is sufficient that R comprises the A
+subspace and few atoms nearby. Defining GR to be the
+Green’s function projected onto the R subspace in anal-
+ogy with Eq. (3), the screened Coulomb interaction at
+the RPA level is given by
+WR =
+�
+I − VRPR
+�−1VR,
+(7)
+where VR is the bare Coulomb interaction
+(VR)ij,kl =
+�
+dr
+�
+dr′ψi (r)ψ∗
+j (r)
+e2
+|r − r′|ψ∗
+k(r′)ψl (r′),
+(8)
+being ψi(r) the orbitals in the R region, and PR is the
+static component of the polarizability
+(PR)ij,kl = −2i
+� dz′
+2π Gik(−z′)Glj(z′).
+(9)
+The projection of WR onto the A subspace then yields
+the static screened interaction WA. Since we aim at per-
+forming many-body simulations of the effective model,
+we need to partially unscreen the Coulomb parameters,
+eliminating from WA the screening channels arising from
+A-A transitions included in PR, which will be treated at
+a more sophisticated level of theory. This can be done
+according to the following prescription
+UA = WA
+�
+I + PAWA
+�−1,
+(10)
+using the polarization PA of the A electrons obtained
+from GA similarly to Eq. (9). The matrix elements in
+UA can therefore be regarded as the effective (partially
+screened) Coulomb parameters.
+C.
+Solutions of the low-energy models
+The Green’s function of Eq. (6), together with the in-
+teractions parameters of Eq. (10), define a low-energy
+model which can be solved with many-body techniques.
+Here, we propose two somewhat complementary strate-
+gies, i.e., exact diagonalization (ED) and the dynamical
+mean-field theory (DMFT) [42] as implemented within its
+real-space generalization (R-DMFT) for inhomogeneous
+systems [43–47].
+1.
+Exact diagonalization
+The ED technique requires a Hamiltonian formulation
+of the effective model.
+If the states of the active and
+embedding subspaces are energetically well-separated, it
+is possible to neglect the dynamical character of the hy-
+bridization function and construct an effective Hamilto-
+nian as
+Heff
+A = HA + ∆A(z = 0).
+(11)
+
+4
+Including the screened Coulomb interaction, the model
+Hamiltonian then reads
+H =
+�
+ij,σ
+�
+Heff
+A − Hdc
+A
+�
+ijc†
+iσcjσ
++ 1
+2
+�
+ijkl,σσ′
+�
+UA
+�
+ij,klc†
+jσc†
+kσ′clσ′ciσ,
+(12)
+where c(†)
+iσ denote the annihilation (creation) operator of
+an electron at LO i with spin σ, and the double-counting
+correction Hdc
+A accounts for the interaction already in-
+cluded at the mean-field level by DFT (see Sec. III D).
+The diagonalization of this Hamiltonian yields the many-
+body spectrum (eigenstates and eigenvalues) which can
+be used to construct the Green’s function GED
+A
+through
+its Lehmann representation [48]. The many-body self-
+energy is obtained from the Dyson equation
+ΣED
+A (z) = z − Heff
+A −
+�
+GED
+A (z)
+�−1,
+(13)
+and it describes both local Σii and non-local Σi̸=j elec-
+tronic correlations in the LO basis. An obvious advan-
+tage of ED over, e.g., quantum Monte Carlo [49], is that it
+provides direct access to retarded self-energy and Green’s
+function, and hence the electron transmission function,
+without the need to perform an analytic continuation
+numerically, which is an intrinsically ill-defined prob-
+lem [50]. Note that within ED, we obtain a many-body
+self-energy which is, by construction, spin-independent,
+i.e., Σσ
+ij = Σ¯σ
+ij since Heff
+A follows from a restricted DFT
+calculation.
+2.
+Real-space DMFT
+The idea behind R-DMFT consists of mapping a many-
+body problem onto a set of auxiliary Anderson impurity
+models (AIMs) —one for each atom α— described by the
+projected Green’s function [44–46]
+gσ
+α(z) = (Gσ
+A(z))α .
+(14)
+The solution of AIM α (see details below) yields a local
+many-body self-energy Σσ
+α(z), so that the self-energy of
+the A subspace is block diagonal in the atomic subspaces
+Σσ
+A(z) = diag(
+�
+Σσ
+α(z) | α ∈ A
+�
+).
+(15)
+The set of auxiliary AIMs are coupled by the Dyson equa-
+tion
+Gσ
+A(z) =
+�
+z+µ−(HA−Hdc
+A )−∆A(z)−Σσ
+A(z)
+�−1, (16)
+where the Green’s function Gσ
+A includes the many-body
+self-energy and the double-counting correction, and the
+chemical potential µ is determined to preserve the DFT
+occupation of the A subspace. Finally, Eqs. (14-16) are
+iterated self-consistently starting with an initial guess
+(typically Σσ
+A = 0) until convergence.
+More in detail, in AIM α the impurity electrons inter-
+act through a screened local Coulomb repulsion projected
+onto atom α, i.e., Uα = (UA)ij,kl | i, j, k, l ∈ α [51].
+Moreover, the impurity is embedded in a self-consistent
+bath of non-interacting electrons, which describes the rest
+of the electronic system, encoded in the hybridization
+function
+∆σ
+α(z) = z +µ−(Hα −Hdc
+α )−
+�
+gσ
+α(z)
+�−1 −Σσ
+α(z). (17)
+Also within R-DMFT, it is convenient to use ED to
+solve the AIMs to have direct access to retarded func-
+tions. This requires to discretize the hybridization func-
+tion with a finite number of bath orbitals, described by
+orbital energies ϵσ
+m and hopping parameters to the impu-
+rity tσ
+mi. The hybridization parameters together with the
+local Coulomb blocks Uα, define the AIM Hamiltonian
+HAIM =
+�
+ij,σ
+�
+Hα − Hdc
+α
+�
+ijc†
+iσcjσ − µ
+�
+iσ
+c†
+iσciσ
++
+�
+m,σ
+ϵσ
+ma†
+mσamσ +
+�
+mi,σ
+tσ
+mi(a†
+mσciσ + c†
+iσamσ)
++ 1
+2
+�
+ijkl,σσ′
+�
+Uα
+�
+ij,klc†
+jσc†
+kσ′clσ′ciσ,
+(18)
+where c(†)
+iσ and a(†)
+mσ denote the annihilation (creation)
+operator of an electron at LO i with spin σ, or at bath
+orbital m with spin σ, respectively. Once the many-body
+spectrum of the AIM is known, the local self-energy is
+evaluated in terms of the local Green’s function Gσ
+α as
+Σσ
+α(z) =
+�
+gσ
+α(z)
+�−1 −
+�
+Gσ
+α(z)
+�−1.
+(19)
+At convergence, we define the R-DMFT self-energy as
+Σσ,R−DMFT
+A
+(z) = Σσ
+A(z) − Hdc
+A − µ,
+(20)
+so that it contains all shifts related to the density matrix.
+In terms of approximations, R-DMFT takes into ac-
+count local electronic correlations (Σii), neglecting non-
+local correlations (i.e., Σij = 0), but some degree of
+non-locality is retained as Σii ̸= Σjj, and the AIMs
+are coupled through the self-consistent Dyson equation.
+Therefore, R-DMFT is suitable to treat intrinsically in-
+homogeneous systems [26, 46, 47, 52–54].
+Moreover,
+R-DMFT is considerably lighter in terms of computa-
+tional complexity with respect to the direct ED of the
+original many-body problem and can treat systems with
+hundreds of atoms in the active space, inaccessible to
+ED [26, 44, 46]. Finally, besides the restricted solution
+Σσ
+A = Σ¯σ
+A, within R-DMFT we also have the freedom
+of breaking the spin degeneracy, and describe magnetic
+solutions [28, 30, 31, 44, 55].
+D.
+Double-counting correction
+The double-counting (DC) correction Hdc
+A
+aims at
+eliminating the correlations in the A subspace included
+
+5
+at a mean-field level by DFT, which are instead to be
+included in a more sophisticated level of theory within
+the many-body simulations. Unfortunately, an analyt-
+ical expression of the correlation effects accounted for
+within DFT is unknown, and therefore several approxi-
+mations [47, 56–58] have been developed in the context of
+DFT+DMFT [59, 60] or DFT+U [61, 62]. For a single-
+orbital AIM (as in the case of the simulations in this
+work) the DC correction can be reasonably approximated
+within the fully localized limit (FFL) [57, 63–65]
+�
+Hdc
+A
+�
+ii = (UA)ii,ii
+�
+nDFT
+i
+− 1
+2
+�
+,
+(21)
+where nDFT
+i
+is the DFT occupation of orbital i. Hence,
+we use this form of DC for the R-DMFT calculations.
+However, there’s no established method for the general
+case of multi-site and multi-orbital Coulomb interaction
+as is the case for ED. Here, we propose a self-consistent
+procedure in which a set of local parameters is optimized
+to fulfill the condition
+(ΣA)ii(|z| → ∞) = 0,
+(22)
+This approach ensures that the electronic properties at
+high-energies, which are well described by a one-particle
+approach, are restored to the DFT level.
+E.
+Correlated quantum transport
+To describe the electronic transport properties, we use
+the non-equilibrium Green’s function (NEGF) approach
+[66, 67].
+In NEGF, we identify a device region sur-
+rounding the nanojunction’s constriction and downfold
+the leads’ electrons by virtue of an efficient recursive al-
+gorithm [68]. The corresponding Green’s function reads
+GD(z) =
+�
+zSD−HD−ΣL(z)−ΣR(z)−ΣD(z)
+�−1, (23)
+where ΣL(R) is the self-energy describing the electrons in
+the left (right) electrodes, and
+ΣD(z) = SDAS−1
+A ΣA(z)S−1
+A SAD
+(24)
+projects the many-body self-energy of the active space
+ΣA (i.e., obtained within either ED or R-DMFT) onto
+the device region.
+Following the generalization of the
+Landauer formula proposed by Meir and Wingreen [69],
+the conductance is given by
+G = G0T(EF ),
+(25)
+where G0 = e2/h is the conductance quantum, and the
+transmission function is computed as
+T(E) = Tr[GD(z)ΓL(z)G†
+D(z)ΓR(z)],
+(26)
+with ΓL(R) the anti-hermitian part of ΣL(R)
+ΓL(R) = i
+�
+ΣL(R) − Σ†
+L(R)
+�
+.
+(27)
+While Eqs. (25)−(27) neglect the incoherent contribu-
+tions (i.e., due to inelastic scattering) to the transmis-
+sion that arises from the many-body self-energy [35, 70–
+74], they provide a good approximation of the low-bias
+transport properties, even in the presence of strong cor-
+relations within the A subspace [34, 69].
+active
+molecule
+screening
+scattering region
+(a)
+(b)
+tip layer
+slab
+pz LOs
+FIG. 1.
+(a) Schematics of the scattering region of the single-
+molecule junction, consisting of the molecular bridge and the
+Au electrodes. The screening region (R) and the active space
+within the molecule (A) are highlighted. (b) Detailed struc-
+ture of pentadienyl and benzyl radical, and Au electrodes. For
+pentadienyl, we also show schematically the mapping onto the
+C and N pz LOs.
+IV.
+COMPUTATIONAL DETAILS
+The structures were set up with the atomic simula-
+tion environment (ASE) software package [75] and the
+DFT calculations were performed with the GPAW pack-
+age [76–78]. We performed a geometry optimization, and
+the atomic positions were relaxed until the forces on each
+atom were below 0.001 Hartree/Bohr−1 (≈ 0.05 eV/˚A).
+For converging the electron density, we used an LCAO
+double-ζ basis set, with a grid spacing of 0.2 ˚A, and
+the Perdew–Burke–Ernzerhof exchange-correlation func-
+tional [79]. For the electron transport calculations, we
+followed the method described in [68]. The leads were
+modeled by a three-layer-thick Au(111) slab sampled
+with a 3×1×1 k-point grid along the transport direction.
+The scattering region also includes one Au slab and an
+additional Au layer terminated by a four-atom Au tip,
+to which the molecule anchoring groups are attached.
+For all structures, the A subspace describing the effec-
+tive model is composed of the pz LOs of the C and N
+atoms of the molecular bridge, while the R subspace for
+the cRPA calculation of the screened interaction includes
+the molecule and also extends to the Au atoms of the tip
+(see Fig. 1).
+
+6
+V.
+INIGHTS FROM AB-INITIO SIMULATIONS
+In order to understand the many-body effects arising
+in the open-shell configuration, it is useful to recall some
+chemical and electronic properties of the pentadienyl and
+benzyl radicals, and how those are reflected by ab-initio
+simulations.
+In particular, we look at the spatial dis-
+tribution of the SOMO and at the ab-initio Coulomb
+parameters projected onto the LOs of the active space.
+A.
+Structure of the SOMO
+The pentadienyl radical (C5H7) is a linear molecule,
+and the shortest polyene radical after allyl. It has three
+resonant structures. In each structure, the unpaired elec-
+tron is hosted on one of the odd C atoms.
+The delo-
+calization of the unpaired electron along the molecular
+backbone contributes to the thermodynamical stability
+of the molecule [80, 81].
+The structure we consider is
+obtained by substituting a hydrogen atom at each end
+of the chain by an amino group. By diagonalization of
+the AOs Hamiltonian in the subspace of the molecule,
+we find an eigenvalue just above the Fermi energy, corre-
+sponding to a partially occupied MO (i.e., the SOMO).
+The pentadienyl resonant structures and the projection
+of the SOMO onto the pz LOs of the active space are
+shown in Figs. 2(a,b), respectively. The SOMO reflects
+the resonant structures, with the largest projection on
+the odd- and nodes at even- C atoms. It also displays a
+significant projection onto the anchoring groups, suggest-
+ing a strong coupling to the electrodes in the junction.
+The benzene molecule (C6H6) is a cyclic aromatic hy-
+brocarbon and the archetypical building block for molec-
+ular electronics. For our analysis, we consider a related
+compound, the benzyl radical (C6H5CH2−), which is ob-
+tained by substituting a hydrogen atom with a methylene
+(CH2) group. The benzyl radical is also stabilized by res-
+onance but, unlike pentadyenil, in both resonant struc-
+tures the unpaired electron is hosted on the benzylic C,
+as illustrated in Fig. 2(c). We focus on the meta con-
+figuration, in which the amino groups are substituted at
+the 1,3-positions of the aromatic ring, while the methy-
+lene group is substituted in the 5-position, i.e., along the
+longer branch of the ring (see also Fig. 1). As expected,
+we find an eigenvalue lying at the Fermi energy, corre-
+sponding to the SOMO shown in Fig. 2(d). The SOMO
+displays the largest projection at the pz LO of the ben-
+zylic C atom and displays nodes at every other C (simi-
+larly to pentadienyl). However, it does not extend to the
+anchoring groups, thus suggesting a weak coupling to the
+electrodes.
+B.
+Coulomb parameters in the LO basis
+The partially screened Coulomb matrix projected onto
+the LO basis of the active space Uij = (UA)ij is shown in
+(c)
+(a)
+SOMO (pz LOs)
+SOMO (pz LOs)
+(b)
+(d)
+FIG. 2. Resonances and SOMO isosurface (from LOs pz) of
+pentadienyl (a,b) and benzyl (c,d) radicals. In pentadienyl,
+the unpaired electron is hosted by one of the odd C of the
+polyene chain, which also display the largest contributions in
+the isosurface, while the even C correspond to nodes. In both
+benzyl resonant structures, the unpaired electron is hosted
+by the benzylic C, and the isosurface displays nodes on every
+other C, similarly as in pentadienyl. Isovalues: ±0.03 au.
+1
+2
+3
+4
+5
+(b)
+N C C C C C C C N
+N
+C
+C
+C
+C
+C
+C
+C
+N
+(a)
+N
+N
+C
+C
+C
+C
+C
+C C C C C
+N
+N
+FIG. 3. Partially screened Coulomb parameters Uij = (UA)ij
+in the LO basis for the pentadienyl (a) and the benzyl (b)
+radicals.
+Figs. 3(a,b) for the pentadienyl and the benzyl radicals,
+respectively. In both cases, the intra-orbital couplings Uii
+are in the range of 4–5 eV and are slightly stronger for
+the atoms farther away from the metallic Au electrons,
+due to the weaker screening effects. Similar values of the
+Coulomb repulsion are found for the anchoring groups.
+However, as we shall see later, while the Cpz LOs are
+close to half-filling the Npz LOs are almost full, resulting
+in weak correlation effects.
+VI.
+ELECTRON TRANSPORT
+We start our analysis by looking at the electron trans-
+port properties of the pentadienyl and benzyl junctions.
+In particular, we compare the predictions of DFT and
+many-body simulations, where the Coulomb repulsion is
+treated at different levels of approximation.
+
+7CH2
+CH2
+CH
+H7CH2
+CH2
+CH
+H7CH2
+CH2
+CH
+H7CH2
+CH2
+CH
+H7CH2
+CH2
+CH
+H7
+A.
+Pentadienyl
+Within DFT, the transmission function displays a res-
+onance close to the Fermi energy (denoted by EF ) corre-
+sponding to ballistic transport through the SOMO. The
+resonance is found at ϵSOMO = 70 meV and has a width
+ΓSOMO ≈ 300 meV, reflecting a significant hybridization
+of the SOMO with the states of the electrodes.
+The
+slight misalignment between the SOMO resonance and
+EF , yield a conductance G = 5.7 × 10−1 G0 in each
+spin channel, see Fig. 4(a), This scenario changes as the
+SOMO resonance is split due to the Coulomb repulsion.
+However, depending on the splitting mechanism, we ob-
+serve fundamentally different transport properties.
+Within spin-unrestricted R-DMFT calculations, the
+spin rotational symmetry is broken.
+The doublet de-
+generacy is lifted as the SOMO is split into an occu-
+pied state in the majority-spin channel (e.g., ↓-SOMO)
+and an unoccupied state in the minority-spin channel (↑-
+SUMO). This approximation yields a magnetic insulator
+with a spin gap ∆s ≈ 1.3 eV and a magnetic moment
+⟨Sz⟩ ≃ 1/2 due to the single unpaired electron.
+The
+spin-dependent splitting of a transmission feature, e.g.,
+a resonance [16, 17, 82] or an antiresonance [30, 31], has
+been suggested as a suitable mechanism for the realiza-
+tion of organic spin filters. For pentadienyl, the splitting
+is approximately symmetric around the Fermi level, thus
+yielding a similar conductance in the two spin channels
+G↑ = 1.9 × 10−2 G0 and G↓ = 1.5 × 10−2 G0 and low
+spin-filtering efficiency. The spin-unrestricted R-DMFT
+transmission functions are shown in Fig. 4(a) .
+Another possible mechanism to split the SOMO is ob-
+tained without lifting the spin degeneracy (i.e., within
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+R-DMFT
+R-DMFT
+10-6
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+R-DMFT
+ED
+DFT
+(a)
+(b)
+node
+FIG. 4. Electron transmission function through the pentadi-
+enyl radical junction. DFT predicts a SOMO resonance close
+to EF . Taking into account the Coulomb repulsion beyond
+restricted DFT yields: (a) a splitting of the resonance into
+↓-SOMO and ↑-SUMO due to spin-symmetry breaking; (b) a
+splitting of the resonance without symmetry breaking and a
+transmission node due to many-body effects.
+either R-DMFT or ED). In this case, we find that the
+SOMO transmission resonance is split, revealing an un-
+derlying transmission node, see Fig. 4(b). Hence, many-
+body calculations predict a strong suppression of the con-
+ductance, by several order of magnitude, in stark contrast
+with the single-particle picture, in which electron trans-
+port is dominated by a nearly-resonant ballistic channel.
+Note that the splitting is substantially larger in ED than
+in R-DMFT, and considering that the antiresonance is
+not aligned with EF , it also results in a much stronger
+suppression of the conductance G = 8.1 × 10−4 G0 (ED)
+versus G = 4.9×10−1 G0 (R-DMFT). This suggests that
+non-local effects play an important role, as it can be ex-
+pected in low-dimensional systems [27, 32].
+Since a linear π-conjugated molecule does not display
+any topological node, the pentadienyl node has been sug-
+gested to arise from destructive interference between dif-
+ferent charged states of the molecule [14]. In Sec. VII, we
+discuss in detail the microscopic mechanism responsible
+for the splitting of the SOMO and for the transmission
+node, and show that they are intertwined.
+B.
+Benzyl
+In the case of benzene single-molecule junctions, there
+is more than one possible configuration for the ring to
+bridge the electrodes, depending on the position of the
+amino anchoring groups. We focus on the meta configu-
+ration (i.e., amino groups substituted at the 1,3-positions
+of the aromatic ring) which is particularly relevant in the
+context of molecular electronics.
+Within DFT, the transmission function displays two
+striking features which can be readily identified in
+Figs. 5(a,b): a narrow asymmetric Fano resonance at
+ϵFano < 10 meV, close to EF , and a wide antiresonance
+at ϵDQI ≈ −0.8 eV. Both features originate from quan-
+tum interference (QI) effects. Clarifying the nature of the
+resonances and highlighting their differences, will prove
+helpful in understanding how electronic correlations af-
+fect the transport properties and to shed light on the
+underlying microscopic mechanism.
+The Fano resonance has a characteristic asymmetric
+line shape and arises from the QI between the SOMO,
+which is mostly localized at the benzylic C atom, and
+the delocalized MOs on the molecular backbone, which
+have a strong overlap with the states of the metallic Au
+electrodes [83–85]. The antiresonance is the hallmark of
+destructive QI in the meta configuration and it is well-
+established in the literature, from both the experimen-
+tal [86–88] and theoretical [89–93] points of view. It arises
+from the interference between the HOMO and LUMO of
+the ring itself [93]. There is a subtle interplay between the
+antiresonance and the functional groups (not necessarily
+radical). It is well-established that substituents and ad-
+sorbates affect the relative position of destructive inter-
+ference features with respect to the Fermi energy. The
+chemical control of the antiresonance can be exploited
+
+8
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+R-DMFT
+ED
+DFT
+R-DMFT
+R-DMFT
+(a)
+(b)
+Fano
+DQI
+FIG. 5. Electron transmission function through the benzyl radical junction, displaying the Fano and antiresonance originating
+by quantum interference effects. (a) Breaking the spin symmetry results in the spin-splitting of both the Fano and the DQI
+features. (b) Including many-body effects beyond DFT, the Fano resonance is split (without symmetry-breaking) while the
+DQI antiresonance is shifted to lower energies.
+for a wide range of applications ranging from nanoelec-
+tronics [94] to chemical sensing [95, 96] In principle, the
+position of the antiresonance is also influenced by the
+substitution position in the ring (see, e.g., [94] and refer-
+ences therein), but this effect is of marginal relevance to
+the scope of the present work.
+The Fano resonance is indeed the transport signa-
+ture of the SOMO. However, in contrast to pentedienyl,
+where the SOMO is delocalized along the molecular back-
+bone and dominates the electron transport, in benzyl,
+the SOMO is mostly localized on the methyl functional
+group. It is therefore interesting to investigate the effect
+of the Coulomb repulsion and highlight the differences
+between the two cases. Within restricted DFT simula-
+tions, the narrow Fano resonance is partially concealed by
+the wider QI antiresonance. Breaking the spin symmetry
+within spin-unrestricted R-DMFT yields a pair of spin-
+split Fano resonances, as shown in Fig. 5(a). In the ma-
+jority spin channel, ϵ↑
+Fano < 0 falls within the transmis-
+sion depletion caused by the antiresonance and the asym-
+metric Fano profile is clearly observable.
+Its counter-
+part in the minority spin channel is found above EF , i.e.,
+ϵ↓
+Fano > 0, and is still mostly concealed by the background
+transmission. Interestingly, the spin-symmetry breaking
+also induces spin-resolved QI antiresonances [30, 31, 97]
+but the splitting ϵ↓
+DQI − ϵ↑
+DQI is however weaker than in
+the Fano case, since the spin imbalance yields ⟨Sz⟩ ≃ 1/2
+on the pz LO of the benzylic C, and a weaker magneti-
+zation in the rest of the molecule.
+Not allowing breaking the spin symmetry in the many-
+body simulations reveal another scenario, as shown in
+Fig. 5(b). The difference is twofold. We observe a split-
+ting of the Fano resonance in both R-DMFT and ED
+(with the ED splitting being significantly larger) but no
+splitting is detected for the QI antiresonance, which is
+rather shifted further away from EF . This suggests that
+the microscopic mechanism behind the splitting with and
+without spin-symmetry breaking are fundamentally dif-
+ferent, as it distinguishes between the two QI features.
+Moreover, in contrast to the case of pentadienyl, the split-
+ting of the SOMO in benzyl does not result in a strong
+suppression of the transmission within the SOMO-SUMO
+gap. The two observations above are deeply connected,
+and eventually, they can both be rationalized in terms of
+the spatial distribution of the SOMO.
+VII.
+MICROSCOPIC MECHANISM
+A.
+Splitting of the SOMO
+So far, we have seen that the Coulomb repulsion in-
+duces a splitting of the SOMO of the organic radicals.
+In order to gain a deeper understanding of the electronic
+mechanism behind the splitting, and how it affects the
+transport properties of the junction, it is useful to look
+at the retarded self-energy in the LO basis Σij = (ΣA)ij,
+corresponding to ΣED
+A
+and Σσ,R−DMFT
+A
+in Eqs. (13, 20),
+respectively. The many-body effects encoded in the self-
+energy can be rationalized by interpreting the real part
+as an energy-dependent level shift, and the imaginary
+part as an effective electron-electron scattering rate. We
+argue that the mechanism discussed in the following is a
+common feature of organic radicals. Therefore, we dis-
+cuss the pentadienyl and benzyl radicals in parallel and
+highlight the differences whenever necessary.
+In order to compare the different approximations, it
+is convenient to look at the trace of the self-energy ma-
+trix. Within spin-unrestricted R-DMFT, which is shown
+in Figs. 6(a,d), the real part of the self-energy is weakly
+energy-dependent around EF , and determines a shift of
+the SOMO resonance in opposite directions for the two
+spin polarizations. The imaginary part is negligible (not
+shown) resulting in highly coherent SOMO and SUMO
+electronic excitations below and above EF .
+Note that
+the ground state of spin-unrestricted R-DMFT is two-
+fold degenerate, and it is invariant under a flip of all
+spins: {σi} → {¯σi}. This picture is qualitatively anal-
+ogous to what one can expect also at the single-particle
+level, i.e., within DFT+U. Many-body effects are weak,
+and the dominant effect arises from the spin-symmetry
+breaking, as both radicals are magnetic insulators with a
+
+9
+-6
+-4
+-2
+0
+2
+-60
+-40
+-20
+0
+20
+-4
+-2
+0
+2
+R-DMFT
+ED
+R-DMFT
+R-DMFT
+-0.5
+0
+0.5
+1
+-30
+-20
+-10
+0
+10
+-80
+-60
+-40
+-20
+0
+20
+40
+-1
+-0.5
+0
+0.5
+-12
+-10
+-8
+-6
+R-DMFT
+ED
+pentadienyl
+benzyl
+(a)
+(d)
+(b)
+(e)
+(c)
+(f)
+R-DMFT
+R-DMFT
+FIG. 6. Trace of the retarded self-energy Tr[Σ(E)] in the LO
+basis for the pentadienyl (a,b,c) and benzyl (d,e,f) radicals
+(the real and imaginary parts are denoted by solid and dashed
+lines, respectively). Within spin-unrestricted R-DMFT (a,d)
+the self-energy displays a weakly energy-dependent real part,
+which is different in each spin sector, while the imaginary part
+is negligible (not shown). Within both R-DMFT (b,e) and
+ED (c,f) the self-energy is dominated by a single resonance at
+energy ϵr (denoted by a solid grey line).
+spin SOMO-SUMO gap.
+The scenario is completely different within restricted
+R-DMFT and ED, as shown in Figs. 6(b,c,e,f). There,
+the self-energy is dominated by a single resonance and its
+energy dependence can be well described within a one-
+pole approximation (OPA)
+ΣOPA(E) =
+a
+E − EF − ϵr + ıγ .
+(28)
+The OPA self-energy has a Lorentzian shape, where ϵr
+and γ denote the resonant energy and the width of
+the resonance, whereas a controls the amplitude of the
+curve. The imaginary part of the self-energy plays the
+role of a giant electron-electron scattering rate and sup-
+presses electronic excitations around ϵr ≃ ϵSOMO, while
+the real part redistributes the spectral weight towards
+higher energies. This many-body mechanism, akin to the
+Mott metal-to-insulator transition as described within
+DMFT [42], is at the origin of the splitting of the SOMO
+resonance.
+In organic radicals, the following hierarchy of emergent
+energy scales is realized: ΓSOMO ≪ ∆ <∼ Uscreened, where
+the typical energy scale associated with the screened
+Coulomb repulsion Uscreened significantly exceeds the nar-
+row width of the SOMO resonance (∼ 10–100 meV),
+and the HOMO-LUMO single-particle gap ∆ controlled
+by the C-C π-bonds (∼ eV). This sets the electrons in
+the SOMO deep within the strongly correlated regime.
+Such a general condition suggests this mechanism to be
+common to organic radicals with a single unpaired elec-
+tron.
+Multi-radical molecules [98] and networks [99],
+may display different electronic and transport properties
+due to effective interactions between the unpaired elec-
+trons [7, 18–20].
+B.
+Spatial structure of the electronic correlations
+While R-DMFT and ED seem to qualitatively describe
+the same many-body mechanism for the splitting of the
+SOMO, it is also interesting to look at the whole self-
+energy matrix.
+As discussed in Sec. III C, within ED
+all elements Σij ̸= 0, whereas within R-DMFT Σij ∝ δij.
+Remarkably, all elements of the self-energy (irrespectively
+of the approximation) are well described by the OPA with
+the same resonant energy ϵr, as shown in Figs. 7(a,e).
+The off-diagonal elements (when non-zero) can have ei-
+ther sign since it is not determined by causality. It is then
+easy to have a comprehensive look at the self-energy by
+plotting the matrix Σij(ϵr), as shown in Figs. 7(c,d,g,h).
+Indeed, looking at the ED self-energy matrix, clear pat-
+terns emerge. Along the diagonal, some elements Σii are
+significantly larger than the others (note the logarithmic
+scale), and this asymmetry is mirrored by the off-diagonal
+elements. Upon close inspection, we can associate them
+with the pz LOs with the largest SOMO projection, thus
+confirming that the strongest many-body effects corre-
+late with the spatial distribution of the SOMO. Within
+R-DMFT, we find an analogous pattern along the diag-
+onal, as indicated in the insets.
+Despite its approximations (local Coulomb interaction,
+local correlations), it seems that R-DMFT tells qualita-
+tively the same story as the full ED simulations. This
+advocates for a substantially local character of the mi-
+croscopic mechanism, that can describe both the splitting
+of the SOMO and its consequences on electron transport,
+whereas non-local effects renormalize the splitting.
+C.
+Implications for electron transport
+The many-body mechanism behind the splitting of the
+SOMO is common to both the pentadienyl and benzyl
+radicals. However, its consequences on electron transport
+are dramatically different. In order to understand why,
+it is necessary to combine the insights from DFT with
+the knowledge about the spatial and energy structure of
+the self-energy.
+In pentadienyl, the SOMO is delocalized throughout
+the molecular backbone, and its large projection on the
+
+10
+0
+1
+2
+3
+4
+5
+6
+1
+2
+3
+4
+5
+0
+6
+-20
+-10
+0
+10
+20
+0
+1
+-10
+-5
+0
+5
+10
+0
+1
+(g)
+(h)
+(c)
+(d)
+N C C C C C C C N
+N
+C
+C
+C
+C
+C
+C
+C
+N
+N C C C C C C C N
+N
+C
+C
+C
+C
+C
+C
+C
+N
+N C C C C C N
+ED
+N C C C C C N
+R-DMFT
+N
+C
+C
+C
+C
+C
+N
+C
+C
+C
+C
+C
+N
+N
+ED
+R-DMFT
+0
+10-1
+101
+-10-1
+-101
+0
+10-1
+101
+-10-1
+-101
+-30
+-20
+-10
+0
+10
+20
+30
+0
+1
+-60
+-40
+-20
+0
+20
+40
+60
+0
+1
+(a)
+(b)
+(e)
+(f)
+ED
+ED
+ED
+ED
+0
+1
+2
+3
+4
+5
+6
+1
+2
+3
+4
+5
+0
+6
+0
+1
+2
+3
+4
+5
+6
+7
+8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+FIG. 7. Component of the ED self-energy Σij(E) and its matrix representation at the resonant energy Im Σij(ϵr) in the LO
+basis for the pentadienyl (a,b,c,d) and benzyl (e,f,g,h) radicals. Each component of the self-energy (grey lines) is dominated by
+a single pole (a,b,e,f) at a resonant energy ϵr. Selected components (i, j) are highlighted (color lines) and are labeled according
+to their index in the matrix. The matrix structure of the self-energy reflects the spatial distribution of the SOMO, i.e., the
+largest local (Σii) and non-local (Σij̸=i) self-energy contributions are found for the LOs with the largest projections to the
+SOMO (denoted by arrows, see also Fig. 2). Within R-DMFT (d,h) the self-energy is diagonal in the LO indices Σij ∝ δij and
+displays the same pattern.
+pz LOs of the anchoring groups (see Fig.2(a)) ensures a
+substantial overlap with the states in the metallic elec-
+trodes. Hence, there is a transmission channel across the
+junction through the SOMO. The pole of the self-energy
+results in a zero of the corresponding Green’s function.
+The suppression of the Green’s function hinders electron
+transport at that energy and is at the origin of the trans-
+mission node [30, 31]. In contrast, in the benzyl radical,
+the SOMO has negligible projection on the amino groups
+(see Fig.2(d)) and transport is dominated by transmis-
+sion channels involving the frontier MOs. Therefore, the
+splitting of the Fano resonance weakly affects those chan-
+nels, and does not prevent the off-resonance transmission
+of electrons across the junction.
+The above picture can be essentially reproduced within
+the following tight-binding (TB) three-orbital model,
+which is schematically represented in Fig. 8(a). Let us
+consider three orbitals (ℓ, c, r) that can be interpreted as
+the amino groups, left (ℓ) and right (r), and the central
+molecule (c). The Hamiltonian in such a basis reads
+H =
+�
+�
+ϵℓ
+t
+t′
+t
+ϵc
+t
+t′
+t
+ϵr
+�
+� .
+(29)
+The hybridization to the electrodes is mediated by the
+external (ℓ, r) orbitals and, for the sake of this discussion,
+it is assumed to be energy-independent:
+ΓL =
+�
+�
+Γ 0 0
+0 0 0
+0 0 0
+�
+� , ΓR =
+�
+�
+0 0 0
+0 0 0
+0 0 Γ
+�
+� .
+(30)
+The Hamiltonian of the isolated system can be diagonal-
+ized to obtain the eigenvalues ϵHOMO, ϵSOMO, and ϵLUMO.
+In light of the results shown in Fig. 7, the Green’s func-
+tion of the device
+GD(z) =
+�
+z − H + ıΓL/2 + ıΓR/2 − ΣD(z)
+�−1
+(31)
+is dressed with an OPA self-energy
+ΣD(z) =
+�
+�
+0
+0
+0
+0 ΣOPA(z) 0
+0
+0
+0
+�
+�
+(32)
+which acts on the central part (see Fig. 7(a,e) for a con-
+nection with the ab-initio simulations) and has a pole at
+ϵSOMO. Within such a three-orbital model, the Landauer
+transmission in Eq. (26) simplifies to
+T(E) = Γ2|Gℓr(E)|2,
+(33)
+where Gℓr = (GD)ℓr is the upper-right element of the
+Green’s function, linking the orbitals connected to the
+
+11
+(c)
+10-8
+10-6
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+10-8
+10-6
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+-1
+0
+-2
+-1.5
+-1
+0
+1
+-0.5
+0
+0.5
+1
+-4
+0
+4
+-2
+-1.5
+-0.4
+0
+0.4
+-0.5
+0
+0.5
+1
+Fano
+splitting
+splitting
+node
+zero
+1
+(g)
+(d)
+(h)
+(e)
+(i)
+TB
+OPA
+TB
+OPA
+1
+(a)
+(b)
+(f)
+FIG. 8. Schematic representation of the three-orbital TB model with its parameter, and form of the OPA self-energy (a).
+Weight distribution and eigenvalues of the TB MOs for scenarios representative of the pentadienyl (b) and benzyl (f) radicals.
+The transmission function (c,g) obtained without (grey lines) and with (blue lines) the OPA self-energy captures all relevant
+features of the DFT and many-body simulations. The Green’s function Gℓr is shown for specific energy ranges, which are
+relevant to explaining the spectral features associated with the HOMOs (d,h) and the SOMOs (e,i), as discussed in the text.
+Model parameters [eV]: ϵ = 0.5, ϵc = 0.25, a = 0.25, Γ = 0.05, γ = 0.003, common to both scenarios, t = 0.5, t′ = 0 (b,c,d) and
+t = 0.1, t′ = 0.5 (e,f,g).
+electrodes, and describes the only transmission channel
+across the junction.
+For the sake of simplicity, one can take −ϵℓ = ϵr = ϵ,
+and ϵc ≪ ϵ, which together with a, Γ, and η are kept
+fixed, whereas we choose the parameters t and t′ to de-
+scribe two scenarios, which are representative of the pen-
+tadienyl and benzyl radicals. The results are shown in
+Fig. 8 and described in the following.
+The physics of the pentadienyl radical can be repro-
+duced by choosing t <∼ ϵ and t′ = 0. The correspond-
+ing TB MOs are fairly delocalized throughout the sys-
+tem, as shown in Fig. 8(b). Hence, electron transport
+happens through sequential hopping processes through
+the c orbital. The transmission function, Fig. 8(c), dis-
+plays a SOMO resonance which is split by including the
+OPA self-energy, revealing a transmission node within
+the SOMO-SUMO gap. The origin of the transmission
+node is ascribed to a zero of the Green’s function at the
+SOMO energy Gℓr(E ≃ ϵSOMO) [30, 31] as demonstrated
+in Fig. 8(e).
+Instead, with the choice of parameters t ≪ t′ <∼ ϵ, one
+can describe the physics of the benzyl radical, charac-
+terized by an orbital c, which is weakly coupled to the
+ℓ − r molecular backbone. The corresponding SOMO is
+fairly localized on the central orbital, see Fig. 8(f). The
+transmission function displays a Fano resonance which is
+split by the OPA self-energy see Fig. 8(g). In contrast to
+the previous case, Gℓr does not have a zero, and trans-
+port is dominated by a transmission channel that bridges
+the electrodes through the direct ℓ-r hopping t′. Finally,
+note that in both scenarios above, many-body effects are
+negligible for the HOMO and LUMO resonances (corre-
+sponding to states which are completely filled and empty,
+respectively) even when the “correlated” c orbital has a
+sizable hybridization with ℓ and r, cfr. Figs. 8(c,d,g,h).
+Hence, the three-orbital model can reproduce all fun-
+damental features of the radical junctions discussed in
+this work, and at the same time, provides a simple inter-
+pretation of the numerical simulations.
+D.
+Non-perturbative nature of the splitting
+Within ED and R-DMFT, the solution of the many-
+body problem (i.e., on the lattice or the auxiliary AIM)
+is numerically exact.
+This means that the Coulomb
+repulsion is taken into account in a non-perturbative
+way.
+It is interesting to compare these results to a
+perturbative approach, e.g., within the GW approxima-
+tion [100, 101], which has been extensively and success-
+fully applied to molecules [102–107]. However, the ques-
+tion arises to which extent many-body perturbation the-
+ory approaches are able to describe the physics of open-
+shell systems [108]. Within GW, the self-energy is com-
+puted to the lowest order in perturbation theory, as a
+convolution of the Green’s function and the screened in-
+
+12
+10-4
+10-2
+1
+-2
+-1
+0
+1
+2
+G0W0
+GW
+(a)
+FIG. 9. Electron transmission function through the pentadi-
+enyl radical junction. Both the G0W0 and the self-consistent
+GW approximations fail to predict the splitting of the SOMO,
+as described within ED and R-DMFT, cfr. Fig 4.
+teraction.
+We compute the GW self-energy correction
+projected onto the A region
+Σ(z) = GA(z)WA,
+(34)
+as described in [68], and we consider the case of the pen-
+tadienyl radical without loss of generality.
+In Fig. 9 we see that neither G0W0 nor the fully self-
+consistent GW approximation is able to induce a split-
+ting of the SOMO resonance, and the numerical simu-
+lations rather result in a shift of the corresponding res-
+onance above the Fermi energy. Hence, the many-body
+techniques we propose to investigate open-shell molecules
+are not only sufficient but also necessary for our goal,
+whereas less sophisticated approaches fall short in de-
+scribing the electronic and transport properties arising
+from the strong electronic correlations within the SOMO.
+VIII.
+CONCLUSIONS
+In this work, we have proposed a numerical method
+that that combines ab-initio with state-of-the-art many-
+body techniques and is able to address the complexity
+of a realistic chemical environment as well as electronic
+correlation effects beyond the single-particle picture.
+The deliverable of this project served to shed light on
+the mechanism governing the electronic and transport
+properties of quantum junctions with organic molecules
+in an open-shell configuration. By considering a linear
+and a cyclic radical molecule, we derive a general under-
+standing of the role of many-body effects in molecular
+radicals with a single unpaired electron, and we show
+that they have dramatic consequences on electron
+transport.
+We establish the microscopic mechanism
+behind the splitting of the SOMO resonance and unravel
+a clear link between the space-time structure of electron-
+electron correlations and the spatial distribution of the
+SOMO. We demonstrate this by proposing a minimal
+model, which is capable of grasping the microscopic
+mechanism and thus reproducing all relevant features of
+the transmission properties. Our work will pave the path
+toward a deeper and more comprehensive understanding
+of strongly correlated electron physics at the nanoscale.
+ACKNOWLEDGEMENTS
+We thank J. M. Tomczak for valuable discussions.
+This research is supported by the Austrian Science Fund
+(FWF) through project P 31631 (A.V., R.S.) and by the
+NCCR MARVEL funded by the Swiss National Science
+Foundation grant 51NF40-205602 (G.G., D.P., M.L).
+Computational support from the Swiss Supercomputing
+Center (CSCS) under project ID s1119 is gratefully ac-
+knowledged.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf,len=1653
+page_content='Strongly correlated physics in organic open-shell quantum systems  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Gandus,1, ∗ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Passerone,2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Stadler,3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Luisier,1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Valli3, 4, †  1Integrated Systems Laboratory, ETH Z¨urich, Gloriastrasse 35, 8092 Z¨urich, Switzerland  2Empa, Swiss Federal Laboratories for Materials Science and Technology,  ¨Uberlandstrasse 129, CH-8600, D¨ubendorf, Switzerland  3Institute for Theoretical Physics, Vienna University of Technology,  Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria  4Department of Theoretical Physics, Institute of Physics,  Budapest University of Technology and Economics, M¨uegyetem rkp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', H-1111 Budapest, Hungary  Strongly correlated physics arises due to electron-electron scattering within partially-filled or-  bitals, and in this perspective, organic molecules in open-shell configuration are good candidates to  exhibit many-body effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' With a focus on neutral organic radicals with a molecular orbital host-  ing a single unpaired electron (SOMO) we investigate many-body effects on electron transport in a  single-molecule junction setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Within a combination of density functional theory and many-body  techniques, we perform numerical simulations for an effective model for which all the parameters,  including the Coulomb tensor, are derived ab-initio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' We demonstrate that the SOMO resonance is  prone towards splitting, and identify a giant electronic scattering rate as the driving many-body  mechanism, akin to a Mott metal-to-insulator transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The nature of the splitting, and thus of  the resulting gap, as well as the spatial distribution of the SOMO and its coupling to the electrodes,  have dramatic effects on the transport properties of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' We argue that the phenomenon  and the underlying microscopic mechanism are general, and apply to a wide family of open-shell  molecular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  INTRODUCTION  Strongly correlated electronic physics arises in par-  tially occupied orbitals in the presence of competing en-  ergy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Due to the Coulomb repulsion, electrons  display a collective behavior, leading to the breakdown  of the single-particle picture and the emergence of com-  plex quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Electronic correlations are  also enhanced due to spatial confinement effects in low-  dimensional and nanoscopic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' While in solid-state  physics the concept of a “strongly-correlated metal” is  well-established, its analog for molecules is not obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In chemistry, the majority of stable organic molecules  have closed-shell electronic configurations, and electrons  are paired in delocalized molecular orbitals (MOs) that  are either completely filled or empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The energy differ-  ence between the frontier MOs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', the highest occupied  (HOMO) and the lowest unoccupied (LUMO) orbitals  defines the spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In particular, π-conjugated sys-  tems display a wide HOMO-LUMO gap (∆ ∼ eV) which  is controlled by the overlap of neighboring pz orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' A  molecular system in an open-shell configuration (radical)  is characterized by unpaired valence electrons residing  in non-bonding singly-occupied MOs (SOMOs) found at  intermediate energies between HOMO and LUMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Rad-  icals can form by breaking bonds or by adding/removing  electrons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', in photoinduced processes) and are inter-  mediate products of chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  While open-shell configurations are typically associ-  ated with high chemical reactivity, there exist also species  ∗ gandusgui@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='com  † valli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='angelo@ttk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='bme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='hu  of relatively stable radicals, which possess interesting  electronic, magnetic, and optical functionalities that are  relevant to technological applications ranging from next-  generation spintronics to quantum information [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Tremendous advances in the synthesis and character-  ization of organic radicals triggered recent experimen-  tal studies with organic species that are stable enough  to be trapped in break-junctions [4, 5] or investigated  with scanning tunneling spectroscopy [6–9], which fu-  eled a revival of interest in the molecular Kondo ef-  fect [4, 6–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  There is a growing experimental and  theoretical effort to unravel how many-body effects can  dramatically influence electronic and transport proper-  ties in light of technological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In the con-  text of molecular electronics, noteworthy organic radicals  include triphenylmethyl [4, 5, 12], Blatter radical [13],  polyacetylene [14, 15], benzyl [16, 17], together with the  whole family of polycyclic hydrocarbons with non-Kekul`e  structure [7, 18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Molecular organic frameworks with  transition-metal centers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', iron-porphyrin) are also  typically open-shell, and have been recently suggested  as molecular transistors [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  From the theoretical point of view, in wide-gap semi-  conductors, the electron-electron scattering rate is low  due to the lack of electronic states at the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The accuracy of ab-inito prediction of the gap is a long-  standing issue [23], and numerical simulations for insula-  tors [24, 25] and molecules [25–32] predict a many-body  renormalization of the spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' However, these ef-  fects do not change qualitatively the transport proper-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In open-shell configurations instead, it can be ex-  pected that electron-electron scattering within the par-  tially filled SOMO and many-body effects have a promi-  nent role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='00282v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='str-el]  31 Dec 2022  2  In computational quantum chemistry,  it is well-  established that open-shell molecular configurations re-  quire careful treatment (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', [33] for an overview)  but the accuracy of quantum chemical methods comes  at a high numerical cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Hence, we recently witnessed  significant advances in developing alternative simulation  schemes, that are suitable to describe complex devices  relevant to molecular electronics [11, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In the en-  deavor to achieve predictive power and allow for a quan-  titative comparison with experiments, a suitable method  should be high-throughput — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', scalable and automa-  tized as much as possible, and able to describe a real-  istic chemical environment and many-body correlations  within an ab-initio framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This would allow a coop-  erative effort between theory and experiments, and pave  the path to future breakthroughs for next-generation  quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  SCOPE OF THIS WORK  The scope of this work is to investigate the emergence  of strongly correlated electron physics in the electronic  and transport properties of single-molecule junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  To this end, we have developed a comprehensive nu-  merical workflow that combines density functional theory  (DFT) with quantum field theoretical methods, and it is  able to address the complexity of a realistic chemical en-  vironment as well as electronic correlation effects beyond  the single-particle picture within an ab-initio framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  With both aspects taken into account, we are able to un-  ravel the origin of many-body transport effects in single-  molecule junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The art of combining ab-initio and many-body com-  putational schemes lies in a transformation from non-  orthogonal atomic orbitals (AOs) to recently introduced  local orbitals (LOs) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The LOs are by construction  orthogonal within the same atom and localized in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  They take over the symmetries of the original AOs, while  inheriting the information of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This al-  lows to represent the electronic wavefunction in a region  of the spectrum close to the Fermi energy with a mini-  mal set of orbitals, making them an ideal basis for many-  body calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' So far, LOs have been employed in the  context of DFT [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In what follows, we also evaluate  the Coulomb integrals that describe the electron-electron  repulsion in the LO basis, and thus map to the origi-  nal Hamiltonian onto an effective many-body problem,  which we can feasibly solve with appropriate numerical  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This recipe is particularly suitable to address  strong correlation effects in the transport properties of  molecular junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In terms of applications, we focus on molecular break-  junctions in which the central molecule bridging the elec-  trodes is in an open-shell configuration, which are strong  candidates to manifest many-body effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Specifically,  we select a linear and a cyclic molecular bridge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', a  polyene radical, and a benzene molecule substituted with  a methylene (CH2) radical group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' While both molecules  are π-radicals with one electron in the SOMO, we show  that many-body effects bring out profound differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  We identify the fingerprint of strong electronic correla-  tions in the splitting of the SOMO resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The details  of the splitting and the spatial distribution of the SOMO  on the molecular backbone have dramatic consequences  on the transport properties of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Finally, we demonstrate that such a splitting cannot  be obtained with less sophisticated techniques, such as  many-body perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' We argue that this phe-  nomenon and the underlying microscopic mechanism are  general, and apply to a wide family of open-shell molec-  ular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  METHODS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Local orbitals and low-energy models  The LOs method [36] is a transformation-based ap-  proach that aims at retrieving hydrogen-like orbitals for  atoms in molecules and solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' By construction, LOs are  locally orthogonal on each atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The starting point is a  DFT calculation in an AOs basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The Hilbert space  H is then spanned by a finite set of non-orthogonal or-  bitals {|i⟩}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', with a overlap matrix ⟨i|j⟩ = (S)ij ̸= δij  for |i⟩ , |j⟩ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' A set of LOs {|m⟩} ∈ M ⊆ H can be  obtained for any atom α in subspace M by a subdiago-  nalization of the corresponding Hamiltonian sub-block  Hα |m⟩ = ϵmSα |m⟩  (1)  The LOs are then linear combinations of AOs and are  by definition orthogonal on each atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This allows for a  more natural physical interpretation of the LOs as atomic  orbitals [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In order to obtain an ab-initio effective  model, we formally separate the Hilbert space into an ac-  tive space (A) and an environment (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The active space  consists of a subset of LOs {|a⟩} = A ⊆ M which are ex-  pected to describe the relevant physics close to the Fermi  energy, and at the same time can be efficiently treated  within quantum many-body techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Insytead, the  environment consists of all the remaining LOs and AOs,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', {|e⟩} ∈ E ≡ H \\A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Embedding the active space into  the environment ensures that the effective model pre-  serves all information of the original single-particle DFT  Hamiltonian [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Finally, it is convenient to perform a  L¨owdin orthogonalization [37] of the LO {|a⟩} states and  redefine the A subspace in terms of this new orthonormal  basis set with elements  ��a⊥�  =  �  a  (S−1/2)aa⊥ |a⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (2)  Since the overlap between LOs on different atoms is typi-  cally low, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', (S)ij ≪ 1, the L¨owdin orthonormalization  of the active space results only in a weak deformation of  the original LOs, which preserves their atomic-like sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  3  In practice, the LO low-energy model is constructed  embedding the active subspace into the environment  through a downfolding procedure [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Taking into  account the non-orthogonality between the A and E sub-  spaces [34], we write the Green’s function projected onto  the A subspace as  GA(z) = S−1  A SAHGH(z)SHAS−1  A ,  (3)  where z = E + iη is a complex energy with an infinites-  imal shift η → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' GH denotes the Green’s function of  the full Hilbert space, and SAH the overlap matrix be-  tween orbitals  ��a⊥�  ∈ A and orbitals |i⟩ ∈ H, while the  overlap SA between the  ��a⊥�  states is, by construction,  the identity matrix and will be omitted in what follows  for notational simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The effect of the environment  on the A subspace is described by the hybridization func-  tion  ∆A(z) = g−1  A (z) − GA(z)−1,  (4)  where  gA =  �  z − HA  �−1  (5)  is Green’s function of the isolated A subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Rewriting  GA in terms of ∆A and using the definition of gA yields  GA(z) =  �  z − HA − ∆A(z)  �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (6)  Then, GA can be seen as the resolvent of an effective  A subspace renormalized by the environment through a  dynamical hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The Green’s function describes  the physics of the whole system, projected onto a sub-  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  For a single-particle Hamiltonian, the partition above  is arbitrary, and the procedure remains valid indepen-  dently of the subset of LOs included in the active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In the context of π-conjugated organic molecules, the  projection onto a single pz LO per C atom (and pos-  sibly other species such as N or S) is usually sufficient  to achieve a faithful representation of the frontier MOs,  and hence suitable to describe the physics close to the  Fermi energy [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The possibility of considering a re-  stricted subset of LOs in the effective model is of pivotal  importance in view of performing computationally-heavy  many-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  cRPA and ab-initio Coulomb parameters  In order to derive the electronic interaction parameters  in the A subspace beyond the semi-local density approx-  imations, we employ the constrained Random Phase Ap-  proximation (cRPA) [34, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Within the cRPA, we  select a region R ⊃ A where the formation of electron-  hole pairs is expected to screen the Coulomb interaction  between the A electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Because of the strong local na-  ture of the LOs, it is sufficient that R comprises the A  subspace and few atoms nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Defining GR to be the  Green’s function projected onto the R subspace in anal-  ogy with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (3), the screened Coulomb interaction at  the RPA level is given by  WR =  �  I − VRPR  �−1VR,  (7)  where VR is the bare Coulomb interaction  (VR)ij,kl =  �  dr  �  dr′ψi (r)ψ∗  j (r)  e2  |r − r′|ψ∗  k(r′)ψl (r′),  (8)  being ψi(r) the orbitals in the R region, and PR is the  static component of the polarizability  (PR)ij,kl = −2i  � dz′  2π Gik(−z′)Glj(z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (9)  The projection of WR onto the A subspace then yields  the static screened interaction WA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Since we aim at per-  forming many-body simulations of the effective model,  we need to partially unscreen the Coulomb parameters,  eliminating from WA the screening channels arising from  A-A transitions included in PR, which will be treated at  a more sophisticated level of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This can be done  according to the following prescription  UA = WA  �  I + PAWA  �−1,  (10)  using the polarization PA of the A electrons obtained  from GA similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The matrix elements in  UA can therefore be regarded as the effective (partially  screened) Coulomb parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Solutions of the low-energy models  The Green’s function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (6), together with the in-  teractions parameters of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (10), define a low-energy  model which can be solved with many-body techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Here, we propose two somewhat complementary strate-  gies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', exact diagonalization (ED) and the dynamical  mean-field theory (DMFT) [42] as implemented within its  real-space generalization (R-DMFT) for inhomogeneous  systems [43–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Exact diagonalization  The ED technique requires a Hamiltonian formulation  of the effective model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  If the states of the active and  embedding subspaces are energetically well-separated, it  is possible to neglect the dynamical character of the hy-  bridization function and construct an effective Hamilto-  nian as  Heff  A = HA + ∆A(z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (11)  4  Including the screened Coulomb interaction, the model  Hamiltonian then reads  H =  �  ij,σ  �  Heff  A − Hdc  A  �  ijc†  iσcjσ  + 1  2  �  ijkl,σσ′  �  UA  �  ij,klc†  jσc†  kσ′clσ′ciσ,  (12)  where c(†)  iσ denote the annihilation (creation) operator of  an electron at LO i with spin σ, and the double-counting  correction Hdc  A accounts for the interaction already in-  cluded at the mean-field level by DFT (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' III D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The diagonalization of this Hamiltonian yields the many-  body spectrum (eigenstates and eigenvalues) which can  be used to construct the Green’s function GED  A  through  its Lehmann representation [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The many-body self-  energy is obtained from the Dyson equation  ΣED  A (z) = z − Heff  A −  �  GED  A (z)  �−1,  (13)  and it describes both local Σii and non-local Σi̸=j elec-  tronic correlations in the LO basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' An obvious advan-  tage of ED over, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', quantum Monte Carlo [49], is that it  provides direct access to retarded self-energy and Green’s  function, and hence the electron transmission function,  without the need to perform an analytic continuation  numerically, which is an intrinsically ill-defined prob-  lem [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Note that within ED, we obtain a many-body  self-energy which is, by construction, spin-independent,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', Σσ  ij = Σ¯σ  ij since Heff  A follows from a restricted DFT  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Real-space DMFT  The idea behind R-DMFT consists of mapping a many-  body problem onto a set of auxiliary Anderson impurity  models (AIMs) —one for each atom α— described by the  projected Green’s function [44–46]  gσ  α(z) = (Gσ  A(z))α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (14)  The solution of AIM α (see details below) yields a local  many-body self-energy Σσ  α(z), so that the self-energy of  the A subspace is block diagonal in the atomic subspaces  Σσ  A(z) = diag(  �  Σσ  α(z) | α ∈ A  �  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (15)  The set of auxiliary AIMs are coupled by the Dyson equa-  tion  Gσ  A(z) =  �  z+µ−(HA−Hdc  A )−∆A(z)−Σσ  A(z)  �−1, (16)  where the Green’s function Gσ  A includes the many-body  self-energy and the double-counting correction, and the  chemical potential µ is determined to preserve the DFT  occupation of the A subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Finally, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (14-16) are  iterated self-consistently starting with an initial guess  (typically Σσ  A = 0) until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  More in detail, in AIM α the impurity electrons inter-  act through a screened local Coulomb repulsion projected  onto atom α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', Uα = (UA)ij,kl | i, j, k, l ∈ α [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Moreover, the impurity is embedded in a self-consistent  bath of non-interacting electrons, which describes the rest  of the electronic system, encoded in the hybridization  function  ∆σ  α(z) = z +µ−(Hα −Hdc  α )−  �  gσ  α(z)  �−1 −Σσ  α(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (17)  Also within R-DMFT, it is convenient to use ED to  solve the AIMs to have direct access to retarded func-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This requires to discretize the hybridization func-  tion with a finite number of bath orbitals, described by  orbital energies ϵσ  m and hopping parameters to the impu-  rity tσ  mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The hybridization parameters together with the  local Coulomb blocks Uα, define the AIM Hamiltonian  HAIM =  �  ij,σ  �  Hα − Hdc  α  �  ijc†  iσcjσ − µ  �  iσ  c†  iσciσ  +  �  m,σ  ϵσ  ma†  mσamσ +  �  mi,σ  tσ  mi(a†  mσciσ + c†  iσamσ)  + 1  2  �  ijkl,σσ′  �  Uα  �  ij,klc†  jσc†  kσ′clσ′ciσ,  (18)  where c(†)  iσ and a(†)  mσ denote the annihilation (creation)  operator of an electron at LO i with spin σ, or at bath  orbital m with spin σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Once the many-body  spectrum of the AIM is known, the local self-energy is  evaluated in terms of the local Green’s function Gσ  α as  Σσ  α(z) =  �  gσ  α(z)  �−1 −  �  Gσ  α(z)  �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (19)  At convergence, we define the R-DMFT self-energy as  Σσ,R−DMFT  A  (z) = Σσ  A(z) − Hdc  A − µ,  (20)  so that it contains all shifts related to the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In terms of approximations, R-DMFT takes into ac-  count local electronic correlations (Σii), neglecting non-  local correlations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', Σij = 0), but some degree of  non-locality is retained as Σii ̸= Σjj, and the AIMs  are coupled through the self-consistent Dyson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Therefore, R-DMFT is suitable to treat intrinsically in-  homogeneous systems [26, 46, 47, 52–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Moreover,  R-DMFT is considerably lighter in terms of computa-  tional complexity with respect to the direct ED of the  original many-body problem and can treat systems with  hundreds of atoms in the active space, inaccessible to  ED [26, 44, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Finally, besides the restricted solution  Σσ  A = Σ¯σ  A, within R-DMFT we also have the freedom  of breaking the spin degeneracy, and describe magnetic  solutions [28, 30, 31, 44, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Double-counting correction  The double-counting (DC) correction Hdc  A  aims at  eliminating the correlations in the A subspace included  5  at a mean-field level by DFT, which are instead to be  included in a more sophisticated level of theory within  the many-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Unfortunately, an analyt-  ical expression of the correlation effects accounted for  within DFT is unknown, and therefore several approxi-  mations [47, 56–58] have been developed in the context of  DFT+DMFT [59, 60] or DFT+U [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' For a single-  orbital AIM (as in the case of the simulations in this  work) the DC correction can be reasonably approximated  within the fully localized limit (FFL) [57, 63–65]  �  Hdc  A  �  ii = (UA)ii,ii  �  nDFT  i  − 1  2  �  ,  (21)  where nDFT  i  is the DFT occupation of orbital i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Hence,  we use this form of DC for the R-DMFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  However, there’s no established method for the general  case of multi-site and multi-orbital Coulomb interaction  as is the case for ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Here, we propose a self-consistent  procedure in which a set of local parameters is optimized  to fulfill the condition  (ΣA)ii(|z| → ∞) = 0,  (22)  This approach ensures that the electronic properties at  high-energies, which are well described by a one-particle  approach, are restored to the DFT level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Correlated quantum transport  To describe the electronic transport properties, we use  the non-equilibrium Green’s function (NEGF) approach  [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In NEGF, we identify a device region sur-  rounding the nanojunction’s constriction and downfold  the leads’ electrons by virtue of an efficient recursive al-  gorithm [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The corresponding Green’s function reads  GD(z) =  �  zSD−HD−ΣL(z)−ΣR(z)−ΣD(z)  �−1, (23)  where ΣL(R) is the self-energy describing the electrons in  the left (right) electrodes, and  ΣD(z) = SDAS−1  A ΣA(z)S−1  A SAD  (24)  projects the many-body self-energy of the active space  ΣA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', obtained within either ED or R-DMFT) onto  the device region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Following the generalization of the  Landauer formula proposed by Meir and Wingreen [69],  the conductance is given by  G = G0T(EF ),  (25)  where G0 = e2/h is the conductance quantum, and the  transmission function is computed as  T(E) = Tr[GD(z)ΓL(z)G†  D(z)ΓR(z)],  (26)  with ΓL(R) the anti-hermitian part of ΣL(R)  ΓL(R) = i  �  ΣL(R) − Σ†  L(R)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (27)  While Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (25)−(27) neglect the incoherent contribu-  tions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', due to inelastic scattering) to the transmis-  sion that arises from the many-body self-energy [35, 70–  74], they provide a good approximation of the low-bias  transport properties, even in the presence of strong cor-  relations within the A subspace [34, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  active  molecule  screening  scattering region  (a)  (b)  tip layer  slab  pz LOs  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (a) Schematics of the scattering region of the single-  molecule junction, consisting of the molecular bridge and the  Au electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The screening region (R) and the active space  within the molecule (A) are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (b) Detailed struc-  ture of pentadienyl and benzyl radical, and Au electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' For  pentadienyl, we also show schematically the mapping onto the  C and N pz LOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  COMPUTATIONAL DETAILS  The structures were set up with the atomic simula-  tion environment (ASE) software package [75] and the  DFT calculations were performed with the GPAW pack-  age [76–78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' We performed a geometry optimization, and  the atomic positions were relaxed until the forces on each  atom were below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='001 Hartree/Bohr−1 (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='05 eV/˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  For converging the electron density, we used an LCAO  double-ζ basis set, with a grid spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='2 ˚A, and  the Perdew–Burke–Ernzerhof exchange-correlation func-  tional [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' For the electron transport calculations, we  followed the method described in [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The leads were  modeled by a three-layer-thick Au(111) slab sampled  with a 3×1×1 k-point grid along the transport direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The scattering region also includes one Au slab and an  additional Au layer terminated by a four-atom Au tip,  to which the molecule anchoring groups are attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  For all structures, the A subspace describing the effec-  tive model is composed of the pz LOs of the C and N  atoms of the molecular bridge, while the R subspace for  the cRPA calculation of the screened interaction includes  the molecule and also extends to the Au atoms of the tip  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  6  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  INIGHTS FROM AB-INITIO SIMULATIONS  In order to understand the many-body effects arising  in the open-shell configuration, it is useful to recall some  chemical and electronic properties of the pentadienyl and  benzyl radicals, and how those are reflected by ab-initio  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In particular, we look at the spatial dis-  tribution of the SOMO and at the ab-initio Coulomb  parameters projected onto the LOs of the active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Structure of the SOMO  The pentadienyl radical (C5H7) is a linear molecule,  and the shortest polyene radical after allyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' It has three  resonant structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In each structure, the unpaired elec-  tron is hosted on one of the odd C atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The delo-  calization of the unpaired electron along the molecular  backbone contributes to the thermodynamical stability  of the molecule [80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The structure we consider is  obtained by substituting a hydrogen atom at each end  of the chain by an amino group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' By diagonalization of  the AOs Hamiltonian in the subspace of the molecule,  we find an eigenvalue just above the Fermi energy, corre-  sponding to a partially occupied MO (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', the SOMO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The pentadienyl resonant structures and the projection  of the SOMO onto the pz LOs of the active space are  shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 2(a,b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The SOMO reflects  the resonant structures, with the largest projection on  the odd- and nodes at even- C atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' It also displays a  significant projection onto the anchoring groups, suggest-  ing a strong coupling to the electrodes in the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The benzene molecule (C6H6) is a cyclic aromatic hy-  brocarbon and the archetypical building block for molec-  ular electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' For our analysis, we consider a related  compound, the benzyl radical (C6H5CH2−), which is ob-  tained by substituting a hydrogen atom with a methylene  (CH2) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The benzyl radical is also stabilized by res-  onance but, unlike pentadyenil, in both resonant struc-  tures the unpaired electron is hosted on the benzylic C,  as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' We focus on the meta con-  figuration, in which the amino groups are substituted at  the 1,3-positions of the aromatic ring, while the methy-  lene group is substituted in the 5-position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', along the  longer branch of the ring (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' As expected,  we find an eigenvalue lying at the Fermi energy, corre-  sponding to the SOMO shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The SOMO  displays the largest projection at the pz LO of the ben-  zylic C atom and displays nodes at every other C (simi-  larly to pentadienyl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' However, it does not extend to the  anchoring groups, thus suggesting a weak coupling to the  electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Coulomb parameters in the LO basis  The partially screened Coulomb matrix projected onto  the LO basis of the active space Uij = (UA)ij is shown in  (c)  (a)  SOMO (pz LOs)  SOMO (pz LOs)  (b)  (d)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Resonances and SOMO isosurface (from LOs pz) of  pentadienyl (a,b) and benzyl (c,d) radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In pentadienyl,  the unpaired electron is hosted by one of the odd C of the  polyene chain, which also display the largest contributions in  the isosurface, while the even C correspond to nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In both  benzyl resonant structures, the unpaired electron is hosted  by the benzylic C, and the isosurface displays nodes on every  other C, similarly as in pentadienyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Isovalues: ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='03 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  1  2  3  4  5  (b)  N C C C C C C C N  N  C  C  C  C  C  C  C  N  (a)  N  N  C  C  C  C  C  C C C C C  N  N  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Partially screened Coulomb parameters Uij = (UA)ij  in the LO basis for the pentadienyl (a) and the benzyl (b)  radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 3(a,b) for the pentadienyl and the benzyl radicals,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In both cases, the intra-orbital couplings Uii  are in the range of 4–5 eV and are slightly stronger for  the atoms farther away from the metallic Au electrons,  due to the weaker screening effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Similar values of the  Coulomb repulsion are found for the anchoring groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  However, as we shall see later, while the Cpz LOs are  close to half-filling the Npz LOs are almost full, resulting  in weak correlation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  ELECTRON TRANSPORT  We start our analysis by looking at the electron trans-  port properties of the pentadienyl and benzyl junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In particular, we compare the predictions of DFT and  many-body simulations, where the Coulomb repulsion is  treated at different levels of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  7CH2  CH2  CH  H7CH2  CH2  CH  H7CH2  CH2  CH  H7CH2  CH2  CH  H7CH2  CH2  CH  H7  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Pentadienyl  Within DFT, the transmission function displays a res-  onance close to the Fermi energy (denoted by EF ) corre-  sponding to ballistic transport through the SOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The  resonance is found at ϵSOMO = 70 meV and has a width  ΓSOMO ≈ 300 meV, reflecting a significant hybridization  of the SOMO with the states of the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The  slight misalignment between the SOMO resonance and  EF , yield a conductance G = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='7 × 10−1 G0 in each  spin channel, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 4(a), This scenario changes as the  SOMO resonance is split due to the Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  However, depending on the splitting mechanism, we ob-  serve fundamentally different transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Within spin-unrestricted R-DMFT calculations, the  spin rotational symmetry is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The doublet de-  generacy is lifted as the SOMO is split into an occu-  pied state in the majority-spin channel (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', ↓-SOMO)  and an unoccupied state in the minority-spin channel (↑-  SUMO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This approximation yields a magnetic insulator  with a spin gap ∆s ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='3 eV and a magnetic moment  ⟨Sz⟩ ≃ 1/2 due to the single unpaired electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The  spin-dependent splitting of a transmission feature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=',  a resonance [16, 17, 82] or an antiresonance [30, 31], has  been suggested as a suitable mechanism for the realiza-  tion of organic spin filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' For pentadienyl, the splitting  is approximately symmetric around the Fermi level, thus  yielding a similar conductance in the two spin channels  G↑ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='9 × 10−2 G0 and G↓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5 × 10−2 G0 and low  spin-filtering efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The spin-unrestricted R-DMFT  transmission functions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 4(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Another possible mechanism to split the SOMO is ob-  tained without lifting the spin degeneracy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', within  10-4  10-2  1  2  1  0  1  2  R-DMFT  R-DMFT  10-6  10-4  10-2  1  2  1  0  1  2  R-DMFT  ED  DFT  (a)  (b)  node  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Electron transmission function through the pentadi-  enyl radical junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' DFT predicts a SOMO resonance close  to EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Taking into account the Coulomb repulsion beyond  restricted DFT yields: (a) a splitting of the resonance into  ↓-SOMO and ↑-SUMO due to spin-symmetry breaking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (b) a  splitting of the resonance without symmetry breaking and a  transmission node due to many-body effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  either R-DMFT or ED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In this case, we find that the  SOMO transmission resonance is split, revealing an un-  derlying transmission node, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Hence, many-  body calculations predict a strong suppression of the con-  ductance, by several order of magnitude, in stark contrast  with the single-particle picture, in which electron trans-  port is dominated by a nearly-resonant ballistic channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Note that the splitting is substantially larger in ED than  in R-DMFT, and considering that the antiresonance is  not aligned with EF , it also results in a much stronger  suppression of the conductance G = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='1 × 10−4 G0 (ED)  versus G = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='9×10−1 G0 (R-DMFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This suggests that  non-local effects play an important role, as it can be ex-  pected in low-dimensional systems [27, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Since a linear π-conjugated molecule does not display  any topological node, the pentadienyl node has been sug-  gested to arise from destructive interference between dif-  ferent charged states of the molecule [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' VII, we  discuss in detail the microscopic mechanism responsible  for the splitting of the SOMO and for the transmission  node, and show that they are intertwined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Benzyl  In the case of benzene single-molecule junctions, there  is more than one possible configuration for the ring to  bridge the electrodes, depending on the position of the  amino anchoring groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' We focus on the meta configu-  ration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', amino groups substituted at the 1,3-positions  of the aromatic ring) which is particularly relevant in the  context of molecular electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Within DFT, the transmission function displays two  striking features which can be readily identified in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 5(a,b): a narrow asymmetric Fano resonance at  ϵFano < 10 meV, close to EF , and a wide antiresonance  at ϵDQI ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Both features originate from quan-  tum interference (QI) effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Clarifying the nature of the  resonances and highlighting their differences, will prove  helpful in understanding how electronic correlations af-  fect the transport properties and to shed light on the  underlying microscopic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The Fano resonance has a characteristic asymmetric  line shape and arises from the QI between the SOMO,  which is mostly localized at the benzylic C atom, and  the delocalized MOs on the molecular backbone, which  have a strong overlap with the states of the metallic Au  electrodes [83–85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The antiresonance is the hallmark of  destructive QI in the meta configuration and it is well-  established in the literature, from both the experimen-  tal [86–88] and theoretical [89–93] points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' It arises  from the interference between the HOMO and LUMO of  the ring itself [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' There is a subtle interplay between the  antiresonance and the functional groups (not necessarily  radical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' It is well-established that substituents and ad-  sorbates affect the relative position of destructive inter-  ference features with respect to the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The  chemical control of the antiresonance can be exploited  8  10-4  10-2  1  2  1  0  1  2  10-4  10-2  1  2  1  0  1  2  R-DMFT  ED  DFT  R-DMFT  R-DMFT  (a)  (b)  Fano  DQI  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Electron transmission function through the benzyl radical junction, displaying the Fano and antiresonance originating  by quantum interference effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (a) Breaking the spin symmetry results in the spin-splitting of both the Fano and the DQI  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (b) Including many-body effects beyond DFT, the Fano resonance is split (without symmetry-breaking) while the  DQI antiresonance is shifted to lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  for a wide range of applications ranging from nanoelec-  tronics [94] to chemical sensing [95, 96] In principle, the  position of the antiresonance is also influenced by the  substitution position in the ring (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', [94] and refer-  ences therein), but this effect is of marginal relevance to  the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The Fano resonance is indeed the transport signa-  ture of the SOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' However, in contrast to pentedienyl,  where the SOMO is delocalized along the molecular back-  bone and dominates the electron transport, in benzyl,  the SOMO is mostly localized on the methyl functional  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' It is therefore interesting to investigate the effect  of the Coulomb repulsion and highlight the differences  between the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Within restricted DFT simula-  tions, the narrow Fano resonance is partially concealed by  the wider QI antiresonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Breaking the spin symmetry  within spin-unrestricted R-DMFT yields a pair of spin-  split Fano resonances, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In the ma-  jority spin channel, ϵ↑  Fano < 0 falls within the transmis-  sion depletion caused by the antiresonance and the asym-  metric Fano profile is clearly observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Its counter-  part in the minority spin channel is found above EF , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=',  ϵ↓  Fano > 0, and is still mostly concealed by the background  transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Interestingly, the spin-symmetry breaking  also induces spin-resolved QI antiresonances [30, 31, 97]  but the splitting ϵ↓  DQI − ϵ↑  DQI is however weaker than in  the Fano case, since the spin imbalance yields ⟨Sz⟩ ≃ 1/2  on the pz LO of the benzylic C, and a weaker magneti-  zation in the rest of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Not allowing breaking the spin symmetry in the many-  body simulations reveal another scenario, as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The difference is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' We observe a split-  ting of the Fano resonance in both R-DMFT and ED  (with the ED splitting being significantly larger) but no  splitting is detected for the QI antiresonance, which is  rather shifted further away from EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This suggests that  the microscopic mechanism behind the splitting with and  without spin-symmetry breaking are fundamentally dif-  ferent, as it distinguishes between the two QI features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Moreover, in contrast to the case of pentadienyl, the split-  ting of the SOMO in benzyl does not result in a strong  suppression of the transmission within the SOMO-SUMO  gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The two observations above are deeply connected,  and eventually, they can both be rationalized in terms of  the spatial distribution of the SOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  MICROSCOPIC MECHANISM  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Splitting of the SOMO  So far, we have seen that the Coulomb repulsion in-  duces a splitting of the SOMO of the organic radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In order to gain a deeper understanding of the electronic  mechanism behind the splitting, and how it affects the  transport properties of the junction, it is useful to look  at the retarded self-energy in the LO basis Σij = (ΣA)ij,  corresponding to ΣED  A  and Σσ,R−DMFT  A  in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (13, 20),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The many-body effects encoded in the self-  energy can be rationalized by interpreting the real part  as an energy-dependent level shift, and the imaginary  part as an effective electron-electron scattering rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' We  argue that the mechanism discussed in the following is a  common feature of organic radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Therefore, we dis-  cuss the pentadienyl and benzyl radicals in parallel and  highlight the differences whenever necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In order to compare the different approximations, it  is convenient to look at the trace of the self-energy ma-  trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Within spin-unrestricted R-DMFT, which is shown  in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 6(a,d), the real part of the self-energy is weakly  energy-dependent around EF , and determines a shift of  the SOMO resonance in opposite directions for the two  spin polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The imaginary part is negligible (not  shown) resulting in highly coherent SOMO and SUMO  electronic excitations below and above EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Note that  the ground state of spin-unrestricted R-DMFT is two-  fold degenerate, and it is invariant under a flip of all  spins: {σi} → {¯σi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This picture is qualitatively anal-  ogous to what one can expect also at the single-particle  level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', within DFT+U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Many-body effects are weak,  and the dominant effect arises from the spin-symmetry  breaking, as both radicals are magnetic insulators with a  9  6  4  2  0  2  60  40  20  0  20  4  2  0  2  R-DMFT  ED  R-DMFT  R-DMFT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5  1  30  20  10  0  10  80  60  40  20  0  20  40  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5  12  10  8  6  R-DMFT  ED  pentadienyl  benzyl  (a)  (d)  (b)  (e)  (c)  (f)  R-DMFT  R-DMFT  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Trace of the retarded self-energy Tr[Σ(E)] in the LO  basis for the pentadienyl (a,b,c) and benzyl (d,e,f) radicals  (the real and imaginary parts are denoted by solid and dashed  lines, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Within spin-unrestricted R-DMFT (a,d)  the self-energy displays a weakly energy-dependent real part,  which is different in each spin sector, while the imaginary part  is negligible (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Within both R-DMFT (b,e) and  ED (c,f) the self-energy is dominated by a single resonance at  energy ϵr (denoted by a solid grey line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  spin SOMO-SUMO gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The scenario is completely different within restricted  R-DMFT and ED, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 6(b,c,e,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' There,  the self-energy is dominated by a single resonance and its  energy dependence can be well described within a one-  pole approximation (OPA)  ΣOPA(E) =  a  E − EF − ϵr + ıγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (28)  The OPA self-energy has a Lorentzian shape, where ϵr  and γ denote the resonant energy and the width of  the resonance, whereas a controls the amplitude of the  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The imaginary part of the self-energy plays the  role of a giant electron-electron scattering rate and sup-  presses electronic excitations around ϵr ≃ ϵSOMO, while  the real part redistributes the spectral weight towards  higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This many-body mechanism, akin to the  Mott metal-to-insulator transition as described within  DMFT [42], is at the origin of the splitting of the SOMO  resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In organic radicals, the following hierarchy of emergent  energy scales is realized: ΓSOMO ≪ ∆ <∼ Uscreened, where  the typical energy scale associated with the screened  Coulomb repulsion Uscreened significantly exceeds the nar-  row width of the SOMO resonance (∼ 10–100 meV),  and the HOMO-LUMO single-particle gap ∆ controlled  by the C-C π-bonds (∼ eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This sets the electrons in  the SOMO deep within the strongly correlated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Such a general condition suggests this mechanism to be  common to organic radicals with a single unpaired elec-  tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Multi-radical molecules [98] and networks [99],  may display different electronic and transport properties  due to effective interactions between the unpaired elec-  trons [7, 18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Spatial structure of the electronic correlations  While R-DMFT and ED seem to qualitatively describe  the same many-body mechanism for the splitting of the  SOMO, it is also interesting to look at the whole self-  energy matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' III C, within ED  all elements Σij ̸= 0, whereas within R-DMFT Σij ∝ δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Remarkably, all elements of the self-energy (irrespectively  of the approximation) are well described by the OPA with  the same resonant energy ϵr, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 7(a,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The off-diagonal elements (when non-zero) can have ei-  ther sign since it is not determined by causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' It is then  easy to have a comprehensive look at the self-energy by  plotting the matrix Σij(ϵr), as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 7(c,d,g,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Indeed, looking at the ED self-energy matrix, clear pat-  terns emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Along the diagonal, some elements Σii are  significantly larger than the others (note the logarithmic  scale), and this asymmetry is mirrored by the off-diagonal  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Upon close inspection, we can associate them  with the pz LOs with the largest SOMO projection, thus  confirming that the strongest many-body effects corre-  late with the spatial distribution of the SOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Within  R-DMFT, we find an analogous pattern along the diag-  onal, as indicated in the insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Despite its approximations (local Coulomb interaction,  local correlations), it seems that R-DMFT tells qualita-  tively the same story as the full ED simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' This  advocates for a substantially local character of the mi-  croscopic mechanism, that can describe both the splitting  of the SOMO and its consequences on electron transport,  whereas non-local effects renormalize the splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Implications for electron transport  The many-body mechanism behind the splitting of the  SOMO is common to both the pentadienyl and benzyl  radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' However, its consequences on electron transport  are dramatically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In order to understand why,  it is necessary to combine the insights from DFT with  the knowledge about the spatial and energy structure of  the self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In pentadienyl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' the SOMO is delocalized throughout  the molecular backbone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' and its large projection on the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+page_content='C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+page_content='R-DMFT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Component of the ED self-energy Σij(E) and its matrix representation at the resonant energy Im Σij(ϵr) in the LO  basis for the pentadienyl (a,b,c,d) and benzyl (e,f,g,h) radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Each component of the self-energy (grey lines) is dominated by  a single pole (a,b,e,f) at a resonant energy ϵr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Selected components (i, j) are highlighted (color lines) and are labeled according  to their index in the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The matrix structure of the self-energy reflects the spatial distribution of the SOMO, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', the  largest local (Σii) and non-local (Σij̸=i) self-energy contributions are found for the LOs with the largest projections to the  SOMO (denoted by arrows, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Within R-DMFT (d,h) the self-energy is diagonal in the LO indices Σij ∝ δij and  displays the same pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  pz LOs of the anchoring groups (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='2(a)) ensures a  substantial overlap with the states in the metallic elec-  trodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Hence, there is a transmission channel across the  junction through the SOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The pole of the self-energy  results in a zero of the corresponding Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The suppression of the Green’s function hinders electron  transport at that energy and is at the origin of the trans-  mission node [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In contrast, in the benzyl radical,  the SOMO has negligible projection on the amino groups  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='2(d)) and transport is dominated by transmis-  sion channels involving the frontier MOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Therefore, the  splitting of the Fano resonance weakly affects those chan-  nels, and does not prevent the off-resonance transmission  of electrons across the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The above picture can be essentially reproduced within  the following tight-binding (TB) three-orbital model,  which is schematically represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Let us  consider three orbitals (ℓ, c, r) that can be interpreted as  the amino groups, left (ℓ) and right (r), and the central  molecule (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The Hamiltonian in such a basis reads  H =  �  �  ϵℓ  t  t′  t  ϵc  t  t′  t  ϵr  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (29)  The hybridization to the electrodes is mediated by the  external (ℓ, r) orbitals and, for the sake of this discussion,  it is assumed to be energy-independent:  ΓL =  �  �  Γ 0 0  0 0 0  0 0 0  �  � , ΓR =  �  �  0 0 0  0 0 0  0 0 Γ  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  (30)  The Hamiltonian of the isolated system can be diagonal-  ized to obtain the eigenvalues ϵHOMO, ϵSOMO, and ϵLUMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In light of the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 7, the Green’s func-  tion of the device  GD(z) =  �  z − H + ıΓL/2 + ıΓR/2 − ΣD(z)  �−1  (31)  is dressed with an OPA self-energy  ΣD(z) =  �  �  0  0  0  0 ΣOPA(z) 0  0  0  0  �  �  (32)  which acts on the central part (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 7(a,e) for a con-  nection with the ab-initio simulations) and has a pole at  ϵSOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Within such a three-orbital model, the Landauer  transmission in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' (26) simplifies to  T(E) = Γ2|Gℓr(E)|2,  (33)  where Gℓr = (GD)ℓr is the upper-right element of the  Green’s function, linking the orbitals connected to the  11  (c)  10-8  10-6  10-4  10-2  1  2  1  0  1  2  10-8  10-6  10-4  10-2  1  2  1  0  1  2  1  0  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5  1  0  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5  1  4  0  4  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5  1  Fano  splitting  splitting  node  zero  1  (g)  (d)  (h)  (e)  (i)  TB  OPA  TB  OPA  1  (a)  (b)  (f)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Schematic representation of the three-orbital TB model with its parameter, and form of the OPA self-energy (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Weight distribution and eigenvalues of the TB MOs for scenarios representative of the pentadienyl (b) and benzyl (f) radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The transmission function (c,g) obtained without (grey lines) and with (blue lines) the OPA self-energy captures all relevant  features of the DFT and many-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The Green’s function Gℓr is shown for specific energy ranges, which are  relevant to explaining the spectral features associated with the HOMOs (d,h) and the SOMOs (e,i), as discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Model parameters [eV]: ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5, ϵc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='25, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='25, Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='05, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='003, common to both scenarios, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5, t′ = 0 (b,c,d) and  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='1, t′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='5 (e,f,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  electrodes, and describes the only transmission channel  across the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  For the sake of simplicity, one can take −ϵℓ = ϵr = ϵ,  and ϵc ≪ ϵ, which together with a, Γ, and η are kept  fixed, whereas we choose the parameters t and t′ to de-  scribe two scenarios, which are representative of the pen-  tadienyl and benzyl radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The results are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 8 and described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The physics of the pentadienyl radical can be repro-  duced by choosing t <∼ ϵ and t′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The correspond-  ing TB MOs are fairly delocalized throughout the sys-  tem, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Hence, electron transport  happens through sequential hopping processes through  the c orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The transmission function, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 8(c), dis-  plays a SOMO resonance which is split by including the  OPA self-energy, revealing a transmission node within  the SOMO-SUMO gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The origin of the transmission  node is ascribed to a zero of the Green’s function at the  SOMO energy Gℓr(E ≃ ϵSOMO) [30, 31] as demonstrated  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 8(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Instead, with the choice of parameters t ≪ t′ <∼ ϵ, one  can describe the physics of the benzyl radical, charac-  terized by an orbital c, which is weakly coupled to the  ℓ − r molecular backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The corresponding SOMO is  fairly localized on the central orbital, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 8(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' The  transmission function displays a Fano resonance which is  split by the OPA self-energy see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 8(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' In contrast to  the previous case, Gℓr does not have a zero, and trans-  port is dominated by a transmission channel that bridges  the electrodes through the direct ℓ-r hopping t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Finally,  note that in both scenarios above, many-body effects are  negligible for the HOMO and LUMO resonances (corre-  sponding to states which are completely filled and empty,  respectively) even when the “correlated” c orbital has a  sizable hybridization with ℓ and r, cfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 8(c,d,g,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Hence, the three-orbital model can reproduce all fun-  damental features of the radical junctions discussed in  this work, and at the same time, provides a simple inter-  pretation of the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Non-perturbative nature of the splitting  Within ED and R-DMFT, the solution of the many-  body problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', on the lattice or the auxiliary AIM)  is numerically exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  This means that the Coulomb  repulsion is taken into account in a non-perturbative  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  It is interesting to compare these results to a  perturbative approach, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', within the GW approxima-  tion [100, 101], which has been extensively and success-  fully applied to molecules [102–107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' However, the ques-  tion arises to which extent many-body perturbation the-  ory approaches are able to describe the physics of open-  shell systems [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Within GW, the self-energy is com-  puted to the lowest order in perturbation theory, as a  convolution of the Green’s function and the screened in-  12  10-4  10-2  1  2  1  0  1  2  G0W0  GW  (a)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Electron transmission function through the pentadi-  enyl radical junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Both the G0W0 and the self-consistent  GW approximations fail to predict the splitting of the SOMO,  as described within ED and R-DMFT, cfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  teraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  We compute the GW self-energy correction  projected onto the A region  Σ(z) = GA(z)WA,  (34)  as described in [68], and we consider the case of the pen-  tadienyl radical without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' 9 we see that neither G0W0 nor the fully self-  consistent GW approximation is able to induce a split-  ting of the SOMO resonance, and the numerical simu-  lations rather result in a shift of the corresponding res-  onance above the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Hence, the many-body  techniques we propose to investigate open-shell molecules  are not only sufficient but also necessary for our goal,  whereas less sophisticated approaches fall short in de-  scribing the electronic and transport properties arising  from the strong electronic correlations within the SOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  CONCLUSIONS  In this work, we have proposed a numerical method  that that combines ab-initio with state-of-the-art many-  body techniques and is able to address the complexity  of a realistic chemical environment as well as electronic  correlation effects beyond the single-particle picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  The deliverable of this project served to shed light on  the mechanism governing the electronic and transport  properties of quantum junctions with organic molecules  in an open-shell configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' By considering a linear  and a cyclic radical molecule, we derive a general under-  standing of the role of many-body effects in molecular  radicals with a single unpaired electron, and we show  that they have dramatic consequences on electron  transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  We establish the microscopic mechanism  behind the splitting of the SOMO resonance and unravel  a clear link between the space-time structure of electron-  electron correlations and the spatial distribution of the  SOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' We demonstrate this by proposing a minimal  model, which is capable of grasping the microscopic  mechanism and thus reproducing all relevant features of  the transmission properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Our work will pave the path  toward a deeper and more comprehensive understanding  of strongly correlated electron physics at the nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We thank J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Tomczak for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  This research is supported by the Austrian Science Fund  (FWF) through project P 31631 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=') and by the  NCCR MARVEL funded by the Swiss National Science  Foundation grant 51NF40-205602 (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  Computational support from the Swiss Supercomputing  Center (CSCS) under project ID s1119 is gratefully ac-  knowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' Sanvito, Chemical Society Reviews 40, 3336 (2011),  URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='1039/c1cs15047b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+page_content=' Sangiovanni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='1103/physrevb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+page_content='org/  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='1103/physrevb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='1103/physrevb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+page_content=' Koval, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content=' S´anchez-  Portal, New Journal of Physics 23, 093027 (2021), URL  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
+page_content='1088/1367-2630/ac1bf3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAyT4oBgHgl3EQfcfcz/content/2301.00282v1.pdf'}
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+arXiv:2301.01030v1  [physics.plasm-ph]  3 Jan 2023
+Analytical study of ion-acoustic solitary waves in a
+magnetized plasma with degenerate electrons
+Moumita Indraa, K. K. Ghoshb, Saibal Rayc
+aSchool of Basic Science, Swami Vivekananda University, Kanthalia, Barrackpore, West
+Bengal, India
+bDepartment of Basic Science & Humanities, Abacus Institute of Engineering and
+Management, Magra, Chinchura, West Bengal, India
+cCentre for Cosmology, Astrophysics and Space Science (CCASS), GLA University,
+Mathura 281406, Uttar Pradesh, India
+Abstract
+The propagation of fully nonlinear ion acoustic solitary waves (IASW) in
+a magneto-plasma with degenerate electrons investigated by Abdelsalam et
+al. [[1] Physics Letters A 372 (2008) 4923]. Based on their work in the present
+work, a rigorous and general analytical study is presented. This confirms
+their implied assumption that (i) only hump and no cavity is possible and
+(ii) for humps, the algebraic equation for the maximum density N obtained
+by them determines it uniquely (naturally assumes N > 1). Here we confirm
+analytically their assertion that N decreases with lx (the direction cosine of
+the wave vector k along the x-axis) and N increases with the increase of the
+Mach number (M).
+Keywords: Waves; Plasmas; MHD; Hydrodynamics; Mach number
+1. Introduction
+Recently, the physics of an electron-positron-ion (EPI) plasma [2, 3, 4, 5, 6,
+7] has received considerable attention, mainly due to its importance in many
+systems in laboratory plasma as well as astrophysical arena. EPI plasma
+Email addresses: moumita.indra93@gmail.com (Moumita Indra),
+kkghosh1954@gmail.com (K. K. Ghosh), saibal.ray@gla.ac.in (Saibal Ray)
+Orcid ID: 0000-0002-7900-7947
+Orcid ID: 0000-0002-5909-0544
+Preprint submitted to Chinese Journal of PhysicsReceived 2022 September 22; accepted 2022 month day
+
+exists in places such as active galactic nuclei (AGNs) [8, 9], pulsar magneto-
+spheres [10, 11] and in many dense astronomical environments, namely, neu-
+tron stars and white dwarfs [12, 13] which is supposed to play a key role in
+understanding the origin and evolution of our entire universe [14]. This kind
+of plasma may also be practically produced in laboratories [15, 16, 17, 18].
+Electrons and positrons are assumed relativistic and degenerate, follow-
+ing the Fermi–Dirac statistics, whereas the warm ions are described by a
+set of classical fluid equations with an individual charge of Zie, (Zi denotes
+the ion-charge state, while e is the electron charge), subject to the influence
+of the electrostatic potential φ. Quantum hydrodynamics
+[23, 24, 25, 26],
+which describes quantum systems within a hydrodynamic framework, was
+first proposed by Madelung [19] and Bohm [20]. Although such a descrip-
+tion is formally accurate for a single particle, Manfredi and Haas [21] later
+expanded the idea to many-particle systems and it gained significant favour
+in the areas of the quantum plasma community.
+Considering the one-dimensional QHD model in the limit of the small
+mass ratio of the charge carriers, Hass et al. [22] were the first to study
+the ion-acoustic waves in unmagnetized quantum plasma. This model has
+been used in various investigations by several authors [23, 24, 25, 26] where
+generally a linear dispersion relation is derived in the linear approximation.
+Thus, ion-acoustic waves (IAW), a fundamental mode in plasma environ-
+ments, have been a subject of extensive research over several decades. Khan
+and Haque [27] showed that in the small (linear) limit of quantum diffraction
+parameter H (ratio of the plasmon energy to the Fermi energy), the system
+behaves as the classical IAW whereas in the non-linear regime the system
+behaves differently. One of the most interesting non-linear features of IAW
+is the existence of ion-acoustic solitary waves (IASW) [28, 29].
+In the weakly nonlinear limit, the quantum plasma is shown to support
+waves described by a deformed Korteweg–de Vries (KdV) equation which
+depends in a non-trivial way on the quantum parameter H. However, in the
+fully non-linear regime, the system exhibits travelling waves which show a
+periodic pattern. Hence there are two main approaches used to investigate
+IASWs, viz., the reductive perturbation technique (KdV method) [30] and
+the pseudo-potential technique for large-amplitude solitary waves (Sagdeev
+method) [31]. The theory of solitons in magneto-plasma was greatly improved
+by an intriguing work by Abdelsalam et al. [1] on completely non-linear IASW
+travelling obliquely to an external magnetic field in a collision-less dense
+Thomas-Fermi magneto-plasma with degenerate electrons.
+2
+
+The degenerate electrons in the above scenario may be described using
+the Thomas-Fermi approximation [32, 33] whereas the ion component can
+be thought of as a classical gas. They have obtained an energy balance-like
+equation involving the Sagdeev potential as follows:
+1
+2
+�dn
+dη
+�2
++ V (n) = 0,
+(1)
+where Sagdeev-like pseudo-potential V (n) is given by
+V (n) =
+9n6
+2(5an8/3 − 3)2[5an8/3 − 2an5/3−3a+1
+M2
++1 − 2n
+M2n2 + cM2(an10/3 − 2an5/3 − 2
+9 + a + 5)],
+(2)
+where
+a =
+3
+5M2,
+(3)
+c = 3lx
+2
+5M2,
+(4)
+η = lxx + lyy − Mt, lx
+2 + ly
+2 = 1 and n = n(η).
+Here M is the Mach number, lx and ly are the direction cosines of the
+wave vector k along the x and y axes respectively, n(= ne
+no) where ne is the
+electron density and no is the unperturbed electron density with ni as the
+ion density.
+As indicated by Abdelsalam et al. [1], the existence of IASW’s for which
+1 ≤ n ≤ N and dn
+dη = 0 at n = 1, N,
+(5)
+requires the following equations and inequality:
+V (n)|n=1 = 0,
+(6)
+dV
+dn |n=1 = 0,
+(7)
+3
+
+d2V
+dn2 |n=1 < 0,
+(8)
+and V (n)|n=N = 0.
+(9)
+Abdelsalam et al. [1] noted that Eqs. (6) and (7) are automatically satis-
+fied by Eq. (2) while the inequality (8) is satisfied if and only if
+lx < M < 1, i.e., c < 0.6 < a,
+in view of Eqs. (3) and (4).
+For the nonlinear dispersion relation, Eq.
+(9), they have numerically
+solved it for several specific values of lx (lx = 0.66, 0.68, 0.7) and on that
+basis argued that if Eq. (9) can be rewritten as
+N = N(lx, M),
+(10)
+where the maximum density N is a decreasing function of lx, i.e., ∂N
+∂lx < 0
+and N is an increasing function of M, i.e., ∂N
+∂M > 0.
+However, in the present work our motivation is to solve the problem of
+Abdelsalam et al. [1] with an analytical methodology under a more general
+treatment. For this we have considered a different format and have shown
+that some of their outcomes can be retrieved with a convincing way and can
+be demonstrated valid in the physical realm.
+2. An analytical methodology
+Putting n = x3 the equations (1) and (2) can be rewritten as
+�dx
+dη
+�2
++ x(1 − x)f(x, lx, M)
+(5ax8 − 3)2
+= 0,
+(11)
+where
+f(x, lx, M) = (1 + x + x2)2
++9lx
+2x6(1 + x + x2 + x3 + x4)2
+25M4
+−3x6(3 + 6x + 4x2 + 2x3)
+5M2
+−3lx
+2x3(2 + 4x + 6x2 + 3x3)
+5M2
+.
+(12)
+4
+
+Equations (5) and (9) are now rewritten as
+dx
+dη = 0 at x = 1, N1/3,
+(13)
+f(N1/3, lx, M) = 0.
+(14)
+The above Eqs. (3) and (4) and the inequality (8) remain unchanged
+except Eq. (9) which is to be replaced by Eq. (14). In other words, the
+question now is whether Eq.
+(14) can determine N (or x) uniquely.
+To
+answer this one needs the following observations on f(x, lx, M).
+3. Observations on f(x, lx, M)
+3.1. Observation 1:
+(i) f(0, lx, M) = 1,
+(ii) f(1) = (3 − 5a)(3 − 5c) < 0,
+(iii) f(∞) > 0.
+Proof: Trivial.
+3.2. Observation 2:
+For given a and c there exist a unique pair of (α, β) such that
+f(x, lx, M) > 0, for 0 < x < β,
+f(β, lx, M) = 0,
+f(x, lx, M) < 0, for β < x < α,
+f(α, lx, M) = 0,
+and f(x, lx, M) > 0 for x > α,
+where 0 < β < 1 < α.
+Corollary:
+∂f(x, lx, M)
+∂x
+> 0 at x = α.
+Proof: Trivial.
+5
+
+3.3. Observation 3:
+f(x, lx, M) < 0 at x = ( 3
+5a)1/8.
+Proof: See Appendix.
+Corollary:
+β < ( 3
+5a)1/8 < 1.
+Proof: Trivial from observation 2.
+3.4. Observation 4:
+∂f(x, lx, M)
+∂lx
+> 0 for x > 1.
+Proof: See Appendix.
+3.5. Observation 5:
+For any α > 1 there exists lx and M that satisfy f(α, lx, M) = 0 and also
+satisfy the inequality (9).
+Proof: See Appendix.
+With these observations one can uniquely determine x (or N) (> 1) satis-
+fying Eq. (12) and also deals with decreasing/increasing feature of x (or N)
+as well as for increase of lx or M. These are answered as follows.
+4. Proof of uniqueness of N
+From the Observation 1, we note that f(1) < 0 and f(∞) > 0. Owing to
+the continuity of f(x) there exists one x, such that f(x1) = 0 and x1 > 1. If
+possible, let there exist x1 and x2 such that
+f(x1) = f(x2) = 0 and x2 > x1 > 1.
+(15)
+6
+
+From Eq. (15), applying Rolle’s theorem, there exists x3 and x4, such
+that
+f ′(x3) = f ′(x4) = 0 and x2 > x4 > x1 > x3 > 1.
+(16)
+But f ′(x) is a polynomial of degree 13 such that f ′(−∞) < 0, f ′(0) > 0,
+f ′(1) < 0 and f ′(∞) > 0. So we can see that f ′(x) vanishes only for x > 1
+which contradicts Eq. (16).
+Hence there exists a unique x(> 1) such that f(x, lx, M) = 0, i.e. there
+exists unique N(> 1) satisfying Eq. (9).
+Now, we have to show analytically that the maximum density N is a
+decreasing function of lx and is an increasing function of M. Differentiating
+both sides of Eq. (12) with respect to M, one gets
+∂f
+∂M < 0, for x > 1.
+(17)
+For
+∂f
+∂M =
+6x3
+25M5[−6l2
+xx3(1 + x + x2 + x3 + x4)2 + 5x3M2(3 + 6x + 4x2 + 2x3)
++5l2
+xM2(2 + 4x + 6x2 + 3x3)]
+<
+6x3
+25M5[−6x3(1 + x + x2 + x3 + x4)2 + 5x3(3 + 6x + 4x2 + 2x3)
++5l2
+x(2 + 4x + 6x2 + 3x3)]
+(since lx < M < 1)
+=
+6x3
+25M5[−18(x9 − x4) − 30(x7 − x2) − 17(x8 − x3) − 7(x8 − x)
+−13(x6 − x) − 10(x10 − 1) − 2(x10 − x5) − x6 − 6x11]
+< 0,
+for x > 1.
+(18)
+From Eqs. (14) and (11), we obtain
+∂N1/3
+∂lx
+= −
+∂f
+∂lx
+∂f
+∂N1/3
+and ∂N1/3
+∂M
+= −
+∂f
+∂M
+∂f
+∂N1/3
+,
+which gives
+∂N1/3
+∂lx
+∂N1/3
+∂M
+=
+∂f
+∂lx
+∂f
+∂M
+, i.e.,
+∂N
+∂lx
+∂N
+∂M
+=
+∂f
+∂lx
+∂f
+∂M
+,
+i.e., ∂N
+∂lx
+< 0 for x > 1,
+(using observation 4 and Eq. (17)).
+7
+
+Again from Eq. (14), we have
+∂N1/3
+∂lx
+∂lx
+∂M
+∂M
+∂N1/3 = −1,
+and 1
+3N−2/3∂N
+∂lx
+5
+3MC = −1
+3N−2/3 ∂N
+∂M ,
+(by Observation (4))
+i.e., ∂N
+∂M > 0 for
+x > 1
+(since, ∂N
+∂lx < 0).
+4.1. Proof of Observation 5:
+From the observation 1 one can see that the equation f(x, lx, M) = 0 has
+at least one root between 0 and 1 and one root greater then 1. Also one can
+note that f(x, lx, M) regarded as a polynomial in x has two changes of sign
+and hence by Descarte’s rule of sign has at most two positive roots. Hence
+equation f(x, lx, M) = 0 has exactly one root between 0 and 1 and exactly
+one root greater than 1 which are called β and α respectively. The continuity
+of f(x, lx, M) ensures that the remaining part of the observation is true.
+5. Conclusion
+In the present work our main motivation was to provide an analytically
+performed rigorous base of the study of Abdelsalam et al. [1] on the propaga-
+tion of fully non-linear ion-acoustic waves in a collision-less magneto-plasma
+with degenerate electrons. The outcomes of the investigation are interesting
+and some explicit features can be exhibited as follows:
+(1) only hump and no cavity is possible;
+(2) for humps, (i) the algebraic equation for the maximum density N ob-
+tained by them determines it uniquely (under the assumption N > 1), (ii) N
+decreases with lx (the direction cosine of the wave vector k along the x-axis)
+and (iii) N increases with the increase of the Mach number (M). All these
+results yield simply from the maximum density N which can be uniquely
+determined by Eq. (9) under the constraint N > 1.
+Another motivation of the present work is related to the astrophysical
+relevance of an EPI plasma, especially in the cases of AGNs [8, 9], pulsar
+magneto-spheres [10, 11], neutron stars and white dwarfs [12, 13].
+A su-
+porting and confirmirmational results of Abdelsalam et al. [1] therefore will
+enhance to understand deeply the structural phenomena occuring in differ-
+ent astrophysical systems. In this connection we would like to mention the
+8
+
+very recent work of Piotrovich et al. [9] where they have hypothesized that
+the AGNs are wormhole mouths rather than supermassive black holes. Es-
+sentially due to bizzare gravitational formation wormholes may emit gamma
+radiation as a result of a collision of accreting flows inside it. Now the in-
+teresting fact is that the radiation has a distinctive spectrum much different
+from those of jets or accretion discs of AGNs. Hopefully an observation of
+such radiation via the EPI and hence IASW would serve as evidence of the
+existence of wormholes.
+Appendix
+Proof of Observation 3
+Let
+� 3
+5a
+�1/8 = γ so that a =
+3
+5γ8
+9
+
+Then at x = γ
+f(x, a, c) = (1 + γ + γ2)2 + 3c
+5γ2(1 + γ + γ2 + γ3 + γ4)2
+− 3
+5γ2(3 + 6γ + 4γ2 + 2γ3) − cγ3(2 + 4γ + 6γ2 + 3γ3
+=
+1
+5γ2[(5γ2(1 + γ + γ2)2 − 3(3 + 6γ + 4γ2 + 2γ3))
++c(3(1 + γ + γ2 + γ3 + γ4)2 − 5γ5(2 + 4γ + 6γ2 + 3γ3))]
+=
+1
+5γ2[(−9 − 18γ − 7γ2 + 4γ3 + 15γ4 + 10γ5 + 5γ6)
++c(3 + 6γ + 9γ2 + 12γ3 + 15γ4 + 2γ5 − 11γ6 − 24γ7 − 12γ8)]
+=
+1
+5γ2[(γ − 1)(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)]
+−c(γ − 1)(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)
+= γ − 1
+5γ2 [(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)
+−c(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)]
+< γ − 1
+5γ2 [(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)
+−3(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)]
+≤ γ − 1
+25γ2 [36 + 108γ + 116γ2 + 60γ3 − 60γ4 − 116γ5 − 108γ6 − 36γ7]
+< 0
+if γ < 1
+Proof of Observation 4
+∂f(x, a, c)
+∂c
+= ax6(1 + x + x2 + x3 + x4)2 − x3(2 + 4x + 6x2 + 3x3)
+≥ x3
+5 [3x3(1 + x + x2 + x3 + x4)2 − 5(2 + 4x + 6x2 + 3x3)] (since, a =
+3
+5M2)
+= x3
+5 [3x3(1 + 2x + 3x+4x3 + 5x4 + 4x5 + 3x6 + 2x7 + x8) − 5(2 + 4x + 6x2 + 3x3)]
+= x3
+5 [−10 − 20x − 30x2 − 12x3 + 6x4 + 9x5 + 12x6 + 15x7 + 12x8 + 9x9 + 6x10 + 3x11)
+> 0 for x > 1
+10
+
+Declaration of competing interest
+The authors declare that they have no known competing financial inter-
+ests or personal relationships that could have appeared to influence the work
+reported in this paper
+acknowledgement
+One of the authors, KKG would like to thank the authority of Abacus
+Institute of Engineering and Management for all the facilities and encourage-
+ment. We all are grateful to the anonymous referee for the useful comments
+which have enhanced the quality of the paper.
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new file mode 100644
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+page_content='plasm-ph]  3 Jan 2023  Analytical study of ion-acoustic solitary waves in a  magnetized plasma with degenerate electrons  Moumita Indraa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Ghoshb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Saibal Rayc  aSchool of Basic Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Swami Vivekananda University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Kanthalia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
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+page_content=' India  bDepartment of Basic Science & Humanities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Abacus Institute of Engineering and  Management,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Magra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
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+page_content=' India  cCentre for Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Astrophysics and Space Science (CCASS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' GLA University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Mathura 281406,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Uttar Pradesh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' India  Abstract  The propagation of fully nonlinear ion acoustic solitary waves (IASW) in  a magneto-plasma with degenerate electrons investigated by Abdelsalam et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [[1] Physics Letters A 372 (2008) 4923].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Based on their work in the present  work, a rigorous and general analytical study is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' This confirms  their implied assumption that (i) only hump and no cavity is possible and  (ii) for humps, the algebraic equation for the maximum density N obtained  by them determines it uniquely (naturally assumes N > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Here we confirm  analytically their assertion that N decreases with lx (the direction cosine of  the wave vector k along the x-axis) and N increases with the increase of the  Mach number (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Keywords: Waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Plasmas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' MHD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Hydrodynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Mach number  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Introduction  Recently, the physics of an electron-positron-ion (EPI) plasma [2, 3, 4, 5, 6,  7] has received considerable attention, mainly due to its importance in many  systems in laboratory plasma as well as astrophysical arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' EPI plasma  Email addresses: moumita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='indra93@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='com (Moumita Indra),  kkghosh1954@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='com (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Ghosh), saibal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='ray@gla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='in (Saibal Ray)  Orcid ID: 0000-0002-7900-7947  Orcid ID: 0000-0002-5909-0544  Preprint submitted to Chinese Journal of PhysicsReceived 2022 September 22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' accepted 2022 month day  exists in places such as active galactic nuclei (AGNs) [8, 9], pulsar magneto-  spheres [10, 11] and in many dense astronomical environments, namely, neu-  tron stars and white dwarfs [12, 13] which is supposed to play a key role in  understanding the origin and evolution of our entire universe [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' This kind  of plasma may also be practically produced in laboratories [15, 16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Electrons and positrons are assumed relativistic and degenerate, follow-  ing the Fermi–Dirac statistics, whereas the warm ions are described by a  set of classical fluid equations with an individual charge of Zie, (Zi denotes  the ion-charge state, while e is the electron charge), subject to the influence  of the electrostatic potential φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Quantum hydrodynamics  [23, 24, 25, 26],  which describes quantum systems within a hydrodynamic framework, was  first proposed by Madelung [19] and Bohm [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Although such a descrip-  tion is formally accurate for a single particle, Manfredi and Haas [21] later  expanded the idea to many-particle systems and it gained significant favour  in the areas of the quantum plasma community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Considering the one-dimensional QHD model in the limit of the small  mass ratio of the charge carriers, Hass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [22] were the first to study  the ion-acoustic waves in unmagnetized quantum plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' This model has  been used in various investigations by several authors [23, 24, 25, 26] where  generally a linear dispersion relation is derived in the linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Thus, ion-acoustic waves (IAW), a fundamental mode in plasma environ-  ments, have been a subject of extensive research over several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Khan  and Haque [27] showed that in the small (linear) limit of quantum diffraction  parameter H (ratio of the plasmon energy to the Fermi energy), the system  behaves as the classical IAW whereas in the non-linear regime the system  behaves differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' One of the most interesting non-linear features of IAW  is the existence of ion-acoustic solitary waves (IASW) [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  In the weakly nonlinear limit, the quantum plasma is shown to support  waves described by a deformed Korteweg–de Vries (KdV) equation which  depends in a non-trivial way on the quantum parameter H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' However, in the  fully non-linear regime, the system exhibits travelling waves which show a  periodic pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Hence there are two main approaches used to investigate  IASWs, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', the reductive perturbation technique (KdV method) [30] and  the pseudo-potential technique for large-amplitude solitary waves (Sagdeev  method) [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' The theory of solitons in magneto-plasma was greatly improved  by an intriguing work by Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1] on completely non-linear IASW  travelling obliquely to an external magnetic field in a collision-less dense  Thomas-Fermi magneto-plasma with degenerate electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  2  The degenerate electrons in the above scenario may be described using  the Thomas-Fermi approximation [32, 33] whereas the ion component can  be thought of as a classical gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' They have obtained an energy balance-like  equation involving the Sagdeev potential as follows:  1  2  �dn  dη  �2  + V (n) = 0,  (1)  where Sagdeev-like pseudo-potential V (n) is given by  V (n) =  9n6  2(5an8/3 − 3)2[5an8/3 − 2an5/3−3a+1  M2  +1 − 2n  M2n2 + cM2(an10/3 − 2an5/3 − 2  9 + a + 5)],  (2)  where  a =  3  5M2,  (3)  c = 3lx  2  5M2,  (4)  η = lxx + lyy − Mt, lx  2 + ly  2 = 1 and n = n(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Here M is the Mach number, lx and ly are the direction cosines of the  wave vector k along the x and y axes respectively, n(= ne  no) where ne is the  electron density and no is the unperturbed electron density with ni as the  ion density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  As indicated by Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1], the existence of IASW’s for which  1 ≤ n ≤ N and dn  dη = 0 at n = 1, N,  (5)  requires the following equations and inequality:  V (n)|n=1 = 0,  (6)  dV  dn |n=1 = 0,  (7)  3  d2V  dn2 |n=1 < 0,  (8)  and V (n)|n=N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (9)  Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1] noted that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (6) and (7) are automatically satis-  fied by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (2) while the inequality (8) is satisfied if and only if  lx < M < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='6 < a,  in view of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  For the nonlinear dispersion relation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (9), they have numerically  solved it for several specific values of lx (lx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='66, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='68, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='7) and on that  basis argued that if Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (9) can be rewritten as  N = N(lx, M),  (10)  where the maximum density N is a decreasing function of lx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', ∂N  ∂lx < 0  and N is an increasing function of M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', ∂N  ∂M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  However, in the present work our motivation is to solve the problem of  Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1] with an analytical methodology under a more general  treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' For this we have considered a different format and have shown  that some of their outcomes can be retrieved with a convincing way and can  be demonstrated valid in the physical realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' An analytical methodology  Putting n = x3 the equations (1) and (2) can be rewritten as  �dx  dη  �2  + x(1 − x)f(x, lx, M)  (5ax8 − 3)2  = 0,  (11)  where  f(x, lx, M) = (1 + x + x2)2  +9lx  2x6(1 + x + x2 + x3 + x4)2  25M4  −3x6(3 + 6x + 4x2 + 2x3)  5M2  −3lx  2x3(2 + 4x + 6x2 + 3x3)  5M2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (12)  4  Equations (5) and (9) are now rewritten as  dx  dη = 0 at x = 1, N1/3,  (13)  f(N1/3, lx, M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (14)  The above Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (3) and (4) and the inequality (8) remain unchanged  except Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (9) which is to be replaced by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' In other words, the  question now is whether Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (14) can determine N (or x) uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  To  answer this one needs the following observations on f(x, lx, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observations on f(x, lx, M)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observation 1:  (i) f(0, lx, M) = 1,  (ii) f(1) = (3 − 5a)(3 − 5c) < 0,  (iii) f(∞) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: Trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observation 2:  For given a and c there exist a unique pair of (α, β) such that  f(x, lx, M) > 0, for 0 < x < β,  f(β, lx, M) = 0,  f(x, lx, M) < 0, for β < x < α,  f(α, lx, M) = 0,  and f(x, lx, M) > 0 for x > α,  where 0 < β < 1 < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Corollary:  ∂f(x, lx, M)  ∂x  > 0 at x = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: Trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observation 3:  f(x, lx, M) < 0 at x = ( 3  5a)1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: See Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Corollary:  β < ( 3  5a)1/8 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: Trivial from observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observation 4:  ∂f(x, lx, M)  ∂lx  > 0 for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: See Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Observation 5:  For any α > 1 there exists lx and M that satisfy f(α, lx, M) = 0 and also  satisfy the inequality (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Proof: See Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  With these observations one can uniquely determine x (or N) (> 1) satis-  fying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (12) and also deals with decreasing/increasing feature of x (or N)  as well as for increase of lx or M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' These are answered as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Proof of uniqueness of N  From the Observation 1, we note that f(1) < 0 and f(∞) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Owing to  the continuity of f(x) there exists one x, such that f(x1) = 0 and x1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' If  possible, let there exist x1 and x2 such that  f(x1) = f(x2) = 0 and x2 > x1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (15)  6  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (15), applying Rolle’s theorem, there exists x3 and x4, such  that  f ′(x3) = f ′(x4) = 0 and x2 > x4 > x1 > x3 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (16)  But f ′(x) is a polynomial of degree 13 such that f ′(−∞) < 0, f ′(0) > 0,  f ′(1) < 0 and f ′(∞) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' So we can see that f ′(x) vanishes only for x > 1  which contradicts Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Hence there exists a unique x(> 1) such that f(x, lx, M) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' there  exists unique N(> 1) satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Now, we have to show analytically that the maximum density N is a  decreasing function of lx and is an increasing function of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Differentiating  both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (12) with respect to M, one gets  ∂f  ∂M < 0, for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (17)  For  ∂f  ∂M =  6x3  25M5[−6l2  xx3(1 + x + x2 + x3 + x4)2 + 5x3M2(3 + 6x + 4x2 + 2x3)  +5l2  xM2(2 + 4x + 6x2 + 3x3)]  <  6x3  25M5[−6x3(1 + x + x2 + x3 + x4)2 + 5x3(3 + 6x + 4x2 + 2x3)  +5l2  x(2 + 4x + 6x2 + 3x3)]  (since lx < M < 1)  =  6x3  25M5[−18(x9 − x4) − 30(x7 − x2) − 17(x8 − x3) − 7(x8 − x)  −13(x6 − x) − 10(x10 − 1) − 2(x10 − x5) − x6 − 6x11]  < 0,  for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (18)  From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (14) and (11), we obtain  ∂N1/3  ∂lx  = −  ∂f  ∂lx  ∂f  ∂N1/3  and ∂N1/3  ∂M  = −  ∂f  ∂M  ∂f  ∂N1/3  ,  which gives  ∂N1/3  ∂lx  ∂N1/3  ∂M  =  ∂f  ∂lx  ∂f  ∂M  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=',  ∂N  ∂lx  ∂N  ∂M  =  ∂f  ∂lx  ∂f  ∂M  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', ∂N  ∂lx  < 0 for x > 1,  (using observation 4 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  7  Again from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (14), we have  ∂N1/3  ∂lx  ∂lx  ∂M  ∂M  ∂N1/3 = −1,  and 1  3N−2/3∂N  ∂lx  5  3MC = −1  3N−2/3 ∂N  ∂M ,  (by Observation (4))  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=', ∂N  ∂M > 0 for  x > 1  (since, ∂N  ∂lx < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Proof of Observation 5:  From the observation 1 one can see that the equation f(x, lx, M) = 0 has  at least one root between 0 and 1 and one root greater then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Also one can  note that f(x, lx, M) regarded as a polynomial in x has two changes of sign  and hence by Descarte’s rule of sign has at most two positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Hence  equation f(x, lx, M) = 0 has exactly one root between 0 and 1 and exactly  one root greater than 1 which are called β and α respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' The continuity  of f(x, lx, M) ensures that the remaining part of the observation is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Conclusion  In the present work our main motivation was to provide an analytically  performed rigorous base of the study of Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1] on the propaga-  tion of fully non-linear ion-acoustic waves in a collision-less magneto-plasma  with degenerate electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' The outcomes of the investigation are interesting  and some explicit features can be exhibited as follows:  (1) only hump and no cavity is possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  (2) for humps, (i) the algebraic equation for the maximum density N ob-  tained by them determines it uniquely (under the assumption N > 1), (ii) N  decreases with lx (the direction cosine of the wave vector k along the x-axis)  and (iii) N increases with the increase of the Mach number (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' All these  results yield simply from the maximum density N which can be uniquely  determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' (9) under the constraint N > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Another motivation of the present work is related to the astrophysical  relevance of an EPI plasma, especially in the cases of AGNs [8, 9], pulsar  magneto-spheres [10, 11], neutron stars and white dwarfs [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  A su-  porting and confirmirmational results of Abdelsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [1] therefore will  enhance to understand deeply the structural phenomena occuring in differ-  ent astrophysical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' In this connection we would like to mention the  8  very recent work of Piotrovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' [9] where they have hypothesized that  the AGNs are wormhole mouths rather than supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Es-  sentially due to bizzare gravitational formation wormholes may emit gamma  radiation as a result of a collision of accreting flows inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Now the in-  teresting fact is that the radiation has a distinctive spectrum much different  from those of jets or accretion discs of AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' Hopefully an observation of  such radiation via the EPI and hence IASW would serve as evidence of the  existence of wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  Appendix  Proof of Observation 3  Let  � 3  5a  �1/8 = γ so that a =  3  5γ8  9  Then at x = γ  f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' c) = (1 + γ + γ2)2 + 3c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2(1 + γ + γ2 + γ3 + γ4)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='− 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2(3 + 6γ + 4γ2 + 2γ3) − cγ3(2 + 4γ + 6γ2 + 3γ3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2[(5γ2(1 + γ + γ2)2 − 3(3 + 6γ + 4γ2 + 2γ3))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='+c(3(1 + γ + γ2 + γ3 + γ4)2 − 5γ5(2 + 4γ + 6γ2 + 3γ3))]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2[(−9 − 18γ − 7γ2 + 4γ3 + 15γ4 + 10γ5 + 5γ6)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='+c(3 + 6γ + 9γ2 + 12γ3 + 15γ4 + 2γ5 − 11γ6 − 24γ7 − 12γ8)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2[(γ − 1)(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='−c(γ − 1)(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='= γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2 [(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='−c(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='< γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='5γ2 [(9 + 27γ + 34γ2 + 30γ3 + 15γ4 + 5γ5)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='−3(3 + 9γ + 18γ2 + 30γ3 + 45γ4 + 47γ5 + 36γ6 + 12γ7)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='≤ γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='25γ2 [36 + 108γ + 116γ2 + 60γ3 − 60γ4 − 116γ5 − 108γ6 − 36γ7]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='< 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='if γ < 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='Proof of Observation 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='∂f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' c)  ∂c  = ax6(1 + x + x2 + x3 + x4)2 − x3(2 + 4x + 6x2 + 3x3)  ≥ x3  5 [3x3(1 + x + x2 + x3 + x4)2 − 5(2 + 4x + 6x2 + 3x3)] (since,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' a =  3  5M2)  = x3  5 [3x3(1 + 2x + 3x+4x3 + 5x4 + 4x5 + 3x6 + 2x7 + x8) − 5(2 + 4x + 6x2 + 3x3)]  = x3  5 [−10 − 20x − 30x2 − 12x3 + 6x4 + 9x5 + 12x6 + 15x7 + 12x8 + 9x9 + 6x10 + 3x11)  > 0 for x > 1  10  Declaration of competing interest  The authors declare that they have no known competing financial inter-  ests or personal relationships that could have appeared to influence the work  reported in this paper  acknowledgement  One of the authors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' KKG would like to thank the authority of Abacus  Institute of Engineering and Management for all the facilities and encourage-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' We all are grateful to the anonymous referee for the useful comments  which have enhanced the quality of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content='  References  [1] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
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+page_content=' Berezhiani, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfGfvE/content/2301.01030v1.pdf'}
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+arXiv:2301.03120v1  [quant-ph]  8 Jan 2023
+On generating r-uniform subspaces with the isometric mapping method
+K. V. Antipin∗
+Faculty of Physics, M. V. Lomonosov Moscow State University,
+Leninskie gory, Moscow 119991, Russia
+(Dated: January 10, 2023)
+We propose a compositional approach to construct subspaces consisting entirely of r-uniform
+states, including the ones in heterogeneous systems. The approach allows one to construct new
+objects from old ones: it combines encoding isometries of pure quantum error correcting codes with
+entangled multipartite states and subspaces. The presented methods can be also used to construct
+new pure quantum error correcting codes from certain combinations of old ones. The approach is
+illustrated with various examples including constructions of 2-, 3-, 4-, 5-uniform subspaces. The
+results are then compared with analogous constructions obtained with the use of orthogonal arrays.
+I.
+INTRODUCTION
+Multipartite entanglement is crucial for realization of
+various protocols of quantum information processing [1–
+4].
+One important manifestation of this phenomenon
+is genuine multipartite entanglement (GME) [5–7].
+In
+GME states entanglement is present in every bipartite
+cut of a compound system, which makes them useful
+in communication protocols such as quantum telepor-
+tation and dense coding [8, 9].
+Another interesting
+form is r-uniform (also known as maximal) entangle-
+ment [3, 10, 11]. Each reduction of an r-uniform state
+to r subsystems is maximally mixed. This property is
+closely related to quantum secret sharing [12, 13] and
+quantum error correcting codes (QECCs) [14, 15].
+Recently the notion of entangled subspaces has been
+attracting much attention owing to its potential use in
+quantum information science. It was first described in
+Ref. [16], where the term “completely entangled sub-
+spaces” was coined.
+Later, depending on the form
+of multipartite entanglement present in each state of
+a subspace, several other types were introduced: gen-
+uinely entangled subspaces (GESs) [17], negative par-
+tial transpose (NPT) subspaces [18], r-uniform sub-
+spaces (rUSs) [15].
+In the present paper we concen-
+trate on construction of r-uniform subspaces, mostly
+for heterogeneous systems, i. e., those having differ-
+ent local dimensions.
+There are a number of tools
+for constructing r-uniform states in homogeneous sys-
+tems: graph states [19], elements of combinatorial de-
+sign such as Latin squares [20], symmetric matrices [21],
+orthogonal arrays (OAs) [22] and their variations [23–
+25].
+For construction of r-uniform states in heteroge-
+neous systems OAs were extended to mixed orthogonal
+arrays (MOAs) [26]. Recent developments of this method
+can be found in Refs. [27, 28].
+The main source for
+r-uniform subspaces in homogeneous systems are pure
+quantum error correcting codes [3, 15]. Little is known
+about construction of r-uniform subspaces in heteroge-
+neous systems (the only method we could find in liter-
+∗ kv.antipin@physics.msu.ru
+ature was based on Proposition 12 of Ref. [28]).
+De-
+velopment of new methods of construction of rUSs for
+this case is our main motivation for the present paper.
+R-uniform subspaces in heterogeneous systems have re-
+lation to QECCs over mixed alphabets [29] and quan-
+tum information masking [30]. To our knowledge, for a
+given system the largest possible dimension of rUSs is
+unknown, so building new instances of such subspaces
+could bring some insights in this question.
+We use compositional tools of diagrammatic reason-
+ing [31–33], which allow us to come up with new con-
+structions and provide further instances of states and
+subspaces with important properties. Tensor diagrams
+are widely used in quantum information theory, in par-
+ticular, in theory of QECCs. Recently a framework for
+the construction of new stabilizer QECCs from old ones
+with the use of tensor networks has been presented in
+Ref. [34].
+The paper is organized as follows. In Section II nec-
+essary definitions and some theoretical background are
+given. The main results of the current paper are provided
+in Section III. In Subsection III A we give diagrammatic
+representation of basic properties of rUSs upon which, in
+Subsection III B, we derive the methods of constructing
+rUSs in heterogeneous systems such as glueing several
+subspaces together, eliminating parties, combining pure
+error correcting codes and maximally entangled states
+and subspaces. In Subsection III C we compare our re-
+sults with the ones obtained with the use of the mixed
+orthogonal arrays method. Finally, in Section IV we con-
+clude with discussing possible directions of further re-
+search.
+II.
+PRELIMINARIES
+Let us first give the definition of r-uniform states of an
+n-partite finite-dimensional system with local dimensions
+d1, . . . , dn. Such a system is usually associated with the
+tensor product Hilbert space Cd1 ⊗. . .⊗Cdn. A state |ψ⟩
+in Cd1 ⊗ . . .⊗ Cdn is called r-uniform if all its reductions
+
+2
+FIG. 1. Doubling notation for the process of action of a linear
+operator V on a pure state ψ
+FIG. 2. Reduction of a bipartite pure state ψ to subsystem A
+at least to r parties are maximally mixed, i. e.,
+Tr{i1, ..., ir}c[|ψ⟩⟨ψ|] =
+1
+di1 · . . . · dir
+Ii1, ..., ir
+(1)
+for
+all
+r-element
+subsets
+{i1, . . . , ir}
+of
+the
+set
+{1, . . . , n}. Here {i1, . . . , ir}c denotes the complement
+of the given set in the set of all parties.
+It is clear
+that r-uniform state is also l-uniform for all l < r. By
+the properties of the Schmidt decomposition, the nec-
+essary condition for r-uniform states to exist is that
+di1 ·. . .·dir ⩽ dir+1 ·. . .·din is satisfied for each bipartition
+i1, . . . , ir|ir+1, . . . , in.
+An r-uniform subspace — a subspace of Cd1 ⊗. . .⊗Cdn
+consisting entirely of r-uniform vectors.
+For homogeneous systems, i. e., those having equal
+local dimensions, the existence of r-uniform subspaces
+can be deduced from the existence of certain quan-
+tum error correcting codes (QECCs).
+Recall that a
+QECC ((n, K, d))D is a special K-dimensional subspace
+of
+�
+CD�⊗n such that for each its state any error affect-
+ing not more than a certain number of subsystems can
+be corrected. For a code with distance d = 2t + 1 the
+number is equal to t. In addition, a code with distance d
+can detect d − 1 errors.
+In addition to the ((n, K, d))D notation for general
+QECCs, we will use the designation [[n, k, d]]D for stabi-
+lizer QECCs. While the symbols n and d from the latter
+notation have the same sense as those in the former one,
+the dimension of the codespace for the code [[n, k, d]]D
+is equal to Dk.
+A quantum error correcting code is called pure if
+⟨i| E |j⟩ = 0
+(2)
+for any states |i⟩ , |j⟩ from an orthonormal set spanning
+the code space and for any error operator E with weight
+strictly less than the distance of the code.
+It is known that each pure QECC ((n, K, d))D yields a
+K-dimensional (d − 1)-uniform subspace of (CD)⊗n, and
+vice versa [15].
+To address the case of heterogeneous systems, in the
+present paper we will use encoding isometries of the exist-
+ing pure QECCs in combination with various states and
+FIG. 3. Diagrammatic representation of the maximally mixed
+state (up to the normalization factor).
+subspaces of lower number of parties. A similar approach
+dealing with isometric mapping to entangled subspaces
+proved to be effective in constructing multipartite gen-
+uinely entangled subspaces [35].
+Throughout the paper we use tensor diagrams, in par-
+ticular, we use doubled-process theory notation adopted
+from Ref. [31]. The doubling notation indicates the pas-
+sage from pure state vectors to their associated density
+operators, as shown on Fig. 1.
+To deal also with mixed states, the discarding sym-
+bol (map) is used.
+Applying the discarding map to a
+subsystem of a multipartite state is equivalent to tracing
+out the subsystem, as shown on Fig. 2.
+The adjoint of the discarding map (see Fig. 3) denotes
+the identity operator, which is proportional to the max-
+imally mixed state.
+III.
+RESULTS
+A.
+Basic properties and their diagrammatic
+representation
+We start with pointing at an important basic prop-
+erty of subspaces under consideration. Let |φ⟩ and |χ⟩
+be two mutually orthogonal normalized vectors in an r-
+uniform subspace W. An arbitrary (normalized) linear
+combination |ψ⟩ = α |φ⟩ + β |χ⟩ is also in W, and hence
+its reduction to some r-element subset S of the set of all
+parties yields
+TrSc[|ψ⟩⟨ψ|] = N IS
+= N IS + αβ∗ TrSc[|φ⟩⟨χ|] + βα∗ TrSc[|χ⟩⟨φ|],
+(3)
+where N =
+��
+i∈S di
+�−1, the normalization factor. Con-
+sequently, the last two terms sum up to zero:
+αβ∗ TrSc[|φ⟩⟨χ|] + βα∗ TrSc[|χ⟩⟨φ|] = 0,
+∀ α, β ∈ C,
+|α|2 + |β|2 = 1.
+(4)
+Setting first α real and β imaginary and then both of
+them real, one can deduce that
+TrSc[|φ⟩⟨χ|] = 0.
+(5)
+Now we can formulate this property as
+Lemma 1. For any orthonormal set {|ψ⟩i} spanning r-
+uniform subspace it follows that
+TrSc[|ψi⟩⟨ψj|] ∼ δij IS
+(6)
+for any r-element subset S of the set of all parties.
+
+1TrB|)V
+A)A3
+FIG. 4. Action of V together with tracing out subsystems Sc
+results in a channel ΦS which discards its input and returns
+the maximally mixed state.
+Eq. (6) is known to hold for codewords of pure QECCs
+of distance r + 1, which correspond to r-uniform sub-
+spaces for homogeneous systems, but it is worth stress-
+ing that it is valid for more general case of heterogeneous
+systems.
+Let us think of r-uniform subspaces in terms of isome-
+tries and quantum channels. To each such subspace W
+with dimension K one can relate an isometry V : CK →
+Cd1 ⊗ . . . ⊗ Cdn, which maps an orthonormal basis {|i⟩}
+of CK to some orthonormal set {|ψ⟩i} spanning W:
+V |i⟩ = |ψi⟩ .
+(7)
+Hence, the range of the isometry V coincides with the
+subspace W. As before, let us choose an r-element subset
+S of the set of n parties. Applying the isometry to a state
+in CK with subsequent tracing out n − r subsystems in
+Sc results in action of a quantum channel on the state:
+TrSc[V |φ⟩⟨φ| V †] = ΦS(|φ⟩⟨φ|),
+|φ⟩ ∈ CK.
+(8)
+We
+thus
+obtain
+a
+family
+of
+quantum
+channels
+ΦS : L(CK) → L(Cdi1 ⊗. . .⊗Cdir ), where {i1, . . . , ir} =
+S; one channel for each choice of S. Since subspace W
+is r-uniform, a channel ΦS maps all states of CK to the
+identity on S:
+ΦS(|φ⟩⟨φ|) =
+1
+di1 · . . . · dir
+Ii1, ..., ir,
+|φ⟩ ∈ CK.
+(9)
+In other words, the channels discard the input and map
+everything to the maximally mixed state:
+ΦS(X) =
+Tr[X]
+di1 · . . . · dir
+IS,
+X ∈ L(CK).
+(10)
+By setting X = |i⟩⟨j| in this expression, with the use of
+Eqs. (7), (8), we recover Eq. (6).
+It should be stressed that the above property of map-
+ping to the maximally mixed state holds when the in-
+put dimension of the isometry (and of the corresponding
+channel) is not greater than the dimension of the range
+of the isometry, i. e., the dimension of the r-uniform sub-
+space. In fact, the input dimension can be strictly less
+than that dimension, in which case the isometry takes the
+input states to some subspace of the r-uniform space.
+FIG. 5.
+Construction of a 2-uniform state from an encod-
+ing isometry V of the ((5, 3, 3))3 pure code and a maximally
+entangled state |ψ⟩ in C2 ⊗ C3.
+Diagrammatic representation of Eqs. (8) and (10) is
+shown on Fig. 4, where the symbol ”∼” means that the
+two diagrams are equal up to a scalar factor (the normal-
+ization constant for the maximally mixed state). This
+construction will be crucial in further considerations.
+Now we can choose V to be an encoding isometry of
+some pure quantum error correcting code. Let us try to
+combine this isometry with some states.
+Example: 2-uniform state in heterogeneous systems
+Consider the ((5, 3, 3))3 pure code [36] and its encod-
+ing isometry V . Application of V to one of the parties
+of a bipartite pure state |ψ⟩ in C2 ⊗ C3 yields a 6-partite
+pure state in C2 ⊗
+�
+C3�⊗5, as shown on Fig. 5. The code
+has distance 3, so the code subspace is 2-uniform.
+In
+addition, the code subspace has dimension equal to 3,
+which matches the local dimension of the second party
+of the state |ψ⟩. Therefore, the property of Fig. 4 holds
+in this case with r = 2.
+If the bipartite state |ψ⟩ is maximally entangled, i. e.
+its reduction to the party with local dimension 2 is max-
+imally mixed, then the resulting state in C2 ⊗
+�
+C3�⊗5
+will be 2-uniform. This can be shown diagrammatically.
+One needs to consider the two cases of producing the
+two-party reduction of the state in question: a) all par-
+ties are traced out except some two output subsystems
+of V ; b) all parties are traced out except the first party
+of |ψ⟩ (with dimension 2) and some output subsystem of
+V .
+FIG. 6. The state |ψ⟩ is completely traced out - the part of
+the diagram on the bottom right is a scalar equal to 1.
+
+2
+3
+3
+J
+2
+3
+2
+32
+3
+3
+3
+3
+3
+V
+2
+3S
+S
+l1
+n
+V
+24
+FIG. 7. By the property on Fig. 4 and the maximal entangle-
+ment of |ψ⟩ the resulting state is proportional to I2 ⊗ I3.
+The case a) is presented on Fig. 6: the property on
+Fig. 4 being used, the state |ψ⟩ gets completely traced
+out and the resulting state is proportional to I3 ⊗ I3.
+The case b) is analyzed on Fig. 7: on the first step the
+property on Fig. 4 is used; the second step is due to the
+fact that |ψ⟩ is maximally entangled.
+It is interesting to note that C2⊗
+�
+C3�⊗5 was the small-
+est possible Hilbert space for which a 2-uniform state
+could be constructed with the methods of Ref. [26].
+B.
+Construction of r-uniform subspaces in
+heterogeneous systems
+The simplest method to produce an r-uniform sub-
+space in heterogeneous systems is to ”glue” together two
+r-uniform subspaces in homogeneous systems. By ”glue-
+ing” we mean taking tensor product of the two subspaces:
+this can be done by taking all possible tensor products
+of the vectors spanning the two subspaces, the resulting
+subspace will be spanned by such combinations.
+Lemma 2. Tensor product of an r-uniform subspace and
+a k-uniform subspace is an l-uniform subspace, where l =
+min(r, k).
+Proof. Let W1 be an r-uniform subspace with n parties
+and let W2 be a k-uniform subspace with m parties. Con-
+sider two isometries V1 : HA → HC1 ⊗ · · · ⊗ HCn and
+V2 : HB → HD1 ⊗· · ·⊗HDm, where dim(HA) = dim(W1)
+and dim(HB) = dim(W2). The first one, V1, maps the
+basis states {|i⟩}A of HA to an orthonormal system of
+vectors spanning W1. Similarly, V2 maps the basis states
+{|j⟩}B of HB to vectors spanning W2. Tensor product
+W1 ⊗ W2 is then spanned by the vectors
+(V1 ⊗ V2) (|i⟩A ⊗ |j⟩B) ,
+(11)
+as shown on Fig. 8. Now the property from Fig. 4 can
+be applied when one traces out any n + m − l of the
+parties C1, . . . , Cn, D1, . . . , Dm. As a result, a general
+state |φ⟩ ∈ HA ⊗HB gets completely traced out, and the
+l-party maximally mixed state is produced.
+FIG. 8.
+On the left: action of the isometry V1 ⊗ V2 on a
+particular basis state |i⟩A ⊗ |j⟩B of HA ⊗ HB. On the right:
+action of the isometry V1 ⊗V2 on a general state φ from HA ⊗
+HB, which is equal to a linear combination of basis states
+{|i⟩A ⊗ |j⟩B}. Action of V1 ⊗ V2 on each state in HA ⊗ HB
+generates W1 ⊗ W2, which is l-uniform.
+R-uniform subspaces can be used for r-uniform quan-
+tum information masking [30]. An operation V is said
+to r-uniformly mask quantum information contained in
+states {|i⟩} if it maps them to multipartite states {|ψi⟩}
+whose all reductions to r parties are identical.
+In the
+proof of Lemma 2 an instance of masking has been pro-
+vided: on the right part of Fig. 8 it is shown how each
+state φ from HA ⊗HB is “masked” by the two isometries
+V1 and V2 as an l-uniform state.
+As an example, combining encoding isometries of
+((5, 2, 3))2 and ((5, 3, 3))3 pure codes, by Lemma 2
+we obtain a 6-dimensional 2-uniform subspace of the
+�
+C2�⊗5 ⊗
+�
+C3�⊗5 Hilbert space.
+Can we reduce the number of parties?
+A structure
+similar to the one on Fig. 5 can be used. Let us take the
+encoding isometry V of the [[6, 2, 3]]3 (stabilizer) pure
+code (Ref. [37], Corollary 3.6). The range of the isome-
+try is a 2-uniform subspace of the
+�
+C3�⊗6 Hilbert space,
+which has dimension equal to 32 = 9. Consider a sub-
+space of C2 ⊗ C9, which consists entirely of states max-
+imally entangled with respect to the first party. Such a
+subspace can be easily constructed with the use of Propo-
+sition 3 of Ref. [38]. The dimension of the subspace is
+equal to ⌊ 9
+2⌋ = 4 (Corollary 4 of Ref. [38]). Now let us
+act with V on the second party (the one with dimension
+9) of each state in the subspace. This procedure will gen-
+erate a 4-dimensional subspace of the C2⊗
+�
+C3�⊗6 Hilbert
+space. The analysis, which is similar to that on Figs. 6
+and 7, shows that the resulting subspace is 2-uniform.
+The above construction can be viewed as an illustra-
+tion of subspace masking: only states that belong to a
+specific subspace of C2⊗C9 are masked by V as 2-uniform
+states in C2 ⊗
+�
+C3�⊗6.
+The next property provides an important way of gen-
+erating r-uniform subspaces from those of larger number
+of parties. It can be seen as the extension of Theorem 20
+of Ref. [39] to the case of heterogeneous systems.
+Theorem 1. Let W be an r-uniform subspace of Hilbert
+space with the set S = {d1, . . . , dn} of local dimensions.
+Let r ⩾ 1 and dim(W) = K. Then, for any di ∈ S,
+there exists an r − 1-uniform subspace of Hilbert space
+with local dimensions S \ {di}. The dimension of this
+
+m
+A
+B2
+35
+subspace is equal to diK.
+Proof. Consider an orthonormal set of vectors {|ψk⟩}
+which span W. Each vector |ψk⟩ is r-uniform, and so,
+in particular, its reduction to party i is proportional to
+the maximally mixed operator I{i}.
+Accordingly, the
+Schmidt decomposition of |ψk⟩ with respect to the bi-
+partition ”party i|the rest” reads
+|ψk⟩ =
+1
+√di
+di−1
+�
+j=0
+���φ(k)
+j
+�
+P ⊗
+���χ(k)
+j
+�
+P ,
+(12)
+where {
+���φ(k)
+j
+�
+P } and {
+���χ(k)
+j
+�
+P }, with j = 0, . . . , di − 1
+and k fixed, are two orthonormal sets of vectors in r − 1-
+partite and 1-partite Hilbert spaces with local parties
+P = {1, . . . , n} \ {i} and P = {i}, respectively. In the
+right part of Eq. (12) vectors with the same upper index
+satisfy the orthonormality condition, for example,
+�
+χ(k)
+m
+���χ(k)
+n
+�
+P = δmn.
+(13)
+Now consider an r − 1-element subset J ⊂ P and, for
+some numbers k, s ∈ {1, 2, . . . , K}, take the reduction
+of |ψk⟩⟨ψs| to the set J ∪ {i}:
+TrP \J [|ψk⟩⟨ψs|]
+= 1
+di
+di−1
+�
+j, l=0
+TrP \J
+����φ(k)
+j
+��
+φ(s)
+l
+���
+P
+�
+⊗
+���χ(k)
+j
+��
+χ(s)
+l
+���
+P .
+(14)
+By
+Lemma
+1,
+this
+reduction
+is
+proportional
+to
+δks IJ∪{i} = δks IJ ⊗ I{i}, and hence
+1
+di
+di−1
+�
+j, l=0
+TrP \J
+����φ(k)
+j
+��
+φ(s)
+l
+���
+P
+�
+⊗
+���χ(k)
+j
+��
+χ(s)
+l
+���
+P
+= δks
+1
+dJ
+IJ ⊗ 1
+di
+I{i},
+(15)
+where dJ is the product of local dimensions of the parties
+in J. Multiplying both parts of this equality by
+�
+χ(k)
+m
+���
+and
+���χ(s)
+n
+�
+, with the use of condition (13), we obtain
+TrP \J
+����φ(k)
+m
+��
+φ(s)
+n
+���
+P
+�
+= δksδmn
+1
+dJ
+IJ.
+(16)
+Taking trace over subsystem J in the last equation, one
+can see that the set of diK vectors {
+���φ(t)
+j
+�
+P }, j =
+0, . . . , di − 1, t = 1, . . . , K is an orthonormal system.
+Since Eq. (16) holds for any choice of an r − 1-element
+subset J ⊂ P, the states {
+���φ(t)
+j
+�
+P } are r − 1-uniform. In
+addition, from the same equation one can see that any lin-
+ear combination of these vectors is an r−1-uniform state.
+Therefore, the system {
+���φ(t)
+j
+�
+P } spans a diK-dimensional
+r − 1-uniform subspace. From Eq. (12) it follows that
+diTr{i}
+� K
+�
+s=1
+|ψs⟩⟨ψs|
+�
+(17)
+is the orthogonal projector on the subspace in question.
+A practical way to obtain an orthonormal system of
+vectors spanning the subspace defined by the projector
+in Eq. (17) is as follows. Let us suppose that party i
+with local dimension di is being eliminated, just as in the
+condition of Theorem 1. Consider an orthonormal system
+of one party vectors {|vj⟩}, j = 0, . . . , di−1, which spans
+Cdi and such that each partial scalar product
+���µ(s)
+j
+�
+≡ ⟨vj|ψs⟩ ,
+j = 0, . . . , di − 1,
+s = 1, . . . , K,
+(18)
+is a non-null vector, where the only input of the
+(co)vector ⟨vj| is joined with the i-th output of the vec-
+tor |ψs⟩ in each partial scalar product. Then diK vectors
+{
+���µ(s)
+j
+�
+} span the subspace in question. Indeed, the vec-
+tors are mutually orthogonal:
+�
+µ(t)
+l
+���µ(s)
+j
+�
+= ⟨ψt|vl⟩ ⟨vj|ψs⟩
+= ⟨vj| TrP [|ψs⟩⟨ψt|] |vl⟩ = 1
+di
+δts ⟨vj|vl⟩
+= 1
+di
+δtsδlj,
+(19)
+where the third equality follows from r-uniformity of the
+original vectors {|ψs⟩} and Lemma 1.
+Consequently,
+{√di
+���µ(s)
+j
+�
+} is an orthonormal system, and the corre-
+sponding projector
+�
+j, s
+�
+di
+���µ(s)
+j
+��
+µ(s)
+j
+���
+�
+di = di
+�
+j, s
+⟨vj|ψs⟩ ⟨ψs|vj⟩
+= diTr{i}
+��
+s
+|ψs⟩⟨ψs|
+�
+(20)
+coincides with the one in Eq. (17).
+As an example, from the obtained above 4-dimensional
+2-uniform subspace of C2 ⊗
+�
+C3�⊗6 Hilbert space one can
+produce a 12-dimensional 1-uniform subspace of C2 ⊗
+�
+C3�⊗5 Hilbert space by eliminating one of the parties
+with local dimension 3.
+When an initial r-uniform subspace is spanned just by
+1 vector, the described above practical method becomes
+similar to Proposition 12 of Ref. [28].
+As we have seen from Lemma 2, glueing two uniform
+subspaces together doesn’t increase the uniformity pa-
+rameter of the resulting subspace. This is in accordance
+
+6
+with the general principle that local operations cannot
+produce any entanglement over that present in origi-
+nal states. Let us show that making use of additional
+resources such as maximally entangled states can lead
+to larger uniformity parameters of the produced states
+and subspaces in comparison with original ones. At first
+we consider uniform subspaces in homogeneous systems,
+i. e., those corresponding to pure quantum error correct-
+ing codes.
+Recall that the parameters of a ((n, K, d))D code sat-
+isfy the inequality [36, 40]
+K ⩽ Dn−2(d−1),
+(21)
+which is called the quantum Singleton bound.
+If pa-
+rameters of a code saturate the bound in Eq. (21),
+the code is called quantum maximum distance separable
+code (QMDS) [36]. It is known that all QMDS codes are
+pure [36].
+In Ref. [15] an important observation about QMDS
+codes was made. We reformulate it here in a more general
+form and provide the proof.
+Lemma 3 (An observation in the proof of Proposition
+7 of Ref. [15]). Let ((n, K, d)) be a QMDS code. Con-
+sider the projector P = �K
+s=1 |ψs⟩⟨ψs| on the codespace,
+where {|ψs⟩} is an orthonormal set of vectors that span
+the codespace. Then each reduction of P to n − (d − 1)
+parties is proportional to the maximally mixed operator
+In−(d−1).
+Proof. According to Theorem 20 of Ref. [39] (or The-
+orem 1 here), tracing out one party yields a projector
+on a KD-dimensional subspace of
+�
+CD�⊗n−1, hence af-
+ter d − 1 such steps of tracing out we have a projec-
+tor on a subspace of
+�
+CD�⊗(n−(d−1)) with dimension
+KDd−1 = Dn−(d−1), i. e.
+the projector on the whole
+space
+�
+CD�⊗(n−(d−1)), the identity operator.
+We stress that this holds only for QMDS codes - those
+with K = Dn−2(d−1), and not for other pure codes.
+With the use of QMDS codes we can now formulate
+the following property.
+Theorem 2. Let ((n1, K1, d1))D1 and ((n2, K2, d2))D2
+be two QMDS codes with K1 = K2 ≡ K > 1. Denote
+r1 ≡ d1 − 1 and r2 ≡ d2 − 1.
+Then there exists an
+l-uniform state in
+�
+CD1�⊗n1 ⊗
+�
+CD2�⊗n2 Hilbert space
+with
+l = min (n1 − r1, n2 − r2, r1 + r2 + 1) .
+(22)
+Proof. Since the two codes are pure, there are two sub-
+spaces related to them: an r1-uniform subspace W1 of
+�
+CD1�⊗n1 and an r2-uniform subspace W2 of
+�
+CD2�⊗n2
+such that dim(W1) = dim(W2) = K.
+Consider two isometries V1 : CK →
+�
+CD1�⊗n1 and
+V2 : CK →
+�
+CD2�⊗n2 whose ranges coincide with W1 and
+FIG. 9. Steps to prove l-uniformity of the state |ψ⟩S1S2.
+W2, respectively. Now let us take any bipartite maxi-
+mally entangled state |φ⟩AB in CK ⊗ CK and construct
+the state
+|ψ⟩S1S2 = (V1 ⊗ V2) |φ⟩AB ,
+(23)
+which belongs to
+�
+CD1�⊗n1 ⊗
+�
+CD2�⊗n2. Here S1 and S2
+denote the sets of the output parties of the isometries V1
+and V2, respectively. We claim that the state |ψ⟩S1S2 is
+l-uniform, with l as in Eq. (22).
+The underlying principle is shown on Fig 9. Let us as-
+sume that all subsystems of |ψ⟩S1S2 are traced out except
+some m1 output subsystems of isometry V1 and some m2
+output subsystems of isometry V2 , as shown on Fig 9,
+a). If, for example, m2 ⩽ r2, the rule from Fig. 4 can be
+applied and we arrive at the situation shown on Fig 9, b),
+where isometry V2 is eliminated and the state |φ⟩AB gets
+partially traced out. Next, the state |φ⟩AB is maximally
+entangled, and so its reduction to A is the maximally
+mixed operator
+1
+K IA. As a result, isometry V1 acts on
+the identity operator IA, as shown on Fig 9, c). The steps
+b − c), without taking into account the trace over output
+parties of V1, can be written as
+V1TrB{|φ⟩⟨φ|AB}V †
+1 = 1
+K V1V †
+1 = 1
+K PW1,
+(24)
+where PW1 – the orthogonal projector on subspace W1,
+the first equality follows from maximal entanglement of
+|φ⟩AB, the second one – from the fact that the isometry
+V1 has subspace W1 as its range. Now if m1 ⩽ n1 − r1
+then, by Lemma 3, performing the trace over n1 − m1
+output subsystems of V1 (Fig 9, c)) produces the maxi-
+mally mixed state of m1 parties (Fig 9, d)) in addition to
+the maximally mixed state of m2 parties obtained earlier
+in the first step. We conclude that if m1 ⩽ n1 − r1 and
+m2 ⩽ r2, the reduced state of m1 + m2 parties is maxi-
+mally mixed. The roles of V1 and V2 can be interchanged,
+and we obtain that if m1 ⩽ r1 and m2 ⩽ n2 − r2, the
+reduced state is maximally mixed.
+Now we need to determine the maximal number l such
+that any partition of l = m1 +m2 into m1 output parties
+
+m2
+m1
+m2
+元.元
+元.元
+元.元
+Vi
+V2
+Vi
+A
+B
+A
+B
+a)
+6)
+m1
+m2
+m1
+m2
+V1
+A
+c)
+d)7
+of V1 and m2 output parties of V2 yields, after perform-
+ing the trace over the rest n1 + n2 − l subsystems, the
+maximally mixed state of l parties. Let us first consider
+partitions in which m2 = 0. In this case the maximal
+value of m1, for which the scheme on Fig. 9 can still be
+applied, is n1 − r1, as it was shown above. This number
+is then an upper bound on l. Interchanging the roles of
+V1 and V2 and setting m1 = 0, we obtain another bound,
+n2 − r2, and hence
+l ⩽ min(n1 − r1, n2 − r2).
+(25)
+Next, let us assume that r2 > r1. Let ⌈x⌉ denote the
+ceiling of x and ⌊x⌋ denote the floor of x. Consider a par-
+tition of l into m1 = ⌊ l
+2⌋ and m2 = ⌈ l
+2⌉. If ⌈ l
+2⌉ > r2 then
+⌊ l
+2⌋ > r1 also holds, and one cannot apply the scheme on
+Fig. 9 since neither V2 nor V1 can be eliminated in the
+first step a)-b) with the use of the rule from Fig. 4. Con-
+sequently, we can take into account only those values of
+l that satisfy ⌈ l
+2⌉ ⩽ r2. Accordingly, consider a partition
+of l into m2 = ⌈ l
+2⌉+α and m1 = ⌊ l
+2⌋−α for some integer
+α > 0 such that ⌈ l
+2⌉ + α = r2 + 1. In this case V2 cannot
+be eliminated in the first step of scheme on Fig. 9. On
+the other hand, the scheme can be initiated by apply-
+ing the rule from Fig. 4 with respect to V1 on condition
+that m1 = ⌊ l
+2⌋ − α ⩽ r1. If the condition is satisfied, in
+the step c)-d) of the scheme (with interchanged V1 and
+V2) the reduction of PW2 to m2 parties will be maximally
+mixed by Lemma 3, since m2 ⩽ l ⩽ n2 − r2 by Eq. (25).
+This principle continues to work for greater values of α
+(but bounded by the condition ⌈ l
+2⌉ + α = m2 ⩽ n2 − r2),
+as m1 gets smaller. It is clear that partitions with α < 0
+will also work, as the step a)-b) will be initiated with the
+use of V2. To sum up, the maximal possible value of l
+satisfies
+� l
+2
+�
++ α = r2 + 1,
+� l
+2
+�
+− α = r1.
+(26)
+Adding these two equalities, we obtain
+l =
+� l
+2
+�
++
+� l
+2
+�
+= r1 + r2 + 1.
+(27)
+When r1 = r2 ≡ r, we can choose (odd) l such that
+⌈ l
+2⌉ = r + 1 and ⌊ l
+2⌋ = r. For partitions with m1 = ⌊ l
+2⌋
+and m2 = ⌈ l
+2⌉ and, vice versa, m1 = ⌈ l
+2⌉ and m2 =
+⌊ l
+2⌋, the scheme on Fig. 9 is initiated with the use of
+V1 and V2, respectively. For all other partitions, which
+can be parameterized with integer α as m1 = ⌈ l
+2⌉ + α
+and m2 = ⌊ l
+2⌋ − α or vice versa, the scheme also works
+by the analysis similar to that in the above paragraph.
+Consequently,
+l =
+� l
+2
+�
++
+� l
+2
+�
+= 2r + 1,
+(28)
+which is just a special case of Eq. (27).
+Immediate application of Theorem 2, with the use
+of the correspondence between r-uniform states and 1-
+dimensional pure quantum codes, produces
+Corollary 2.1. Let ((n, K, d))D be a QMDS code with
+K > 1. Then there exists a pure ((2n, 1, d′))D code with
+distance
+d′ = min(n − d + 2, 2d).
+(29)
+As an example, consider a ((4, 4, 2))2 code, which can
+be the stabilizer [[4, 2, 2]]2 code obtained from the well-
+known [[5, 1, 3]]2 code with the use of Theorem 20 of
+Ref. [39]. Combining [[4, 2, 2]]2 with itself produces an
+8-qubit 3-uniform state. It is known that ⌊ n
+2 ⌋-uniform
+states of n qubits (absolutely maximally entangled (AME)
+states) don’t exist for n > 6 [3, 39, 41, 42], so the con-
+structed state has maximal possible uniformity parame-
+ter.
+To give an example with heterogeneous systems, let
+us combine pure codes ((4, 4, 2))2 and ((5, 4, 3))4. The
+latter code can be produced by tensoring ((5, 2, 3))2 with
+itself (Theorem 14 of Ref. [36]). By the construction in
+the proof of Theorem 2, the two codes yield a 3-uniform
+state in
+�
+C2�⊗4 ⊗
+�
+C4�⊗5. We can then obtain r-uniform
+subspaces by eliminating some parties of this state, but
+in this case some produced subspaces will demonstrate
+better values of r than those predicted by Theorem 1
+owing to the following observation.
+Corollary
+2.2.
+Let
+((n1, K1, d1))D1
+and
+((n2, K2, d2))D2
+be
+two
+QMDS
+codes
+with
+K1 = K2
+≡ K
+> 1.
+Denote r1
+≡ d1 − 1 and
+r2 ≡ d2 − 1.
+Then for any integers 0 ⩽ α ⩽ r1 and
+0 ⩽ β ⩽ r2 there exists an l-uniform subspace W of
+�
+CD1�⊗n1−α ⊗
+�
+CD2�⊗n2−β Hilbert space such that
+dim(W) = D α
+1 Dβ
+2 ,
+l = min(n1 − r1 − α, n2 − r2 − β,
+r1 + r2 + 1 − α − β).
+(30)
+Proof. By Theorem 2, we can construct a state in
+�
+CD1�⊗n1 ⊗
+�
+CD2�⊗n2 Hilbert space with the uniformity
+parameter given by Eq. (22). Next, we eliminate α + β
+parties of this state by the procedure described after the
+proof of Theorem 1. Let us take α orthonormal systems
+of vectors {
+���v(µ)
+i
+�
+}, µ = 1, . . . , α, i = 0, . . . , D1−1, each
+system being a basis for the corresponding CD1 Hilbert
+FIG. 10. Construction of the states which span the l-uniform
+subspace W of
+�
+CD1�⊗n1−α ⊗
+�
+CD2�⊗n2−β Hilbert space.
+
+Vi8
+space. Similarly, we take β orthonormal systems of vec-
+tors {
+���w(ν)
+j
+�
+}, ν = 1, . . . , β, j = 0, . . . , D2 − 1, each in
+its own CD2 Hilbert space. Next, we pick some specific
+vectors
+���v(1)
+i1
+�
+, . . . ,
+���v(α)
+iα
+�
+, one from each system, and
+eliminate α output parties of the isometry V1 by joining
+them with the inputs of the chosen vectors. Similarly,
+we pick β specific vectors
+���w(1)
+j1
+�
+, . . . ,
+���w(β)
+jβ
+�
+and elim-
+inate β output parties of the isometry V2 (see Fig. 10,
+the indices of vectors v, w are omitted). As a result, we
+obtain
+�
+v(1)
+i1
+��� . . .
+�
+v(α)
+iα
+���
+�
+w(1)
+j1
+��� . . .
+�
+w(β)
+jβ
+��� (V1 ⊗ V2) |φ⟩AB , (31)
+one of the D α
+1 Dβ
+2 states that span the subspace W of
+�
+CD1�⊗n1−α⊗
+�
+CD2�⊗n2−β Hilbert space. All such states
+are hence indexed by the numbers i1, . . . , iα, j1, . . . , jβ,
+which represent the correspondence between tuples of
+vectors v, w and the basis states of W.
+The uniformity of the state in Eq. (31) can be analyzed
+with the use of Fig. 10 and the same reasoning as in the
+proof of Theorem 2. The vectors v and w take up α and β
+positions out of n1 and n2 output parties of the isometries
+V1 and V2, respectively. The parties in these positions
+cannot be traced out, and this results in modifying the
+bounds on l in Eq. (25):
+l ⩽ min(n1 − r1 − α, n2 − r2 − β).
+(32)
+Eq. (26) is also modified, with r1 and r2 replaced by
+r1 − α and r2 − β, respectively. As a result, we obtain
+the expression for l in Eq. (30).
+Earlier a 3-uniform state in
+�
+C2�⊗4 ⊗
+�
+C4�⊗5 was
+obtained with the use of Theorem 2 from pure codes
+((4, 4, 2))2 and ((5, 4, 3))4. Eliminating one party with
+dimension 2 and one party with dimension 4, or, in terms
+of Corollary 2.2, setting α = β = 1, we produce an
+8-dimensional 2-uniform subspace of
+�
+C2�⊗3 ⊗
+�
+C4�⊗4
+Hilbert space. We stress that the original state has spe-
+cial structure and, as a result, after the elimination of 2
+parties the produced subspace has higher value l = 2 in
+comparison with l = 1 predicted by Theorem 1.
+C.
+Comparison with mixed orthogonal arrays
+method and further constructions
+Mixed orthogonal arrays (MOAs) [43, 44] in its specific
+form, irredundant MOAs (IrMOAs) [26], have become a
+powerful tool in construction of r-uniform states in het-
+erogeneous systems [26–28]. In this subsection we present
+several applications of the compositional approach that
+allow us to reproduce or extend some results obtained
+with the use of IrMOAs (in terms of the minimal num-
+ber of parties for a given uniformity parameter). Such a
+comparison also reveals some weaknesses of the presented
+in this paper approach.
+In general it becomes more difficult to find examples
+of r-uniform states in heterogeneous systems when the
+number of parties gets smaller.
+Let us consider some
+results from Ref. [27].
+Proposition 1 (Corollary 3.2 of Ref. [27].). 2-
+uniform states exist for the following configura-
+tions:
+1. C3 ⊗
+�
+C2�⊗N for N ⩾ 8.
+2.
+�
+C3�⊗2 ⊗
+�
+C2�⊗N for N ⩾ 12.
+3.
+�
+C3�⊗3⊗
+�
+C2�⊗N for N ⩾ 11 and
+�
+C3�⊗4⊗
+�
+C2�⊗N
+for N ⩾ 10.
+We can reproduce the first result for N = 8. The pro-
+cedure is as follows. The pure code ((5, 2, 3))2 is com-
+bined with itself by the construction of Theorem 2, which
+results in a 10-qubit 3-uniform state, i. e. a ((10, 1, 4))2
+pure code (Corollary 2.1). Next, by eliminating two par-
+ties, by Corollary 2.2 we obtain a 4-dimensional 8-qubit
+2-uniform subspace, i. e. a pure ((8, 4, 3))2 code. Now we
+have an encoding isometry which maps vectors from C4
+to the 2-uniform code space (briefly, the ”8-qubit isome-
+try”). Finally, we can take a maximally entangled state
+in C3⊗C3 and act on one of its parties with the obtained
+isometry, and the construction here will be similar to the
+one presented on Fig. 5. The resulting state, which be-
+longs to C3⊗
+�
+C2�⊗8, is 2-uniform. For larger values of N
+we can use the same auxiliary state and various combina-
+tions of isometries and, if necessary, glue them together
+with the use of Lemma 2. As an example, the isometry
+for N = 10, which maps C4 to 10-qubit 2-uniform sub-
+space, can be obtained from glueing the subspace of the
+code [[5, 1, 3]]2 with itself. In other words, 10-qubit isom-
+etry is obtained from glueing 5-qubit isometry with itself.
+Next, by eliminating one party of a ((8, 1, 4))2 state, we
+obtain an isometry which maps vectors from C2 to the
+2-uniform 7-qubit space (the ”7-qubit isometry”).
+By
+the same procedure, from the state ((10, 1, 4))2 we ob-
+tain the 9-qubit isometry. Now, the isometry for N = 12
+can be obtained from glueing 7-qubit and 5-qubit isome-
+tries, for N = 13 – from 5-qubit and 8-qubit ones, and
+so forth. We haven’t found any appropriate isometries to
+construct the states with N = 9 and N = 11 (those that
+we’ve found have input dimension 2, which is less than
+the local dimension of the second party of the auxiliary
+state). To conclude, we cannot reproduce the first result
+of Proposition 1 only for N = 9 and N = 11 with the
+current approach.
+The second result from Proposition 1 is harder to re-
+produce. The reason for that is as follows: we can take
+a 4-qutrit 2-uniform state, which can be, for example,
+the graph state of Ref. [19] or a QMDS code [[4, 0, 3]]3
+from Corollary 3.6 of Ref. [37], but now we need to act
+with an isometry on its two parties, i. e., on a compound
+subsystem with local dimension 3 × 3 = 9, as shown on
+Fig. 11. The 8-qubit isometry that was used before is
+
+9
+FIG. 11. An isometry V acting on a joint subsystem of two
+parties with local dimensions 3.
+not appropriate here since it has input dimension equal
+to 4, which is less than the output dimension of the two
+combined parties. A proper isometry can be constructed
+from other error correcting codes with the use of the split-
+ting property for r-uniform subspaces, which is a direct
+consequence of the splitting method for r-uniform states
+appeared earlier in Refs. [26, 28].
+Lemma 4. Let W be an r-uniform subspace of Hilbert
+space with the set S = {d1, . . . , dn} of local dimensions.
+Let di = d′
+id′′
+i for some i: 1 ⩽ i ⩽ n and some integer
+d′
+i, d′′
+i > 1. Then there exists an r-uniform subspace of
+Hilbert space with the set of local dimensions given by
+{d′
+i, d′′
+i } ∪ [S \ {di}] and having the same dimension as
+the original subspace.
+Proof. The subspace in question can be obtained from
+the original one by splitting the i-th subsystem of each
+state in W into two smaller ones, i′ and i′′, with local
+dimensions d′
+i and d′′
+i , respectively. Each newly obtained
+state is r-uniform, as follows from the splitting method
+described in Refs. [26, 28].
+Consequently, a subspace,
+which consists of such states, is r-uniform.
+Now we can return to the construction of a 2-uniform
+state in
+�
+C3�⊗2 ⊗
+�
+C2�⊗12. Consider a pure ((6, 16, 3))4
+code (Corollary 3.6 of Ref. [37]).
+By splitting each
+ququart into 2 qubits, by Lemma 4, the code is converted
+into a pure ((12, 16, 3))2 code. Since its encoding isome-
+try (the ”12-qubit isometry”) has input dimension equal
+to 16, we can act with it on a compound subsystem con-
+sisting of two combined parties of the state [[4, 0, 3]]3 (see
+Fig. 11). The resulting state is 2-uniform. The isome-
+tries for larger N can be obtained from glueing the 12-
+qubit isometry with the described above isometries. As
+an example, a 17-qubit isometry is obtained from glue-
+ing the 12-qubit and the 5-qubit ones (it doesn’t matter
+that the input dimension of the 5-qubit isometry is 2 –
+the input dimension of the 12-qubit isometry is 16, and
+the resulting one will have the input dimension equal to
+16 × 2 = 32).
+N = 16 is obtained from glueing the
+8-qubit isometry with itself. All other numbers N ⩾ 18
+can be otained similarly. In addition, N = 14 can be con-
+structed from splitting the code [[7, 3, 3]]4 (Corollary 3.6
+of Ref. [37]). We cannot reproduce the second result of
+Proposition 1 only for N = 13 and N = 15.
+As for the third result of Proposition 1, 2-uniform
+states in
+�
+C3�⊗3⊗
+�
+C2�⊗N can be obtained with action of
+the described above isometries on one party of the state
+[[4, 0, 3]]3. As earlier, the cases N = 9 and N = 11 are
+not covered by our approach, but we can construct a state
+with N = 8, which extends the proposition. The result
+for uniform states in
+�
+C3�⊗4 ⊗
+�
+C2�⊗N can be substan-
+tially extended. Consider a code [[4, 0, 3]]6, for example,
+from Corollary 3.6 of Ref. [37]. By Lemma 4, by split-
+ting each subsystem with local dimension 6 into qubit
+and qutrit subsystems, we obtain a 2-uniform state in
+�
+C3�⊗4 ⊗
+�
+C2�⊗4. The case N ⩾ 5 is trivial: we can glue
+the state [[4, 0, 3]]3 with a 2-uniform state of N qubits,
+which exists for N ⩾ 5 and can be obtained, for exam-
+ple, from graph states constructions. These observations
+extend the proposition from N = 10 to N = 4.
+Gathering the above results, we can formulate
+Proposition 2 (Combination of Corollary 3.2 of Ref. [27]
+with the current approach). 2-uniform states exist for the
+following configurations:
+1. C3 ⊗
+�
+C2�⊗N for N ⩾ 8.
+2.
+�
+C3�⊗2 ⊗
+�
+C2�⊗N for N ⩾ 12.
+3.
+�
+C3�⊗3 ⊗
+�
+C2�⊗N for N = 8 and N ⩾ 10 and
+�
+C3�⊗4 ⊗
+�
+C2�⊗N for N ⩾ 4.
+Let us also analyze some results of Ref. [28].
+Proposition 3 (Theorem 9 of Ref. [28]). For any d > 2,
+the following holds.
+1. There exists a 2-uniform state in
+�
+C2�⊗2 ⊗
+�
+Cd�⊗N
+for any N ⩾ 7 and N ̸= 4d + 2, 4d + 3.
+2. There exists a 2-uniform state in C2 ⊗
+�
+Cd�⊗N for
+any N ⩾ 5.
+We can start with the 2-uniform subspace of the code
+[[5, 1, 3]]2 and act with a proper isometry on three sub-
+systems, i. e. on a joint system of dimension 8, of each
+vector in the code. Therefore, in addition to having a
+2-uniform subspace as its range, an appropriate isometry
+must have input dimension greater or equal 8. The code
+family [[N, N − 4, 3]]d, 4 ⩽ N ⩽ d2 + 1, d > 2 (Corol-
+lary 3.6 of Ref. [37]) provides us with proper isometries
+for 6 ⩽ N ⩽ 10. In addition, the isometry with 5 out-
+put parties, which corresponds to [[5, 1, 3]]d, only works
+when d ⩾ 8, since in this case its input dimension is equal
+to d. Isometries for all other numbers, N > 10, can be
+obtained by glueing the codes with N < 10 (Lemma 2).
+As a result, we lift the constraint N ̸= 4d + 2, 4d + 3 and
+obtain 2-uniform subspaces instead of just states.
+We can only reproduce the second result of Proposi-
+tion 3. All the described above isometries, this time in-
+cluding the one with N = 5, can be used to act on one
+party of a maximally entangled state in C2 ⊗ C2.
+Instead of a maximally entangled state in C2 ⊗ C2 we
+could use maximally entangled subspaces of C2 ⊗ Cp,
+
+3
+3
+2
+2
+2
+A
+3
+3
+3
+310
+which have dimension equal to ⌊ p
+2⌋, by Corollary 4 of
+Ref. [7]. Now the isometry, which corresponds to code
+[[N, N − 4, 3]]d, acts on a party with local dimension p
+of each state in the maximally entangled subspace. The
+input dimension of the isometry, dN−4, hence must be
+greater or equal p, and we have the condition
+N ⩾ 4 + logd p.
+(33)
+Summing the results, we can formulate the extension
+of Proposition 3
+Proposition 4. The following holds.
+1. For 2 < d ⩽ 8 there exists a 2-uniform subspace of
+�
+C2�⊗2 ⊗
+�
+Cd�⊗N with dimension 2 for any N ⩾ 6.
+2. For d > 8 there exists a 2-uniform subspace of
+�
+C2�⊗2 ⊗
+�
+Cd�⊗N with dimension 2 for any N ⩾ 5.
+3. for d > 2 and p ⩾ 2 there exists a 2-uniform sub-
+space of C2 ⊗
+�
+Cd�⊗N with dimension ⌊ p
+2⌋ for any
+N ⩾ 4 + logd p.
+The above examples show that the presented approach
+is more effective in constructing r-uniform states and
+subspaces with larger local dimensions, i. e., qutrits or
+higher. Indeed, there are not many qubit isometries for
+a given value of the uniformity parameter, and, in repro-
+ducing some results of Proposition 1, we had to resort
+to splitting the codes of higher dimensionality. A simi-
+lar tendency was observed in Ref. [35] where genuinely
+entangled subspaces were constructed with the isometric
+mapping method: when local dimension goes to infinity,
+the dimension of the obtained subspaces asymptotically
+approaches the maximal possible value.
+Finally, let us provide some constructions with higher
+values of the uniformity parameter.
+Consider the pure QMDS code [[10, 4, 4]]3 from The-
+orem 13 of Ref. [45].
+From Corollary 2.2 with α =
+β = 1, we obtain a 5-uniform 9-dimensional subspace
+of
+�
+C3�⊗18. The corresponding isometry V has the input
+dimension equal to 9. Let us take a maximally entan-
+gled subspace of C2 ⊗ C9, which has dimension equal to
+⌊ 9
+2⌋ = 4 (Corollary 4 of Ref. [7]). Action of V on the sec-
+ond party of each state in this subspace yields a 5-uniform
+4-dimensional subspace of C2 ⊗
+�
+C3�⊗18. With the same
+isometry V we could act instead on the joint subsystem
+of three parties of each state in the code space [[5, 1, 3]]2,
+and this procedure yields a 5-uniform 2-dimensional sub-
+space of
+�
+C2�⊗2 ⊗
+�
+C3�⊗18.
+Consider the [[10, 0, 6]]4 code, which can be obtained
+from the classical [10, 5, 6] MDS code over GF(16) of
+Ref. [46] by the correspondence between stabilizer QMDS
+codes and self-dual classical MDS codes (Theorem 15 of
+Ref. [47], see also Proposition 15 of Ref. [15]). By elimina-
+tion of one party a code ((9, 4, 5))4 is constructed (The-
+orem 20 of Ref. [39]).
+By splitting the latter code we
+obtain a ((18, 4, 5))2 code whose encoding isometry has
+the input dimension equal to 4. Applying this isometry
+to one of the parties of a maximally entangled state in
+C3 ⊗ C3 produces a 4-uniform state in C3 ⊗
+�
+C2�⊗18.
+IV.
+DISCUSSION
+In this paper we’ve shown how new r-uniform states
+and subspaces can be constructed from combining al-
+ready known quantum error correcting codes, (maxi-
+mally) entangled states and subspaces.
+The isometric
+mapping method played the key role here: one takes an
+isometry, which, as its range, has a subspace with some
+useful property, and applies it to states or subspaces,
+perhaps with some other interesting property. This ap-
+proach allowed us to complement some results which were
+obtained with the mixed orthogonal arrays method. It
+would be interesting to continue this parallel with OAs.
+One example in this direction could be analyzing encod-
+ing isometries of the QECCs obtained with OAs, for in-
+stance, the ones from Ref. [25]. This could potentially
+lead to new OA and MOA constructions.
+The advantage of the presented approach is its exper-
+imental accessibility: whenever one can realize encoding
+isometries of QECCs as well as prepare auxiliary entan-
+gled states, one can construct uniform states in accor-
+dance with the described above procedures.
+The dis-
+advantage of the approach is that it doesn’t utilize the
+internal structure of the combined objects beyond their
+uniformity property. Taking more structural properties
+into account could result in constructing more classes of
+useful states and subspaces such as, for example, AME
+states, the ones we couldn’t produce with the current
+approach.
+This observation suggests another direction
+of further research: how to combine several QECCs in
+the most efficient way, with taking their specific proper-
+ties into account, to obtain a new QECC with “good”
+characteristics (in the sense similar to the recent “good
+quantum codes” constructions [48, 49]). We stress that
+the distance of the codes composed by the procedure of
+Theorem 2 doesn’t scale with the number of codes be-
+ing combined: the distance of the resulting code will al-
+ways be upper-bounded by the minimum of the number
+in Eq. (25) taken over all the codes being combined.
+It also would be interesting to apply the isometric map-
+ping method to construction of multipartite subspaces
+with another useful property – distillability and closely
+related to it non-positivity of partial transpose across
+each bipartition (distillable and NPT subspaces). This
+direction of research could complement the results ob-
+tained in Refs. [18, 50–52].
+ACKNOWLEDGMENTS
+The author thanks M. V. Lomonosov Moscow State
+University for supporting this work.
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diff --git a/FdE1T4oBgHgl3EQfXAQH/content/tmp_files/load_file.txt b/FdE1T4oBgHgl3EQfXAQH/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,975 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf,len=974
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='03120v1  [quant-ph]  8 Jan 2023  On generating r-uniform subspaces with the isometric mapping method  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Antipin∗  Faculty of Physics, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Lomonosov Moscow State University,  Leninskie gory, Moscow 119991, Russia  (Dated: January 10, 2023)  We propose a compositional approach to construct subspaces consisting entirely of r-uniform  states, including the ones in heterogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The approach allows one to construct new  objects from old ones: it combines encoding isometries of pure quantum error correcting codes with  entangled multipartite states and subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The presented methods can be also used to construct  new pure quantum error correcting codes from certain combinations of old ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The approach is  illustrated with various examples including constructions of 2-, 3-, 4-, 5-uniform subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The  results are then compared with analogous constructions obtained with the use of orthogonal arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  INTRODUCTION  Multipartite entanglement is crucial for realization of  various protocols of quantum information processing [1–  4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  One important manifestation of this phenomenon  is genuine multipartite entanglement (GME) [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  In  GME states entanglement is present in every bipartite  cut of a compound system, which makes them useful  in communication protocols such as quantum telepor-  tation and dense coding [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Another interesting  form is r-uniform (also known as maximal) entangle-  ment [3, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Each reduction of an r-uniform state  to r subsystems is maximally mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' This property is  closely related to quantum secret sharing [12, 13] and  quantum error correcting codes (QECCs) [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Recently the notion of entangled subspaces has been  attracting much attention owing to its potential use in  quantum information science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' It was first described in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [16], where the term “completely entangled sub-  spaces” was coined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Later, depending on the form  of multipartite entanglement present in each state of  a subspace, several other types were introduced: gen-  uinely entangled subspaces (GESs) [17], negative par-  tial transpose (NPT) subspaces [18], r-uniform sub-  spaces (rUSs) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  In the present paper we concen-  trate on construction of r-uniform subspaces, mostly  for heterogeneous systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=', those having differ-  ent local dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  There are a number of tools  for constructing r-uniform states in homogeneous sys-  tems: graph states [19], elements of combinatorial de-  sign such as Latin squares [20], symmetric matrices [21],  orthogonal arrays (OAs) [22] and their variations [23–  25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  For construction of r-uniform states in heteroge-  neous systems OAs were extended to mixed orthogonal  arrays (MOAs) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Recent developments of this method  can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The main source for  r-uniform subspaces in homogeneous systems are pure  quantum error correcting codes [3, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Little is known  about construction of r-uniform subspaces in heteroge-  neous systems (the only method we could find in liter-  ∗ kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='antipin@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='ru  ature was based on Proposition 12 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  De-  velopment of new methods of construction of rUSs for  this case is our main motivation for the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  R-uniform subspaces in heterogeneous systems have re-  lation to QECCs over mixed alphabets [29] and quan-  tum information masking [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' To our knowledge, for a  given system the largest possible dimension of rUSs is  unknown, so building new instances of such subspaces  could bring some insights in this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  We use compositional tools of diagrammatic reason-  ing [31–33], which allow us to come up with new con-  structions and provide further instances of states and  subspaces with important properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Tensor diagrams  are widely used in quantum information theory, in par-  ticular, in theory of QECCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Recently a framework for  the construction of new stabilizer QECCs from old ones  with the use of tensor networks has been presented in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In Section II nec-  essary definitions and some theoretical background are  given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The main results of the current paper are provided  in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In Subsection III A we give diagrammatic  representation of basic properties of rUSs upon which, in  Subsection III B, we derive the methods of constructing  rUSs in heterogeneous systems such as glueing several  subspaces together, eliminating parties, combining pure  error correcting codes and maximally entangled states  and subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In Subsection III C we compare our re-  sults with the ones obtained with the use of the mixed  orthogonal arrays method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Finally, in Section IV we con-  clude with discussing possible directions of further re-  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  PRELIMINARIES  Let us first give the definition of r-uniform states of an  n-partite finite-dimensional system with local dimensions  d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Such a system is usually associated with the  tensor product Hilbert space Cd1 ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='⊗Cdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' A state |ψ⟩  in Cd1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='⊗ Cdn is called r-uniform if all its reductions  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Doubling notation for the process of action of a linear  operator V on a pure state ψ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Reduction of a bipartite pure state ψ to subsystem A  at least to r parties are maximally mixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=',  Tr{i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=', ir}c[|ψ⟩⟨ψ|] =  1  di1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' · dir  Ii1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=', ir  (1)  for  all  r-element  subsets  {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , ir}  of  the  set  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Here {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , ir}c denotes the complement  of the given set in the set of all parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  It is clear  that r-uniform state is also l-uniform for all l < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' By  the properties of the Schmidt decomposition, the nec-  essary condition for r-uniform states to exist is that  di1 ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='·dir ⩽ dir+1 ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='·din is satisfied for each bipartition  i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , ir|ir+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  An r-uniform subspace — a subspace of Cd1 ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='⊗Cdn  consisting entirely of r-uniform vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  For homogeneous systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=', those having equal  local dimensions, the existence of r-uniform subspaces  can be deduced from the existence of certain quan-  tum error correcting codes (QECCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Recall that a  QECC ((n, K, d))D is a special K-dimensional subspace  of  �  CD�⊗n such that for each its state any error affect-  ing not more than a certain number of subsystems can  be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' For a code with distance d = 2t + 1 the  number is equal to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In addition, a code with distance d  can detect d − 1 errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  In addition to the ((n, K, d))D notation for general  QECCs, we will use the designation [[n, k, d]]D for stabi-  lizer QECCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' While the symbols n and d from the latter  notation have the same sense as those in the former one,  the dimension of the codespace for the code [[n, k, d]]D  is equal to Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  A quantum error correcting code is called pure if  ⟨i| E |j⟩ = 0  (2)  for any states |i⟩ , |j⟩ from an orthonormal set spanning  the code space and for any error operator E with weight  strictly less than the distance of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  It is known that each pure QECC ((n, K, d))D yields a  K-dimensional (d − 1)-uniform subspace of (CD)⊗n, and  vice versa [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  To address the case of heterogeneous systems, in the  present paper we will use encoding isometries of the exist-  ing pure QECCs in combination with various states and  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Diagrammatic representation of the maximally mixed  state (up to the normalization factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  subspaces of lower number of parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' A similar approach  dealing with isometric mapping to entangled subspaces  proved to be effective in constructing multipartite gen-  uinely entangled subspaces [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Throughout the paper we use tensor diagrams, in par-  ticular, we use doubled-process theory notation adopted  from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The doubling notation indicates the pas-  sage from pure state vectors to their associated density  operators, as shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  To deal also with mixed states, the discarding sym-  bol (map) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Applying the discarding map to a  subsystem of a multipartite state is equivalent to tracing  out the subsystem, as shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The adjoint of the discarding map (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 3) denotes  the identity operator, which is proportional to the max-  imally mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Basic properties and their diagrammatic  representation  We start with pointing at an important basic prop-  erty of subspaces under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let |φ⟩ and |χ⟩  be two mutually orthogonal normalized vectors in an r-  uniform subspace W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' An arbitrary (normalized) linear  combination |ψ⟩ = α |φ⟩ + β |χ⟩ is also in W, and hence  its reduction to some r-element subset S of the set of all  parties yields  TrSc[|ψ⟩⟨ψ|] = N IS  = N IS + αβ∗ TrSc[|φ⟩⟨χ|] + βα∗ TrSc[|χ⟩⟨φ|],  (3)  where N =  ��  i∈S di  �−1, the normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Con-  sequently, the last two terms sum up to zero:  αβ∗ TrSc[|φ⟩⟨χ|] + βα∗ TrSc[|χ⟩⟨φ|] = 0,  ∀ α, β ∈ C,  |α|2 + |β|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (4)  Setting first α real and β imaginary and then both of  them real, one can deduce that  TrSc[|φ⟩⟨χ|] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (5)  Now we can formulate this property as  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' For any orthonormal set {|ψ⟩i} spanning r-  uniform subspace it follows that  TrSc[|ψi⟩⟨ψj|] ∼ δij IS  (6)  for any r-element subset S of the set of all parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  1TrB|)V  A)A3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Action of V together with tracing out subsystems Sc  results in a channel ΦS which discards its input and returns  the maximally mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (6) is known to hold for codewords of pure QECCs  of distance r + 1, which correspond to r-uniform sub-  spaces for homogeneous systems, but it is worth stress-  ing that it is valid for more general case of heterogeneous  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Let us think of r-uniform subspaces in terms of isome-  tries and quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' To each such subspace W  with dimension K one can relate an isometry V : CK →  Cd1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' ⊗ Cdn, which maps an orthonormal basis {|i⟩}  of CK to some orthonormal set {|ψ⟩i} spanning W:  V |i⟩ = |ψi⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (7)  Hence, the range of the isometry V coincides with the  subspace W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' As before, let us choose an r-element subset  S of the set of n parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Applying the isometry to a state  in CK with subsequent tracing out n − r subsystems in  Sc results in action of a quantum channel on the state:  TrSc[V |φ⟩⟨φ| V †] = ΦS(|φ⟩⟨φ|),  |φ⟩ ∈ CK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (8)  We  thus  obtain  a  family  of  quantum  channels  ΦS : L(CK) → L(Cdi1 ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='⊗Cdir ), where {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , ir} =  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' one channel for each choice of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Since subspace W  is r-uniform, a channel ΦS maps all states of CK to the  identity on S:  ΦS(|φ⟩⟨φ|) =  1  di1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' · dir  Ii1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=', ir,  |φ⟩ ∈ CK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (9)  In other words, the channels discard the input and map  everything to the maximally mixed state:  ΦS(X) =  Tr[X]  di1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' · dir  IS,  X ∈ L(CK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (10)  By setting X = |i⟩⟨j| in this expression, with the use of  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (7), (8), we recover Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  It should be stressed that the above property of map-  ping to the maximally mixed state holds when the in-  put dimension of the isometry (and of the corresponding  channel) is not greater than the dimension of the range  of the isometry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=', the dimension of the r-uniform sub-  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In fact, the input dimension can be strictly less  than that dimension, in which case the isometry takes the  input states to some subspace of the r-uniform space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Construction of a 2-uniform state from an encod-  ing isometry V of the ((5, 3, 3))3 pure code and a maximally  entangled state |ψ⟩ in C2 ⊗ C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Diagrammatic representation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (8) and (10) is  shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 4, where the symbol ”∼” means that the  two diagrams are equal up to a scalar factor (the normal-  ization constant for the maximally mixed state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' This  construction will be crucial in further considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Now we can choose V to be an encoding isometry of  some pure quantum error correcting code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let us try to  combine this isometry with some states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Example: 2-uniform state in heterogeneous systems  Consider the ((5, 3, 3))3 pure code [36] and its encod-  ing isometry V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Application of V to one of the parties  of a bipartite pure state |ψ⟩ in C2 ⊗ C3 yields a 6-partite  pure state in C2 ⊗  �  C3�⊗5, as shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The code  has distance 3, so the code subspace is 2-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  In  addition, the code subspace has dimension equal to 3,  which matches the local dimension of the second party  of the state |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Therefore, the property of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 4 holds  in this case with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  If the bipartite state |ψ⟩ is maximally entangled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  its reduction to the party with local dimension 2 is max-  imally mixed, then the resulting state in C2 ⊗  �  C3�⊗5  will be 2-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' This can be shown diagrammatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  One needs to consider the two cases of producing the  two-party reduction of the state in question: a) all par-  ties are traced out except some two output subsystems  of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' b) all parties are traced out except the first party  of |ψ⟩ (with dimension 2) and some output subsystem of  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The state |ψ⟩ is completely traced out - the part of  the diagram on the bottom right is a scalar equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  2  3  3  J  2  3  2  32  3  3  3  3  3  V  2  3S  S  l1  n  V  24  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' By the property on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 4 and the maximal entangle-  ment of |ψ⟩ the resulting state is proportional to I2 ⊗ I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The case a) is presented on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 6: the property on  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 4 being used, the state |ψ⟩ gets completely traced  out and the resulting state is proportional to I3 ⊗ I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The case b) is analyzed on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 7: on the first step the  property on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 4 is used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' the second step is due to the  fact that |ψ⟩ is maximally entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  It is interesting to note that C2⊗  �  C3�⊗5 was the small-  est possible Hilbert space for which a 2-uniform state  could be constructed with the methods of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Construction of r-uniform subspaces in  heterogeneous systems  The simplest method to produce an r-uniform sub-  space in heterogeneous systems is to ”glue” together two  r-uniform subspaces in homogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' By ”glue-  ing” we mean taking tensor product of the two subspaces:  this can be done by taking all possible tensor products  of the vectors spanning the two subspaces, the resulting  subspace will be spanned by such combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Tensor product of an r-uniform subspace and  a k-uniform subspace is an l-uniform subspace, where l =  min(r, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let W1 be an r-uniform subspace with n parties  and let W2 be a k-uniform subspace with m parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Con-  sider two isometries V1 : HA → HC1 ⊗ · · · ⊗ HCn and  V2 : HB → HD1 ⊗· · ·⊗HDm, where dim(HA) = dim(W1)  and dim(HB) = dim(W2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The first one, V1, maps the  basis states {|i⟩}A of HA to an orthonormal system of  vectors spanning W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Similarly, V2 maps the basis states  {|j⟩}B of HB to vectors spanning W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Tensor product  W1 ⊗ W2 is then spanned by the vectors  (V1 ⊗ V2) (|i⟩A ⊗ |j⟩B) ,  (11)  as shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Now the property from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 4 can  be applied when one traces out any n + m − l of the  parties C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , Cn, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , Dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' As a result, a general  state |φ⟩ ∈ HA ⊗HB gets completely traced out, and the  l-party maximally mixed state is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  On the left: action of the isometry V1 ⊗ V2 on a  particular basis state |i⟩A ⊗ |j⟩B of HA ⊗ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' On the right:  action of the isometry V1 ⊗V2 on a general state φ from HA ⊗  HB, which is equal to a linear combination of basis states  {|i⟩A ⊗ |j⟩B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Action of V1 ⊗ V2 on each state in HA ⊗ HB  generates W1 ⊗ W2, which is l-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  R-uniform subspaces can be used for r-uniform quan-  tum information masking [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' An operation V is said  to r-uniformly mask quantum information contained in  states {|i⟩} if it maps them to multipartite states {|ψi⟩}  whose all reductions to r parties are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  In the  proof of Lemma 2 an instance of masking has been pro-  vided: on the right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 8 it is shown how each  state φ from HA ⊗HB is “masked” by the two isometries  V1 and V2 as an l-uniform state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  As an example, combining encoding isometries of  ((5, 2, 3))2 and ((5, 3, 3))3 pure codes, by Lemma 2  we obtain a 6-dimensional 2-uniform subspace of the  �  C2�⊗5 ⊗  �  C3�⊗5 Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Can we reduce the number of parties?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  A structure  similar to the one on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 5 can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let us take the  encoding isometry V of the [[6, 2, 3]]3 (stabilizer) pure  code (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [37], Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The range of the isome-  try is a 2-uniform subspace of the  �  C3�⊗6 Hilbert space,  which has dimension equal to 32 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Consider a sub-  space of C2 ⊗ C9, which consists entirely of states max-  imally entangled with respect to the first party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Such a  subspace can be easily constructed with the use of Propo-  sition 3 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The dimension of the subspace is  equal to ⌊ 9  2⌋ = 4 (Corollary 4 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Now let us  act with V on the second party (the one with dimension  9) of each state in the subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' This procedure will gen-  erate a 4-dimensional subspace of the C2⊗  �  C3�⊗6 Hilbert  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The analysis, which is similar to that on Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 6  and 7, shows that the resulting subspace is 2-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The above construction can be viewed as an illustra-  tion of subspace masking: only states that belong to a  specific subspace of C2⊗C9 are masked by V as 2-uniform  states in C2 ⊗  �  C3�⊗6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The next property provides an important way of gen-  erating r-uniform subspaces from those of larger number  of parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' It can be seen as the extension of Theorem 20  of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [39] to the case of heterogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let W be an r-uniform subspace of Hilbert  space with the set S = {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , dn} of local dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Let r ⩾ 1 and dim(W) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Then, for any di ∈ S,  there exists an r − 1-uniform subspace of Hilbert space  with local dimensions S \\ {di}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The dimension of this  m  A  B2  35  subspace is equal to diK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Consider an orthonormal set of vectors {|ψk⟩}  which span W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Each vector |ψk⟩ is r-uniform, and so,  in particular, its reduction to party i is proportional to  the maximally mixed operator I{i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Accordingly, the  Schmidt decomposition of |ψk⟩ with respect to the bi-  partition ”party i|the rest” reads  |ψk⟩ =  1  √di  di−1  �  j=0  ���φ(k)  j  �  P ⊗  ���χ(k)  j  �  P ,  (12)  where {  ���φ(k)  j  �  P } and {  ���χ(k)  j  �  P }, with j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , di − 1  and k fixed, are two orthonormal sets of vectors in r − 1-  partite and 1-partite Hilbert spaces with local parties  P = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , n} \\ {i} and P = {i}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In the  right part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (12) vectors with the same upper index  satisfy the orthonormality condition, for example,  �  χ(k)  m  ���χ(k)  n  �  P = δmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (13)  Now consider an r − 1-element subset J ⊂ P and, for  some numbers k, s ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , K}, take the reduction  of |ψk⟩⟨ψs| to the set J ∪ {i}:  TrP \\J [|ψk⟩⟨ψs|]  = 1  di  di−1  �  j, l=0  TrP \\J  ����φ(k)  j  ��  φ(s)  l  ���  P  �  ⊗  ���χ(k)  j  ��  χ(s)  l  ���  P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (14)  By  Lemma  1,  this  reduction  is  proportional  to  δks IJ∪{i} = δks IJ ⊗ I{i}, and hence  1  di  di−1  �  j, l=0  TrP \\J  ����φ(k)  j  ��  φ(s)  l  ���  P  �  ⊗  ���χ(k)  j  ��  χ(s)  l  ���  P  = δks  1  dJ  IJ ⊗ 1  di  I{i},  (15)  where dJ is the product of local dimensions of the parties  in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Multiplying both parts of this equality by  �  χ(k)  m  ���  and  ���χ(s)  n  �  , with the use of condition (13), we obtain  TrP \\J  ����φ(k)  m  ��  φ(s)  n  ���  P  �  = δksδmn  1  dJ  IJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (16)  Taking trace over subsystem J in the last equation, one  can see that the set of diK vectors {  ���φ(t)  j  �  P }, j =  0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , di − 1, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , K is an orthonormal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (16) holds for any choice of an r − 1-element  subset J ⊂ P, the states {  ���φ(t)  j  �  P } are r − 1-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In  addition, from the same equation one can see that any lin-  ear combination of these vectors is an r−1-uniform state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Therefore, the system {  ���φ(t)  j  �  P } spans a diK-dimensional  r − 1-uniform subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (12) it follows that  diTr{i}  � K  �  s=1  |ψs⟩⟨ψs|  �  (17)  is the orthogonal projector on the subspace in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  A practical way to obtain an orthonormal system of  vectors spanning the subspace defined by the projector  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (17) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let us suppose that party i  with local dimension di is being eliminated, just as in the  condition of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Consider an orthonormal system  of one party vectors {|vj⟩}, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , di−1, which spans  Cdi and such that each partial scalar product  ���µ(s)  j  �  ≡ ⟨vj|ψs⟩ ,  j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , di − 1,  s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , K,  (18)  is a non-null vector, where the only input of the  (co)vector ⟨vj| is joined with the i-th output of the vec-  tor |ψs⟩ in each partial scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Then diK vectors  {  ���µ(s)  j  �  } span the subspace in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Indeed, the vec-  tors are mutually orthogonal:  �  µ(t)  l  ���µ(s)  j  �  = ⟨ψt|vl⟩ ⟨vj|ψs⟩  = ⟨vj| TrP [|ψs⟩⟨ψt|] |vl⟩ = 1  di  δts ⟨vj|vl⟩  = 1  di  δtsδlj,  (19)  where the third equality follows from r-uniformity of the  original vectors {|ψs⟩} and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Consequently,  {√di  ���µ(s)  j  �  } is an orthonormal system, and the corre-  sponding projector  �  j, s  �  di  ���µ(s)  j  ��  µ(s)  j  ���  �  di = di  �  j, s  ⟨vj|ψs⟩ ⟨ψs|vj⟩  = diTr{i}  ��  s  |ψs⟩⟨ψs|  �  (20)  coincides with the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  As an example, from the obtained above 4-dimensional  2-uniform subspace of C2 ⊗  �  C3�⊗6 Hilbert space one can  produce a 12-dimensional 1-uniform subspace of C2 ⊗  �  C3�⊗5 Hilbert space by eliminating one of the parties  with local dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  When an initial r-uniform subspace is spanned just by  1 vector, the described above practical method becomes  similar to Proposition 12 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  As we have seen from Lemma 2, glueing two uniform  subspaces together doesn’t increase the uniformity pa-  rameter of the resulting subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' This is in accordance  6  with the general principle that local operations cannot  produce any entanglement over that present in origi-  nal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let us show that making use of additional  resources such as maximally entangled states can lead  to larger uniformity parameters of the produced states  and subspaces in comparison with original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' At first  we consider uniform subspaces in homogeneous systems,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=', those corresponding to pure quantum error correct-  ing codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Recall that the parameters of a ((n, K, d))D code sat-  isfy the inequality [36, 40]  K ⩽ Dn−2(d−1),  (21)  which is called the quantum Singleton bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  If pa-  rameters of a code saturate the bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (21),  the code is called quantum maximum distance separable  code (QMDS) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' It is known that all QMDS codes are  pure [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [15] an important observation about QMDS  codes was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' We reformulate it here in a more general  form and provide the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Lemma 3 (An observation in the proof of Proposition  7 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let ((n, K, d)) be a QMDS code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Con-  sider the projector P = �K  s=1 |ψs⟩⟨ψs| on the codespace,  where {|ψs⟩} is an orthonormal set of vectors that span  the codespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Then each reduction of P to n − (d − 1)  parties is proportional to the maximally mixed operator  In−(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' According to Theorem 20 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [39] (or The-  orem 1 here), tracing out one party yields a projector  on a KD-dimensional subspace of  �  CD�⊗n−1, hence af-  ter d − 1 such steps of tracing out we have a projec-  tor on a subspace of  �  CD�⊗(n−(d−1)) with dimension  KDd−1 = Dn−(d−1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  the projector on the whole  space  �  CD�⊗(n−(d−1)), the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  We stress that this holds only for QMDS codes - those  with K = Dn−2(d−1), and not for other pure codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  With the use of QMDS codes we can now formulate  the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let ((n1, K1, d1))D1 and ((n2, K2, d2))D2  be two QMDS codes with K1 = K2 ≡ K > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Denote  r1 ≡ d1 − 1 and r2 ≡ d2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Then there exists an  l-uniform state in  �  CD1�⊗n1 ⊗  �  CD2�⊗n2 Hilbert space  with  l = min (n1 − r1, n2 − r2, r1 + r2 + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (22)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Since the two codes are pure, there are two sub-  spaces related to them: an r1-uniform subspace W1 of  �  CD1�⊗n1 and an r2-uniform subspace W2 of  �  CD2�⊗n2  such that dim(W1) = dim(W2) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Consider two isometries V1 : CK →  �  CD1�⊗n1 and  V2 : CK →  �  CD2�⊗n2 whose ranges coincide with W1 and  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Steps to prove l-uniformity of the state |ψ⟩S1S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  W2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Now let us take any bipartite maxi-  mally entangled state |φ⟩AB in CK ⊗ CK and construct  the state  |ψ⟩S1S2 = (V1 ⊗ V2) |φ⟩AB ,  (23)  which belongs to  �  CD1�⊗n1 ⊗  �  CD2�⊗n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Here S1 and S2  denote the sets of the output parties of the isometries V1  and V2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' We claim that the state |ψ⟩S1S2 is  l-uniform, with l as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The underlying principle is shown on Fig 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let us as-  sume that all subsystems of |ψ⟩S1S2 are traced out except  some m1 output subsystems of isometry V1 and some m2  output subsystems of isometry V2 , as shown on Fig 9,  a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' If, for example, m2 ⩽ r2, the rule from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 4 can be  applied and we arrive at the situation shown on Fig 9, b),  where isometry V2 is eliminated and the state |φ⟩AB gets  partially traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Next, the state |φ⟩AB is maximally  entangled, and so its reduction to A is the maximally  mixed operator  1  K IA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' As a result, isometry V1 acts on  the identity operator IA, as shown on Fig 9, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The steps  b − c), without taking into account the trace over output  parties of V1, can be written as  V1TrB{|φ⟩⟨φ|AB}V †  1 = 1  K V1V †  1 = 1  K PW1,  (24)  where PW1 – the orthogonal projector on subspace W1,  the first equality follows from maximal entanglement of  |φ⟩AB, the second one – from the fact that the isometry  V1 has subspace W1 as its range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Now if m1 ⩽ n1 − r1  then, by Lemma 3, performing the trace over n1 − m1  output subsystems of V1 (Fig 9, c)) produces the maxi-  mally mixed state of m1 parties (Fig 9, d)) in addition to  the maximally mixed state of m2 parties obtained earlier  in the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' We conclude that if m1 ⩽ n1 − r1 and  m2 ⩽ r2, the reduced state of m1 + m2 parties is maxi-  mally mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The roles of V1 and V2 can be interchanged,  and we obtain that if m1 ⩽ r1 and m2 ⩽ n2 − r2, the  reduced state is maximally mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Now we need to determine the maximal number l such  that any partition of l = m1 +m2 into m1 output parties  m2  m1  m2  元.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='元  元.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='元  元.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='元  Vi  V2  Vi  A  B  A  B  a)  6)  m1  m2  m1  m2  V1  A  c)  d)7  of V1 and m2 output parties of V2 yields, after perform-  ing the trace over the rest n1 + n2 − l subsystems, the  maximally mixed state of l parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let us first consider  partitions in which m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In this case the maximal  value of m1, for which the scheme on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 9 can still be  applied, is n1 − r1, as it was shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' This number  is then an upper bound on l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Interchanging the roles of  V1 and V2 and setting m1 = 0, we obtain another bound,  n2 − r2, and hence  l ⩽ min(n1 − r1, n2 − r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (25)  Next, let us assume that r2 > r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let ⌈x⌉ denote the  ceiling of x and ⌊x⌋ denote the floor of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Consider a par-  tition of l into m1 = ⌊ l  2⌋ and m2 = ⌈ l  2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' If ⌈ l  2⌉ > r2 then  ⌊ l  2⌋ > r1 also holds, and one cannot apply the scheme on  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 9 since neither V2 nor V1 can be eliminated in the  first step a)-b) with the use of the rule from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Con-  sequently, we can take into account only those values of  l that satisfy ⌈ l  2⌉ ⩽ r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Accordingly, consider a partition  of l into m2 = ⌈ l  2⌉+α and m1 = ⌊ l  2⌋−α for some integer  α > 0 such that ⌈ l  2⌉ + α = r2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In this case V2 cannot  be eliminated in the first step of scheme on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' On  the other hand, the scheme can be initiated by apply-  ing the rule from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 4 with respect to V1 on condition  that m1 = ⌊ l  2⌋ − α ⩽ r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' If the condition is satisfied, in  the step c)-d) of the scheme (with interchanged V1 and  V2) the reduction of PW2 to m2 parties will be maximally  mixed by Lemma 3, since m2 ⩽ l ⩽ n2 − r2 by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  This principle continues to work for greater values of α  (but bounded by the condition ⌈ l  2⌉ + α = m2 ⩽ n2 − r2),  as m1 gets smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' It is clear that partitions with α < 0  will also work, as the step a)-b) will be initiated with the  use of V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' To sum up, the maximal possible value of l  satisfies  � l  2  �  + α = r2 + 1,  � l  2  �  − α = r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (26)  Adding these two equalities, we obtain  l =  � l  2  �  +  � l  2  �  = r1 + r2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (27)  When r1 = r2 ≡ r, we can choose (odd) l such that  ⌈ l  2⌉ = r + 1 and ⌊ l  2⌋ = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' For partitions with m1 = ⌊ l  2⌋  and m2 = ⌈ l  2⌉ and, vice versa, m1 = ⌈ l  2⌉ and m2 =  ⌊ l  2⌋, the scheme on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 9 is initiated with the use of  V1 and V2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' For all other partitions, which  can be parameterized with integer α as m1 = ⌈ l  2⌉ + α  and m2 = ⌊ l  2⌋ − α or vice versa, the scheme also works  by the analysis similar to that in the above paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Consequently,  l =  � l  2  �  +  � l  2  �  = 2r + 1,  (28)  which is just a special case of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Immediate application of Theorem 2, with the use  of the correspondence between r-uniform states and 1-  dimensional pure quantum codes, produces  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let ((n, K, d))D be a QMDS code with  K > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Then there exists a pure ((2n, 1, d′))D code with  distance  d′ = min(n − d + 2, 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (29)  As an example, consider a ((4, 4, 2))2 code, which can  be the stabilizer [[4, 2, 2]]2 code obtained from the well-  known [[5, 1, 3]]2 code with the use of Theorem 20 of  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Combining [[4, 2, 2]]2 with itself produces an  8-qubit 3-uniform state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' It is known that ⌊ n  2 ⌋-uniform  states of n qubits (absolutely maximally entangled (AME)  states) don’t exist for n > 6 [3, 39, 41, 42], so the con-  structed state has maximal possible uniformity parame-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  To give an example with heterogeneous systems, let  us combine pure codes ((4, 4, 2))2 and ((5, 4, 3))4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The  latter code can be produced by tensoring ((5, 2, 3))2 with  itself (Theorem 14 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' By the construction in  the proof of Theorem 2, the two codes yield a 3-uniform  state in  �  C2�⊗4 ⊗  �  C4�⊗5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' We can then obtain r-uniform  subspaces by eliminating some parties of this state, but  in this case some produced subspaces will demonstrate  better values of r than those predicted by Theorem 1  owing to the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Corollary  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Let  ((n1, K1, d1))D1  and  ((n2, K2, d2))D2  be  two  QMDS  codes  with  K1 = K2  ≡ K  > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Denote r1  ≡ d1 − 1 and  r2 ≡ d2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Then for any integers 0 ⩽ α ⩽ r1 and  0 ⩽ β ⩽ r2 there exists an l-uniform subspace W of  �  CD1�⊗n1−α ⊗  �  CD2�⊗n2−β Hilbert space such that  dim(W) = D α  1 Dβ  2 ,  l = min(n1 − r1 − α, n2 − r2 − β,  r1 + r2 + 1 − α − β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (30)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' By Theorem 2, we can construct a state in  �  CD1�⊗n1 ⊗  �  CD2�⊗n2 Hilbert space with the uniformity  parameter given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Next, we eliminate α + β  parties of this state by the procedure described after the  proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let us take α orthonormal systems  of vectors {  ���v(µ)  i  �  }, µ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , α, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , D1−1, each  system being a basis for the corresponding CD1 Hilbert  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Construction of the states which span the l-uniform  subspace W of  �  CD1�⊗n1−α ⊗  �  CD2�⊗n2−β Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Vi8  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Similarly, we take β orthonormal systems of vec-  tors {  ���w(ν)  j  �  }, ν = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , β, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , D2 − 1, each in  its own CD2 Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Next, we pick some specific  vectors  ���v(1)  i1  �  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' ,  ���v(α)  iα  �  , one from each system, and  eliminate α output parties of the isometry V1 by joining  them with the inputs of the chosen vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Similarly,  we pick β specific vectors  ���w(1)  j1  �  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' ,  ���w(β)  jβ  �  and elim-  inate β output parties of the isometry V2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 10,  the indices of vectors v, w are omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' As a result, we  obtain  �  v(1)  i1  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  �  v(α)  iα  ���  �  w(1)  j1  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  �  w(β)  jβ  ��� (V1 ⊗ V2) |φ⟩AB , (31)  one of the D α  1 Dβ  2 states that span the subspace W of  �  CD1�⊗n1−α⊗  �  CD2�⊗n2−β Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' All such states  are hence indexed by the numbers i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , iα, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , jβ,  which represent the correspondence between tuples of  vectors v, w and the basis states of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The uniformity of the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (31) can be analyzed  with the use of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 10 and the same reasoning as in the  proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The vectors v and w take up α and β  positions out of n1 and n2 output parties of the isometries  V1 and V2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The parties in these positions  cannot be traced out, and this results in modifying the  bounds on l in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (25):  l ⩽ min(n1 − r1 − α, n2 − r2 − β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (32)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (26) is also modified, with r1 and r2 replaced by  r1 − α and r2 − β, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' As a result, we obtain  the expression for l in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Earlier a 3-uniform state in  �  C2�⊗4 ⊗  �  C4�⊗5 was  obtained with the use of Theorem 2 from pure codes  ((4, 4, 2))2 and ((5, 4, 3))4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Eliminating one party with  dimension 2 and one party with dimension 4, or, in terms  of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='2, setting α = β = 1, we produce an  8-dimensional 2-uniform subspace of  �  C2�⊗3 ⊗  �  C4�⊗4  Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' We stress that the original state has spe-  cial structure and, as a result, after the elimination of 2  parties the produced subspace has higher value l = 2 in  comparison with l = 1 predicted by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Comparison with mixed orthogonal arrays  method and further constructions  Mixed orthogonal arrays (MOAs) [43, 44] in its specific  form, irredundant MOAs (IrMOAs) [26], have become a  powerful tool in construction of r-uniform states in het-  erogeneous systems [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In this subsection we present  several applications of the compositional approach that  allow us to reproduce or extend some results obtained  with the use of IrMOAs (in terms of the minimal num-  ber of parties for a given uniformity parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Such a  comparison also reveals some weaknesses of the presented  in this paper approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  In general it becomes more difficult to find examples  of r-uniform states in heterogeneous systems when the  number of parties gets smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Let us consider some  results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Proposition 1 (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 2-  uniform states exist for the following configura-  tions:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' C3 ⊗  �  C2�⊗N for N ⩾ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  �  C3�⊗2 ⊗  �  C2�⊗N for N ⩾ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  �  C3�⊗3⊗  �  C2�⊗N for N ⩾ 11 and  �  C3�⊗4⊗  �  C2�⊗N  for N ⩾ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  We can reproduce the first result for N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The pro-  cedure is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The pure code ((5, 2, 3))2 is com-  bined with itself by the construction of Theorem 2, which  results in a 10-qubit 3-uniform state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' a ((10, 1, 4))2  pure code (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Next, by eliminating two par-  ties, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='2 we obtain a 4-dimensional 8-qubit  2-uniform subspace, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' a pure ((8, 4, 3))2 code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Now we  have an encoding isometry which maps vectors from C4  to the 2-uniform code space (briefly, the ”8-qubit isome-  try”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Finally, we can take a maximally entangled state  in C3⊗C3 and act on one of its parties with the obtained  isometry, and the construction here will be similar to the  one presented on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The resulting state, which be-  longs to C3⊗  �  C2�⊗8, is 2-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' For larger values of N  we can use the same auxiliary state and various combina-  tions of isometries and, if necessary, glue them together  with the use of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' As an example, the isometry  for N = 10, which maps C4 to 10-qubit 2-uniform sub-  space, can be obtained from glueing the subspace of the  code [[5, 1, 3]]2 with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In other words, 10-qubit isom-  etry is obtained from glueing 5-qubit isometry with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Next, by eliminating one party of a ((8, 1, 4))2 state, we  obtain an isometry which maps vectors from C2 to the  2-uniform 7-qubit space (the ”7-qubit isometry”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  By  the same procedure, from the state ((10, 1, 4))2 we ob-  tain the 9-qubit isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Now, the isometry for N = 12  can be obtained from glueing 7-qubit and 5-qubit isome-  tries, for N = 13 – from 5-qubit and 8-qubit ones, and  so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' We haven’t found any appropriate isometries to  construct the states with N = 9 and N = 11 (those that  we’ve found have input dimension 2, which is less than  the local dimension of the second party of the auxiliary  state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' To conclude, we cannot reproduce the first result  of Proposition 1 only for N = 9 and N = 11 with the  current approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The second result from Proposition 1 is harder to re-  produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The reason for that is as follows: we can take  a 4-qutrit 2-uniform state, which can be, for example,  the graph state of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [19] or a QMDS code [[4, 0, 3]]3  from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='6 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [37], but now we need to act  with an isometry on its two parties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=', on a compound  subsystem with local dimension 3 × 3 = 9, as shown on  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The 8-qubit isometry that was used before is  9  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' An isometry V acting on a joint subsystem of two  parties with local dimensions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  not appropriate here since it has input dimension equal  to 4, which is less than the output dimension of the two  combined parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' A proper isometry can be constructed  from other error correcting codes with the use of the split-  ting property for r-uniform subspaces, which is a direct  consequence of the splitting method for r-uniform states  appeared earlier in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [26, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let W be an r-uniform subspace of Hilbert  space with the set S = {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' , dn} of local dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Let di = d′  id′′  i for some i: 1 ⩽ i ⩽ n and some integer  d′  i, d′′  i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Then there exists an r-uniform subspace of  Hilbert space with the set of local dimensions given by  {d′  i, d′′  i } ∪ [S \\ {di}] and having the same dimension as  the original subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The subspace in question can be obtained from  the original one by splitting the i-th subsystem of each  state in W into two smaller ones, i′ and i′′, with local  dimensions d′  i and d′′  i , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Each newly obtained  state is r-uniform, as follows from the splitting method  described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [26, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Consequently, a subspace,  which consists of such states, is r-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Now we can return to the construction of a 2-uniform  state in  �  C3�⊗2 ⊗  �  C2�⊗12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Consider a pure ((6, 16, 3))4  code (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='6 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  By splitting each  ququart into 2 qubits, by Lemma 4, the code is converted  into a pure ((12, 16, 3))2 code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Since its encoding isome-  try (the ”12-qubit isometry”) has input dimension equal  to 16, we can act with it on a compound subsystem con-  sisting of two combined parties of the state [[4, 0, 3]]3 (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The resulting state is 2-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The isome-  tries for larger N can be obtained from glueing the 12-  qubit isometry with the described above isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' As  an example, a 17-qubit isometry is obtained from glue-  ing the 12-qubit and the 5-qubit ones (it doesn’t matter  that the input dimension of the 5-qubit isometry is 2 –  the input dimension of the 12-qubit isometry is 16, and  the resulting one will have the input dimension equal to  16 × 2 = 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  N = 16 is obtained from glueing the  8-qubit isometry with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' All other numbers N ⩾ 18  can be otained similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In addition, N = 14 can be con-  structed from splitting the code [[7, 3, 3]]4 (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='6  of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' We cannot reproduce the second result of  Proposition 1 only for N = 13 and N = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  As for the third result of Proposition 1, 2-uniform  states in  �  C3�⊗3⊗  �  C2�⊗N can be obtained with action of  the described above isometries on one party of the state  [[4, 0, 3]]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' As earlier, the cases N = 9 and N = 11 are  not covered by our approach, but we can construct a state  with N = 8, which extends the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The result  for uniform states in  �  C3�⊗4 ⊗  �  C2�⊗N can be substan-  tially extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Consider a code [[4, 0, 3]]6, for example,  from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='6 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' By Lemma 4, by split-  ting each subsystem with local dimension 6 into qubit  and qutrit subsystems, we obtain a 2-uniform state in  �  C3�⊗4 ⊗  �  C2�⊗4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The case N ⩾ 5 is trivial: we can glue  the state [[4, 0, 3]]3 with a 2-uniform state of N qubits,  which exists for N ⩾ 5 and can be obtained, for exam-  ple, from graph states constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' These observations  extend the proposition from N = 10 to N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Gathering the above results, we can formulate  Proposition 2 (Combination of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [27]  with the current approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' 2-uniform states exist for the  following configurations:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' C3 ⊗  �  C2�⊗N for N ⩾ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  �  C3�⊗2 ⊗  �  C2�⊗N for N ⩾ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  �  C3�⊗3 ⊗  �  C2�⊗N for N = 8 and N ⩾ 10 and  �  C3�⊗4 ⊗  �  C2�⊗N for N ⩾ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Let us also analyze some results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Proposition 3 (Theorem 9 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' For any d > 2,  the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' There exists a 2-uniform state in  �  C2�⊗2 ⊗  �  Cd�⊗N  for any N ⩾ 7 and N ̸= 4d + 2, 4d + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' There exists a 2-uniform state in C2 ⊗  �  Cd�⊗N for  any N ⩾ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  We can start with the 2-uniform subspace of the code  [[5, 1, 3]]2 and act with a proper isometry on three sub-  systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' on a joint system of dimension 8, of each  vector in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Therefore, in addition to having a  2-uniform subspace as its range, an appropriate isometry  must have input dimension greater or equal 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The code  family [[N, N − 4, 3]]d, 4 ⩽ N ⩽ d2 + 1, d > 2 (Corol-  lary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='6 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [37]) provides us with proper isometries  for 6 ⩽ N ⩽ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' In addition, the isometry with 5 out-  put parties, which corresponds to [[5, 1, 3]]d, only works  when d ⩾ 8, since in this case its input dimension is equal  to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Isometries for all other numbers, N > 10, can be  obtained by glueing the codes with N < 10 (Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  As a result, we lift the constraint N ̸= 4d + 2, 4d + 3 and  obtain 2-uniform subspaces instead of just states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  We can only reproduce the second result of Proposi-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' All the described above isometries, this time in-  cluding the one with N = 5, can be used to act on one  party of a maximally entangled state in C2 ⊗ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Instead of a maximally entangled state in C2 ⊗ C2 we  could use maximally entangled subspaces of C2 ⊗ Cp,  3  3  2  2  2  A  3  3  3  310  which have dimension equal to ⌊ p  2⌋, by Corollary 4 of  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Now the isometry, which corresponds to code  [[N, N − 4, 3]]d, acts on a party with local dimension p  of each state in the maximally entangled subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The  input dimension of the isometry, dN−4, hence must be  greater or equal p, and we have the condition  N ⩾ 4 + logd p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  (33)  Summing the results, we can formulate the extension  of Proposition 3  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' For 2 < d ⩽ 8 there exists a 2-uniform subspace of  �  C2�⊗2 ⊗  �  Cd�⊗N with dimension 2 for any N ⩾ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' For d > 8 there exists a 2-uniform subspace of  �  C2�⊗2 ⊗  �  Cd�⊗N with dimension 2 for any N ⩾ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' for d > 2 and p ⩾ 2 there exists a 2-uniform sub-  space of C2 ⊗  �  Cd�⊗N with dimension ⌊ p  2⌋ for any  N ⩾ 4 + logd p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The above examples show that the presented approach  is more effective in constructing r-uniform states and  subspaces with larger local dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=', qutrits or  higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Indeed, there are not many qubit isometries for  a given value of the uniformity parameter, and, in repro-  ducing some results of Proposition 1, we had to resort  to splitting the codes of higher dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' A simi-  lar tendency was observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [35] where genuinely  entangled subspaces were constructed with the isometric  mapping method: when local dimension goes to infinity,  the dimension of the obtained subspaces asymptotically  approaches the maximal possible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Finally, let us provide some constructions with higher  values of the uniformity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Consider the pure QMDS code [[10, 4, 4]]3 from The-  orem 13 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  From Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='2 with α =  β = 1, we obtain a 5-uniform 9-dimensional subspace  of  �  C3�⊗18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' The corresponding isometry V has the input  dimension equal to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Let us take a maximally entan-  gled subspace of C2 ⊗ C9, which has dimension equal to  ⌊ 9  2⌋ = 4 (Corollary 4 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Action of V on the sec-  ond party of each state in this subspace yields a 5-uniform  4-dimensional subspace of C2 ⊗  �  C3�⊗18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' With the same  isometry V we could act instead on the joint subsystem  of three parties of each state in the code space [[5, 1, 3]]2,  and this procedure yields a 5-uniform 2-dimensional sub-  space of  �  C2�⊗2 ⊗  �  C3�⊗18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  Consider the [[10, 0, 6]]4 code, which can be obtained  from the classical [10, 5, 6] MDS code over GF(16) of  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [46] by the correspondence between stabilizer QMDS  codes and self-dual classical MDS codes (Theorem 15 of  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [47], see also Proposition 15 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' By elimina-  tion of one party a code ((9, 4, 5))4 is constructed (The-  orem 20 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  By splitting the latter code we  obtain a ((18, 4, 5))2 code whose encoding isometry has  the input dimension equal to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Applying this isometry  to one of the parties of a maximally entangled state in  C3 ⊗ C3 produces a 4-uniform state in C3 ⊗  �  C2�⊗18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  DISCUSSION  In this paper we’ve shown how new r-uniform states  and subspaces can be constructed from combining al-  ready known quantum error correcting codes, (maxi-  mally) entangled states and subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The isometric  mapping method played the key role here: one takes an  isometry, which, as its range, has a subspace with some  useful property, and applies it to states or subspaces,  perhaps with some other interesting property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' This ap-  proach allowed us to complement some results which were  obtained with the mixed orthogonal arrays method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' It  would be interesting to continue this parallel with OAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  One example in this direction could be analyzing encod-  ing isometries of the QECCs obtained with OAs, for in-  stance, the ones from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' This could potentially  lead to new OA and MOA constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The advantage of the presented approach is its exper-  imental accessibility: whenever one can realize encoding  isometries of QECCs as well as prepare auxiliary entan-  gled states, one can construct uniform states in accor-  dance with the described above procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  The dis-  advantage of the approach is that it doesn’t utilize the  internal structure of the combined objects beyond their  uniformity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Taking more structural properties  into account could result in constructing more classes of  useful states and subspaces such as, for example, AME  states, the ones we couldn’t produce with the current  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  This observation suggests another direction  of further research: how to combine several QECCs in  the most efficient way, with taking their specific proper-  ties into account, to obtain a new QECC with “good”  characteristics (in the sense similar to the recent “good  quantum codes” constructions [48, 49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' We stress that  the distance of the codes composed by the procedure of  Theorem 2 doesn’t scale with the number of codes be-  ing combined: the distance of the resulting code will al-  ways be upper-bounded by the minimum of the number  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' (25) taken over all the codes being combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  It also would be interesting to apply the isometric map-  ping method to construction of multipartite subspaces  with another useful property – distillability and closely  related to it non-positivity of partial transpose across  each bipartition (distillable and NPT subspaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' This  direction of research could complement the results ob-  tained in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' [18, 50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The author thanks M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
+page_content=' Lomonosov Moscow State  University for supporting this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfXAQH/content/2301.03120v1.pdf'}
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diff --git a/GtAyT4oBgHgl3EQffPh0/content/tmp_files/2301.00336v1.pdf.txt b/GtAyT4oBgHgl3EQffPh0/content/tmp_files/2301.00336v1.pdf.txt
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+From discrete to continuous: Monochromatic
+3-term arithmetic progressions
+Torin Greenwood∗, Jonathan Kariv†, Noah Williams‡
+December 31, 2022
+Abstract
+We prove a known 2-coloring of the integers [N] := {1, 2, 3, ..., N}
+minimizes the number of monochromatic arithmetic 3-progressions under
+certain restrictions. A monochromatic arithmetic progression is a set of
+equally-spaced integers that are all the same color.
+Previous work by
+Parrilo, Robertson and Saracino conjectured an optimal coloring for large
+N that involves 12 colored blocks. Here, we prove that the conjecture is
+optimal among anti-symmetric colorings with 12 or fewer colored blocks.
+We leverage a connection to the coloring of the continuous interval [0, 1]
+used by Parrilo, Robertson, and Saracino as well as by Butler, Costello
+and Graham. Our proof identifies classes of colorings with permutations,
+then counts the permutations using mixed integer linear programming.
+1
+Introduction
+Consider coloring each of the integers in [N] with one of r colors. A κ-term
+arithmetic progression is any subset of κ equally-spaced integers, denoted a κ-
+AP. An arithmetic progression is monochromatic if every term is colored the
+same color. Can we color [N] in a way that avoids all monochromatic κ-APs?
+A classic result is van der Waerden’s Theorem:
+Theorem 1.1 (van der Waerden, [17]). For any integers r, κ ≥ 1, there exists
+a number N such that every r-coloring of [N] has a monochromatic κ-AP.
+Given that monochromatic κ-APs are guaranteed to exist when enough num-
+bers are colored, we ask a refined question: what is the minimum number of
+monochromatic κ-APs that could exist? To be more precise, define Cr(N) to
+be the set of r-colorings of [N]. For any c ∈ Cr(N), let mκ(c) be the number of
+∗Department
+of
+Mathematics,
+North
+Dakota
+State
+University,
+Fargo,
+ND
+USA,
+torin.greenwood@ndsu.edu
+†Isazi Consulting, Johannesburg, South Africa, jkariv@isaziconsulting.co.za
+‡Department of Mathematical Sciences, Appalachian State University, Boone, NC USA,
+williamsnn@appstate.edu
+1
+arXiv:2301.00336v1  [math.CO]  1 Jan 2023
+
+monochromatic κ-APs induced by c. Finally, let APκ(N) be the total number
+of κ-APs in [N], regardless of whether they are monochromatic or not. Then,
+we look at
+Pr,κ(N) :=
+min
+c∈Cr(N)
+mκ(c)
+APκ(N).
+The focus of this paper is to examine the minimum for monochromatic 3-APs
+within 2-colorings, P(N) := P2,3(N).
+In 1999, Ron Graham proposed that
+limn→∞ P(n) = β for some constant, β, and offered a $100 prize for finding
+β.
+Originally, it was not clear whether colorings could perform better than
+random in the long run: for large values of N, is it possible to color [N] so
+the probability that a randomly selected 3-AP is monochromatic is less than
+(1/2)3 + (1/2)3 = 1/4? It is notable that the analogous question for 2-colorings
+of Zp is answered negatively for p prime. Indeed, Lu and Peng [12] show that for
+a given 2-coloring of Zp, the fraction of 3-APs that are monochromatic depends
+only on the fraction of each color present in the coloring.
+For our question concerning 2-colorings of [N], Parrilo et al. [14] and Butler
+et al. [2] verified independently but nearly simultaneously that it is possible
+to do better than random, and they found upper and lower bounds for the
+minimum monochromatic APs. The upper bound was attained through simu-
+lating good colorings and finding one that performed well. They landed on the
+following 12-block coloring:
+Explicitly, when coloring [N], the blocks would be approximately of the following
+sizes:
+�28N
+548 , 6N
+548, 28N
+548 , 37N
+548 , 59N
+548 , 116N
+548 , 116N
+548 , 59N
+548 , 37N
+548 , 28N
+548 , 6N
+548, 28N
+548
+�
+(1)
+Due to this coloring, P(N) ≤
+117
+548 + o(1).
+Note that this coloring is anti-
+symmetric: the left half of the coloring is a mirror image of the right half
+but uses opposite colors. In [2], Butler et al. performed many computer sim-
+ulations using genetic algorithms to find the optimal coloring, and noted that
+this same 12-block coloring consistently appeared regardless of the seed coloring
+with which they started. They noted that a remaining challenge would be to
+analyze the case of rapidly alternating colorings.
+The goal of this paper is to show that as N → ∞, the 2-coloring of [N] that
+has alternating color blocks with sizes given in Equation (1) is globally optimal
+among anti-symmetric colorings with at most 12 blocks. As far as the authors
+are aware, this is the first result of optimality under any restrictions. Here, we
+let ˜C2(N) be the 2-colorings of [N] that are anti-symmetric and have at most
+12 contiguous segments of red or blue. Then, define
+˜P(N) =
+min
+c∈ ˜C2(N)
+m3(c)
+AP3(N).
+Our main result is as follows:
+2
+
+Theorem 1.2. Consider coloring each integer in [N] with either red or blue
+such that the coloring is anti-symmetric and has at most 12 contiguous blocks.
+Then, as N increases the minimum possible fraction of arithmetic progressions
+approaches 117
+548. That is, limN→∞ ˜P(N) = 117
+548.
+Below, we provide a proof sketch that outlines the sections in the paper.
+Sketch of proof. First, we will convert from discrete colorings of [N] to continu-
+ous colorings of [0, 1] with at most 12 contiguous segments, referred to as block
+colorings. After restricting the number of color changes that can occur within
+a coloring, it turns out that optimizing the discrete colorings is the same as
+optimizing the continuous colorings, as described rigorously in Lemma 3.6.
+When switching to the continuous realm, we let a continuous coloring be a
+function c : [0, 1] → {0, 1}, where 0 and 1 (in the range) represent red and blue,
+respectively. Then, we let f[0,1](c) be the fraction of arithmetic progressions in
+the coloring c that are monochromatic. We can represent this fraction geomet-
+rically by a BCG diagram, described by Butler, Costello, and Graham in [2] and
+illustrated in Figure 1 below. When c consists of 12 contiguous segments, we
+label the endpoints of the coloring as (x0 = 0, x1, . . . , x12 = 1). As we allow the
+coloring c to vary, f[0,1](c) is a piecewise quadratic function in the xi. Moreover,
+each piece of f[0,1](c) is determined completely by the relative ordering of the
+pairs of sums {xi + xj}, as described in Lemma 4.1.
+Next, we aim to identify every piece of the quadratic function over all color-
+ings c of [0, 1] with 12 intervals. Using the GNU Linear Programming Kit [9],
+we count 371, 219 possible arrangements of {xi + xj} that could give distinct
+quadratics in f[0,1], as proved in Lemma 4.2 with the help of our code available
+online at https://cocalc.com/TorinGreenwood/MonochromeSequences/Mo
+nochromaticProgressions.
+Finally, once we have identified the 371, 219 possible pieces in the quadratic
+function, we search for the global minimum of f[0,1] among all these pieces.
+Fortunately, from Lemma 4.3, it turns out that f[0,1] is a continuous function
+with continuous partial derivatives. Thus, we can minimize f[0,1] by search-
+ing for all critical points within each piece of the quadratic. Because f[0,1] is
+piecewise quadratic, its critical points are determined by systems of linear in-
+equalities (defining the domain of a piece of f[0,1]) and equalities (setting the
+partial derivatives of f[0,1] to zero), allowing us again to use linear programming
+to identify the critical points. We describe our search for these critical points
+in Lemma 4.4, completing the proof.
+A byproduct of our proof structure is that among colorings with a fixed
+number of contiguous blocks, there exist optimal colorings with rational end-
+points, as described in Corollary 4.5. In Section 5, we show that with respect
+to 2-colorings of the continuous unit circle S1, the fraction of monochromatic
+APs depends only on the measure of points colored red. This is analogous to
+the results in [5, 12] that concern colorings of Zp for p prime.
+3
+
+2
+Background
+When searching for bounds on the number of monochromatic arithmetic pro-
+gressions in [N], Frankl, Graham, and R¨odl developed the following theorem:
+Theorem 2.1 (Frankl, Graham, R¨odl, [7]). For fixed r and κ, there exists ℓ > 0
+so that the number of monochromatic κ-APs in any r-coloring of {1, 2, . . . , N}
+is at least ℓN 2 + o(N 2).
+This proved that a positive fraction of APs must be monochromatic in the
+long run, but gave no indication of how small ℓ could be.
+Datskovsky made progress on a related problem in [5], analyzing the minimal
+number of monochromatic Schur triples in [N]. A Schur triple (a, b, c) from
+[N] is any triple of integers where a + b = c.
+Datskovsky investigated the
+minimum possible number of monochromatic Schur triples when coloring each
+integer red or blue, and proved that asymptotically, the minimum is N 2/11. The
+proof relied on using a discrete Fourier transform, which yielded a combinatorial
+identity that broke down counts of Schur triples into a few easier to analyze
+sets. Although our proof does not use the discrete Fourier transform, it also
+will transform a discrete problem into a continuous space.
+In [14], Parrilo et al. applied some of the tools from Datskovsky’s work
+to arithmetic progressions. Again, the authors found a combinatorial identity
+breaking down sets of arithmetic progressions into simpler sets, but it was no
+longer possible to enumerate these sets exactly. Instead, the authors ended up
+with bounds on the minimum number of monochromatic progressions possible in
+[N]. They also identified the coloring shown in Equation (1) in the introduction
+above, and verified it was locally optimal among colorings with 12 intervals
+that are antisymmetric. Our paper aims to prove that this coloring is optimal
+globally among the same set of colorings.
+Constellations are a generalization of APs studied in [2], where instead of
+all points being equally spaced like in an AP, the consecutive differences of
+terms must satisfy some fixed proportions. Butler et al. analyzed constellations
+by representing sets of monochromatic constellations using integrals of indicator
+functions. This led them to represent monochromatic regions in two-dimensional
+diagrams which we refer to as BCG diagrams, as illustrated in Figure 1. Visu-
+alizing progressions via these diagrams is crucial to our proof, and provides the
+connection we need between discrete and continuous realms.
+One important aspect of our proof is enumerating the number of ways
+pairwise sums {xi + xj} can be ordered for a list of positive real numbers
+x0 ≤ x1 ≤ . . . ≤ xn with n even and xi + xn−i = 1.
+This problem could
+be framed as counting the number of chambers in a hyperplane arrangement,
+and there already exists a rich set of tools for counting chambers, as seen for ex-
+ample in [16]. However, in this paper, we use mixed integer linear programming,
+which is well-suited to determining whether a system of linear inequalities has
+a solution. This coding approach was also employed by Miller and Peterson in
+[13] when they counted more sums than differences sets, and also by Laaksonen
+4
+
+in [10] when he counted closely-related arrangements of sums of pairs. More
+details on this approach are given in Section 4.2 below.
+The current best known bounds on the minimum number of monochromatic
+κ-APs in the general (non-antisymmetric) case for κ > 3 are found using an
+“unrolling” strategy, described in [12] and [3].
+Here, an optimal coloring of
+some interval {1, . . . , ℓ} for ℓ ≪ N is found explicitly, and then repeated to fill
+the interval [N]. Although this strategy works well for κ > 3, when κ = 3, the
+colorings do no better than random in the long run.
+3
+Relationship between discrete and continuous
+case
+In this section, we define a precise connection between discrete 2-colorings of
+[N], and a natural continuous analogue of 2-coloring [0, 1]. First, we pause to
+define a 3-AP in [N] formally: a 3-AP is any set of 3 terms (a, a+d, a+2d) each
+in [N] where d is any integer including negative values or zero. It is convenient
+for us to include the case where d ≤ 0 in our arguments, although this choice
+ultimately does not change which colorings minimize monochromatic APs nor
+the minimum they attain.
+For the interval [0, 1], we identify any 3-AP (a, a + d, a + 2d) by its first and
+last term (a, a + 2d) in [0, 1] × [0, 1], now allowing d to be any real number.
+We obtain a measure on the set of 3-APs in [0, 1] by choosing the starting and
+ending point of the progressions uniformly. A coloring of the interval is defined
+to be a function c : [0, 1] → {0, 1}.
+In this section, we begin by discussing measurable colorings of [0, 1], which
+can be approximated in a standard way by bead colorings, defined below. Then,
+we show that minimizing monochromatic APs over all measurable colorings of
+[0, 1] is the same as minimizing all APs over just bead colorings, as formalized
+in Lemmas 3.1, 3.2, and 3.3 below.
+Next, we justify that every discrete coloring of [N] has a corresponding con-
+tinuous coloring of [0, 1], and that the fraction of monochromatic APs in the
+continuous coloring is a function of both the monochromatic APs and monochro-
+matic off-by-1 APs in the discrete coloring, as explained above and in Lemma
+3.4. Using this connection, we find that when the number of blocks of contiguous
+runs of colors in a coloring is bounded by n, the fraction of APs in a continuous
+coloring versus its discrete analogue is small as N grows large, formalized in
+Lemma 3.5. Finally, this allows us to prove our main result of the section: that
+minimizing over discrete colorings with a fixed number of blocks is the same as
+minimizing over continuous colorings with the same number of blocks, stated
+rigorously in Lemma 3.6.
+Now, we begin stating our results formally, starting with the definition of a
+Lebesgue-measurable coloring.
+Definition 1. A coloring of [0, 1] is Lebesgue-measurable if c−1(0) is Lebesgue-
+measurable (or equivalently c−1(1) is Lebesgue-measurable).
+5
+
+Definition 2. A bead coloring of [0, 1] is a coloring where for some ℓ, each
+of the intervals ( i
+ℓ, i+1
+ℓ ) is monochrome for i = 0, 1, . . . , ℓ − 1. Each interval
+( i
+ℓ, i+1
+ℓ ) is called a bead, and we sometimes refer to such a coloring as an ℓ-bead
+coloring.
+We introduce bead colorings because they are the continuous analogue of
+coloring the integers [N] obtained by fattening each integer into an interval.
+Our goal is to show that when optimizing colorings over the interval [0, 1], we
+may restrict our attention to bead colorings. We call the set of bead colorings B
+and the set of Lebesgue-measurable colorings M. Observe that B ⊂ M. Finally
+we define a difference between two colorings as follows.
+Definition 3. For two colorings ca ∈ M and cb ∈ M we define d(ca, cb) :=
+µ({x | ca(x) ̸= cb(x)}), where µ is the usual Lebesgue measure on R.
+Recall that we identify an arithmetic progression in [0, 1] by the pair of
+starting and ending points in [0, 1]. For a coloring c on [N], we define f[N](c) to
+be the fraction of arithmetic progressions that are monochromatic. Analogously,
+when c is a coloring of [0, 1], we have the following definition:
+Definition 4. For a coloring c : [0, 1] → {0, 1}, let f[0,1](c) be the Lebesgue
+measure of the set of monochromatic arithmetic 3-term progressions (viewed as
+a subset of [0, 1]2) induced by the coloring c.
+We justify our restriction to bead colorings with the following standard
+measure-theoretic lemmas (proved for completeness momentarily):
+Lemma 3.1. For any two measurable colorings c1 and c2 of [0, 1], if d(c1, c2) <
+ϵ, then |f[0,1](c1) − f[0,1](c2)| < 4ϵ.
+Lemma 3.2. For any measurable coloring cm of [0, 1] and any ϵ > 0 there exists
+a bead coloring cb such that cm and cb disagree on a set of measure at most ϵ.
+As B ⊂ M, Lemma 3.2 immediately implies the following:
+Lemma 3.3. Optimizing monochromatic 3-APs over bead colorings is the same
+as optimizing over all measurable colorings in the following sense:
+inf
+cb∈B f[0,1](cb) =
+inf
+cm∈M f[0,1](cm).
+We begin with the proof of Lemma 3.1.
+Proof of Lemma 3.1. Let A ⊂ [0, 1] be a set of measure ϵ, and consider flipping
+the colors of all elements in A. There are three classes of monochromatic 3-APs
+that could be created or destroyed: the APs where the first, middle or last
+element is flipped (where some APs may belong to more than one class). We
+consider the measure of each of these three classes. As the first and last elements
+of a progression are chosen uniformly, the corresponding classes have measure
+ϵ. The middle element is the average of two uniform random variables, and so
+6
+
+has a triangular distribution on [0, 1] with maximum density 2. Therefore the
+set of monochrome progressions whose middle term is in A would have measure
+at most 2ϵ. Summing the measures of these three classes yields an upper bound
+for their union of 4ϵ.
+We now justify Lemma 3.2, whose proof is a standard measure-theoretic
+argument.
+Proof of Lemma 3.2. By hypothesis, the set Xbl := c−1
+m (0) of blue-colored el-
+ements of [0, 1] is measurable with finite measure. So, a standard result from
+measure theory (e.g.
+[15, Theorem 12]) establishes the existence of a finite
+disjoint collection of open intervals I1, . . . , Iℓ ⊂ [0, 1] satisfying
+µ
+�� ℓ�
+i=1
+Ii
+�
+\ Xbl
+�
++ µ
+�
+Xbl \
+ℓ�
+i=1
+Ii
+�
+< ϵ
+2.
+Since the rationals are dense in [0, 1], we can perturb the 2ℓ endpoints of the
+intervals {Ii}, each by some amount less than
+ϵ
+4ℓ, to find a disjoint collection
+I′
+1, I′
+2, . . . , I′
+ℓ of open intervals with rational endpoints. Let Ubl be the union of
+these intervals. Then, Ubl and Xbl have a symmetric difference of measure at
+most ϵ. It follows that the coloring cb defined by coloring each interval of Ubl
+blue is a bead coloring for which d(cb, cm) < ϵ.
+Call a progression an off-by-1 AP if it is of the form (a, a + d, a + 2d ± 1).
+We will show that we can easily compute f[0,1](cb) for a bead coloring cb with N
+beads by considering the colored beads as an integer coloring of [N], computing
+the number of 3-term APs in this sequence, and adding half of the off-by-1 APs.
+Recall that for a discrete coloring c, m3(c) is the number of monochromatic
+3-APs induced by c. Let m′
+3(c) be the number of monochromatic off-by-1 APs.
+Then, we have the following comparison between colorings of [0, 1] with exactly
+N beads (of not necessarily alternating colors) and corresponding colorings of
+[N].
+Lemma 3.4. Let cb be an N-bead coloring of [0, 1], and let c∗
+b be the discrete
+coloring of [N] corresponding to cb, where the number i is colored blue if and
+only if the ith bead in cb is colored blue. Then,
+f[0,1](cb) = m3(c∗
+b) + m′
+3(c∗
+b)/2
+N 2
+.
+Proof. Consider a randomly chosen progression in [0, 1] identified by its end-
+points (a, b), and a fixed N-bead coloring cb. We use a probabilistic proof, so
+we rewrite
+(µ × µ)((a, b) ∈ [0, 1]2 : (a, b) is monochromatic) =: P((a, b) monochromatic),
+where µ × µ is the usual Lebesgue measure on R2. We will condition on which
+beads S and E contain a and b. Let M be the bead containing the middle
+7
+
+element of the progression. Given a bead coloring of [0, 1], it is useful to define
+the distance between two beads A and B, db(A, B) as 0 when A = B and as one
+more than the number of other beads strictly between A and B otherwise. Note
+that when db(S, E) is even, then S, M, and E must form a 3-AP of beads. On the
+other hand, when db(S, E) is odd, S, M, and E form an off-by-one progression
+and M could be two possible beads depending on the internal positioning of a
+and b within S and E. Formally, letting {Bi}N
+i=1 be the set of beads,
+f[0,1](cb) =
+�
+i,j
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)
+=
+�
+d(Bi,Bj) even
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)
++
+�
+d(Bi,Bj) odd
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)
+(2)
+Now, P(a ∈ Bi, b ∈ Bj) = 1/N 2 for each i and j since a and b are indepen-
+dently and uniformly distributed among the beads. Also, because our coloring
+is fixed, when db(Bi, Bj) is even P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) is 0 or
+1 depending on whether or not the beads S, M, and E form a monochromatic
+3-term AP. Thus,
+�
+d(Bi,Bj) even
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)
+= m3(c∗
+b) · 1
+N 2 .
+(3)
+When db(Bi, Bj) is odd, there are two choices for M: B(i+j−1)/2 or B(i+j+1)/2.
+Thus, we can condition on these two choices:
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj)
+= P((a, b) mono.|a ∈ Bi, b ∈ Bj, M = B(i+j−1)/2) · P(M = B(i+j−1)/2|a ∈ Bi, b ∈ Bj)
++ P((a, b) mono.|a ∈ Bi, b ∈ Bj, M = B(i+j+1)/2) · P(M = B(i+j+1)/2|a ∈ Bi, b ∈ Bj)
+Here, P(M = B(i+j−1)/2|a ∈ Bi, b ∈ Bj) = P(M = B(i+j+1)/2|a ∈ Bi, b ∈ Bj) =
+1/2 because a and b are positioned uniformly within S and E. Additionally,
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj, M = B(i+j−1)/2)
+is 0 or 1 depending on whether the off-by-1 progression in c∗
+b is monochromatic
+or not. Hence, these two terms combined simplify to
+�
+d(Bi,Bj) odd
+P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)
+=
+1
+2N 2 m′
+3(c∗
+b).
+(4)
+8
+
+Plugging in Equations (3) and (4) into Equation (2) completes the proof.
+Much of the rest of this paper will deal with a particular class of colorings
+called “block colorings” which we now define. Informally, they are partitions of
+I into disjoint intervals which are alternately colored red and blue.
+Definition 5. For a finite collection of endpoints {xi} such that 0 = x0 <
+x1 < x2 < · · · < xn−1 < xn = 1, we define the associated “block” coloring as
+the coloring where the n intervals Ji = (xi−1, xi) (for i ∈ {1, 2, . . . , n}) are all
+monochrome and alternate in color.
+Note that the colors assigned to the endpoints {xi} (or indeed to any points
+within a measure zero set) do not matter. With these definitions, we can now
+compare the performance of discrete colorings with their continuous analogues.
+Lemma 3.5. Let C(N, n) be the set of 2-colorings of [N] with at most n contigu-
+ous blocks of colors. For any coloring c ∈ C(N, n), let c∗ be the corresponding
+block coloring of [0, 1] where the interval [(i − 1)/N, i/N) is colored blue by c∗ if
+and only if i ∈ [N] is colored blue by c. Then,
+max
+c∈C(N,n)
+��f[0,1](c∗) − f[N](c)
+�� = O
+� n
+N
+�
+.
+Here, there exists a C > 0 independent of N and n such that |O(n/N)| < Cn/N
+for all positive integers n and N.
+Proof. Our proof will use Lemma 3.4 to rewrite f[0,1](c∗) in terms of f[N](c).
+Before proceeding with this, we will interpret the number of off-by-1 monochro-
+matic APs induced by c, m′
+3(c), in terms of the regular monochromatic APs,
+m3(c). We claim the following:
+m′
+3(c) = 2m3(c) + O(nN).
+(5)
+To verify this, note that each AP (a, a + d, a + 2d) in [N] corresponds almost
+bijectively to a pair of off-by-1 APs by moving the first or last endpoint inwards
+by one: (a + 1, a + d, a + 2d) or (a, a + d, a + 2d − 1). (When N is odd, this
+misses exactly two off-by-1 APs: (1, (N −1)/2, N) and (1, (N +1)/2, N). When
+N is even, this is truly a bijection.)
+Using this near bijection, we now compare when APs and off-by-1 APs are
+monochromatic. Under this contraction action, the only time an AP (a, a +
+d, a + 2d) is monochromatic while one of its corresponding off-by-1 APs is not
+monochromatic is when a or a + 2d is adjacent to a number of the opposite
+color, and the same could be said if the original AP is not monochromatic but
+the off-by-1 AP is. If our coloring only has n intervals total, there are only n−1
+ways to position a immediately before a color change, and similarly only n − 1
+ways to position a + 2d immediately after a color change. Since d can still be
+chosen freely, there are O(nN) possible off-by-1 APs that disagree with their
+corresponding APs on being monochromatic, verifying our claim.
+9
+
+Now, in the notation of Lemma 3.4, we see that (c∗)∗ = c. Thus,
+f[0,1](c∗) = m3(c)
+N 2
++ m′
+3(c)/2
+N 2
+= 2m3(c) + O(nN)
+N 2
+= m3(c)
+N 2/2 + O
+� n
+N
+�
+.
+The proof of the lemma will be complete if we can verify the following:
+m3(c)
+N 2/2 = f[N](c) + O
+� 1
+N
+�
+.
+To see this, recall that by definition f[N](c) = m3(c)/ AP3(N), so that
+m3(c)
+N 2/2 − f[N](c) =
+m3(c)
+AP3(N) · AP3(N) − N 2/2
+N 2/2
+.
+(6)
+We have m3(c) ≤ AP3(N), so that the first fraction on the right in Equation (6)
+is at most 1. Next, note that AP3(N) = N 2/2+O(N): it is easy to compute this
+explicitly for when N is even or odd. But, intuitively, if we pick two numbers x
+and y from [N] at random, there are N 2 ways to do this, and about half the time
+x − y is even and these correspond to the start and end of a 3-AP. Therefore,
+AP3(N) − N 2/2 = O(N), and plugging this into Equation (6) completes the
+proof with
+m3(c)
+N 2/2 − f[N](c) = O
+� 1
+N
+�
+.
+Finally, we end this section with the result rigorously justifying our conver-
+sion between discrete and continuous colorings.
+Lemma 3.6. Let Sn be the block 2-colorings of [0, 1] with at most n blocks, and
+let C(N, n) be the 2-colorings of [N] with at most n contiguous blocks, where
+n = o(N) as N approaches infinity.
+Then, minimizing monochromatic APs
+over Sn is the same as minimizing monochromatic APs over C(N, n) in the
+following sense:
+lim
+N→∞
+���� inf
+c∈Sn f[0,1](c) −
+min
+c∈C(N,n) f[N](c)
+���� = 0.
+Here, we consider block colorings of [0, 1] where the edge of a block is at a
+possibly irrational number. However, as we will see later, all optimal colorings
+of [0, 1] with a fixed number of blocks must have rational endpoints.
+10
+
+Proof. This is mostly a standard ϵ argument, so let ϵ > 0 be given. We aim to
+show for all N sufficiently large,
+���� inf
+c∈Sn f[0,1](c) −
+min
+c∈C(N,n) f[N](c)
+���� ≤ ϵ.
+We prove this in two halves, first proving the infimum is nearly bounded above
+by the minimum, and then arguing the reverse. Consider any coloring ˜c ∈ Sn.
+Then, by Lemma 3.1, for every N sufficiently large, we can find a n-block
+coloring ˜cN of [0, 1] with endpoints of the form r/N for r an integer such that
+��f[0,1](˜c) − f[0,1](˜cN)
+�� < ϵ/4.
+(7)
+This is true because we can round each endpoint to the nearest 1/N. Then,
+we define ˜c∗
+N to be the coloring of [N] where i is colored blue if and only if the
+ith block of ˜cN is blue. Note that ˜c∗
+N still only has at most n blocks, and that
+using the notation from Lemma 3.5, (˜c∗
+N)∗ = ˜cN. So, from Lemma 3.5, for N
+sufficiently large (independent of the colorings ˜c, ˜cN, ˜c∗
+N),
+|f[0,1](˜cN) − f[N](˜c∗
+N)| = O(n/N).
+(8)
+By choosing N sufficiently large (independent of the colorings ˜c, ˜cN, ˜c∗
+N), Equa-
+tions (7) and (8) imply
+min
+c∈C(N,n) f[N](c) ≤ f[N](˜c∗
+N) < f[0,1](˜c) + ϵ
+where this bound holds for all N sufficiently large and for all ˜c ∈ Sn. Therefore,
+for all N sufficiently large,
+min
+c∈C(N,n) f[N](c) ≤ inf
+c∈Sn f[0,1](c) + ϵ.
+Now, we prove the reverse inequality: consider any coloring ˆc of [N], and
+let ˆc∗ be the coloring of [0, 1] induced by ˆc. Again, from Lemma 3.5, for N
+sufficiently large,
+��f[N](ˆc) − f[0,1](ˆc∗)
+�� < ϵ,
+and since this is true for any coloring ˆc ∈ C(N, n), this proves that for N
+sufficiently large,
+inf
+c∈Sn f[0,1](c) ≤
+min
+c∈C(N,n) f[N](c) + ϵ.
+Combining this with the complementary inequality above completes the proof.
+At this point, we have justified that once bounding the number of blocks
+in our coloring, optimizing colorings of [N] is the same as optimizing colorings
+of [0, 1]. We only make use of this result when the number of blocks n = 12
+because that is the conjectured global optimal number of blocks. But, the same
+proof shows that switching to the continuous realm works whenever n = o(N)
+as N → ∞.
+11
+
+first term in progression
+third term in progression
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+coloring
+of [0, 1]
+Figure 1: Below the horizontal axis a coloring, c, is depicted. The horizontal
+axis represents the first term a in an arithmetic progression, and the vertical
+axis represents the third term a+2d in the progression. Whenever a point in the
+diagram is colored red (blue), this corresponds to the progression (a, a+d, a+2d)
+being colored red (blue) by c.
+4
+Proofs for the continuous case
+4.1
+Colorings can be represented by BCG diagrams
+Consider any block coloring c : [0, 1] → {0, 1} of the interval with endpoints
+of the blocks given by {x0, x1, . . . , xn} with x0 = 0 and xn = 1. Without loss
+of generality, assume that the first block (x0, x1) is colored blue, and alternate
+colors for each remaining interval. Recall that the colors of the endpoints of the
+blocks can be assigned in any way, since this does not change the probability of
+selecting a monochromatic progression.
+In [2], Butler, Costello, and Graham proposed a method of visualizing the
+monochromatic arithmetic progressions associated to a coloring in terms of di-
+agrams like in Figure 1. Any arithmetic progression (a, a + d, a + 2d) can be
+identified uniquely by its first and last coordinates, which are represented by
+the horizontal and vertical axes of such a diagram. Note that the diagram is
+divided into vertical strips, horizontal strips, and northwest/southeast diagonal
+strips. Consider any region identified as the intersection of one horizontal, one
+vertical, and one diagonal strip. For a block coloring, this region corresponds
+12
+
+0.8
+0.6
+0.4
+0.2 
+0
+0
+0.2
+0.4
+0.6
+0.8
+xto a collection of monochromatic arithmetic progressions if and only if the in-
+dices of the vertical, horizontal, and diagonal strips defining the region all have
+matching parities.
+Because the total area of the square in any diagram like Figure 1 is one, the
+measure of the set of monochromatic sequences is equal to the sum of the areas
+of the red and blue regions. In Theorem 2.1 of [2], Butler et al. express the
+total colored area as the sum of two integrals involving an indicator function.
+Their work applied to constellations, a generalization of arithmetic progressions.
+Here, we instead derive explicit polynomial equations for the areas. Consider
+any one colored region in such a diagram. As the endpoints xi are perturbed
+slightly, the region remains the same type of polygon although its dimensions
+may change.
+This implies that the area of each region can be represented
+locally as a quadratic in the variables {xi}. Denote a block coloring c by its list
+of endpoints x := (x0, . . . , xn). Then, summing over all monochromatic regions
+shows that the measure of the monochromatic progressions, f[0,1](x), is locally
+quadratic in the {xi}, too. We now denote f(x) := f[0,1](x). When we restrict
+f to act on colorings with exactly n blocks, we will write f(xn).
+As x varies, the regions in the diagram change polygon type. Thus, for each
+n, f(xn) is a piecewise function that is locally quadratic. In order to minimize
+f globally, we wish to identify the boundaries of these pieces in terms of x. The
+following lemma describes how to identify the polygons in such a diagram.
+Lemma 4.1. The region that is the intersection of the ith vertical strip, jth
+horizontal strip, and kth diagonal strip of a diagram is empty or forms a closed
+polygon. The type of polygon is determined by testing whether each of the four
+values {xi + xj, xi + xj+1, xi+1 + xj, xi+1 + xj+1} is greater than or less than
+the two values {2xk, 2xk+1}.
+If this ordering is known, the area of the cor-
+responding region can be expressed as a quadratic polynomial in the variables
+{xi, xi+1, xj, xj+1, xk, xk+1}.
+Proof. In the diagrams like in Figure 1, the horizontal lines all are given by
+{y = xi}n
+i=0 and the vertical lines by {x = xi}n
+i=0.
+At any point (x, y) in
+the diagram, the middle value in the corresponding arithmetic progression is
+(x+y)/2, and setting this equal to any endpoint in our coloring implies that the
+diagonal lines are given by {y = 2xi − x}n
+i=0. As described above, for any triple
+(i, j, k) where i, j, k ∈ {0, . . . , 12} all have matching parities, the intersection of
+the ith vertical strip, jth horizontal strip, and kth diagonal strip corresponds
+to a region of monochromatic arithmetic progressions.
+To determine the shape of the region of the monochromatic progressions,
+first consider the rectangle formed by the intersection of the ith vertical strip
+and jth horizontal strip. The corners of this rectangle have coordinates (xi, xj),
+(xi+1, xj), (xi, xj+1), and (xi+1, xj+1), as labelled in Figure 2. In order for the
+intersection of this rectangle with the kth diagonal strip to be non-empty, we
+need the upper diagonal line y = 2xk+1 − x to be above the lower left corner of
+the rectangle, (xi, xj), and the lower diagonal line y = 2xk − x to be below the
+upper right corner of the rectangle, (xi+1, xj+1). This is the same as requiring
+the inequalities 2xk+1 ≥ xi + xj and 2xk ≤ xi+1 + xj+1.
+13
+
+ith vertical strip
+jth
+horizontal
+strip
+kth diagonal strip
+(x    , x    )
+i+1
+j+1
+(x    , x )
+i+1
+j
+(x , x )
+i
+j
+(x , x    )
+i
+j+1
+y = 2x  - x
+k
+y = 2x     - x
+k+1
+Figure 2: Above is the intersection of the ith vertical strip, jth horizontal
+strip, and kth diagonal strip determined by a block coloring with endpoints
+x = (x0, x1, . . . , xn). Whether the intersection is empty can be determined by
+comparing the diagonal lines {y = 2xk − x, y = 2xk+1 − x} to the corners of
+the box {(xi, xj), (xi+1, xj+1)}.
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi+1 + xj ≤ 2xk ≤ xi + xj+1 ≤ xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(xi+1 − xi)(xj+1 + xi/2 + xi+1/2 − 2xk)
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj,
+max(xi + xj+1, xi+1 + xj) ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(xi+1 − xi)(xj+1 − xj) − (2xk+1 − xj+1 − xi+1)2/2
+Figure 3: Illustrated here are two different ways that the kth diagonal strip
+can intersect with the ith horizontal and jth vertical strip in a coloring. The
+resulting monochromatic region is shaded in gray, and the area of the region is
+given as a quadratic in x below the diagram. The type of polygon is determined
+by the partial permutation given below each diagram. The other 18 possibilities
+are enumerated in Appendix A.
+14
+
+Additionally, the type of polygon formed by the intersection of the strips is
+determined by whether the two diagonal lines y = 2xk − x and y = 2xk+1 − x
+are above or below each of the four corners of the box. For any specific relation-
+ship between the lines and the four corners, some basic geometric arguments
+allow us to find the area of the polygon enclosed by the strips in terms of
+{xi, xi+1, xj, xj+1, xk, xk+1}. It turns out that there are 20 possible arrange-
+ments of the lines that yield distinct polygons. In Figure 3, two possibilities are
+given, along with the corresponding quadratic equations for their areas. The
+full list of 20 polygons is given in Appendix A.
+4.2
+Enumerating BCG diagrams
+Now that we have identified criteria that allow us to determine the shape of each
+monochromatic region in a diagram, we wish to compute how many collections
+of shapes are possible between all diagrams.
+In other words, we now know
+that for each fixed n the function f(xn) is a piecewise quadratic function in
+the endpoints xn, but we would like to identify how many pieces it has. From
+Lemma 4.1, we have that the ordering of the pairwise sums {xi + xj}0≤i<j≤n
+completely determines the shapes in the diagram. This is sequence A237749
+in the On-Line Encyclopedia of Integer Sequences. Currently only 9 elements
+in the sequence are known, ending with 771, 505, 180 possible orderings for the
+pairwise sums with n = 8. Thus, this sequence grows much too quickly to be
+useful in checking every piece of f(xn) for n = 12.
+Note that if f(xn) were everywhere concave up, it could only have a single
+local minimum, which would necessarily be the global minimum as well. Since
+the conjectured optimum solution is a local minimum, the proof would be com-
+plete for any coloring with a finite number of intervals regardless of whether
+the coloring is antisymmetric. Additionally, a gradient descent algorithm would
+quickly lead to the global minimum even if it were unknown in advance. Unfor-
+tunately, through computational search, it is easy to find pieces of f(xn) that
+are not concave up. For this reason, we must search for local minima on each
+piece of f(xn) individually in order to guarantee the conjectured coloring is
+globally minimal.
+Counting pairwise orderings of {xi + xj}0≤i<j≤n is closely related to other
+combinatorial problems. In Figure 4, we find that any ordering of {xi + xj}
+could be encoded within a standard Young tableau of inverse staircase shape.
+Here, the filling of the (r, s) entry of the tableau (where the top of the tableau
+is the (0, 0) entry) is equal to the position of xr + xs when all pairs {xi + xj}
+are placed in increasing order.
+To simplify our computations, we make two restrictions. First, because of
+numerical simulations in [2] and our own, we consider only colorings that are
+antisymmetric: when reflected about the middle of the unit interval, almost
+every point in the coloring is swapped to the opposite color. Equivalently, we
+require that xk +xn−k = 1 for 0 ≤ k ≤ n. Additionally, we do not need to know
+all of the relations in the total ordering of {xi+xj}0≤i<j≤n in order to determine
+a configuration. Instead, for each pair (i, j) with i ̸= j, we search for the value of
+15
+
+x0
+x1
+x2
+x3
+x4
+x5
+x0
+x1
+x2
+x3
+x4
+x5
+1
+2
+4
+5
+3
+6
+11
+9
+7
+8 10
+14
+13
+15
+12 17
+16
+19 20
+18
+21
+Figure 4: Above is pictured a standard Young diagram corresponding to the
+choice of endpoints x0 = 0, x1 = 0.19, x2 = 0.9, x3 = 0.6, x4 = 0.65, and x5 = 1.
+The entry labelled 3 tells us that x0 + x2 is the third smallest in the ordering
+of pairs {xi + xj}5
+i,j=0. Additionally, the non-crossing lattice paths from the
+line y = −x to the lower left of the Young diagram partition the diagram into
+regions where all the corresponding pairwise sums are between two consecutive
+values 2xk−1 and 2xk for some k.
+16
+
+k where 2xk−1 ≤ xi + xj ≤ 2xk. Thus, we look for the number of ways to insert
+the pairwise sums {xi + xj}i̸=j into the line, 0 = 2x0 ≤ 2x1 ≤ · · · ≤ 2xn = 2.
+In Figure 4, this reframing corresponds to not needing to know the entire filling
+of the diagram, but instead having a family of non-crossing lattice paths each
+starting at different points down the diagonal. This set-up is very similar to the
+Lindstr¨om-Gessel-Viennot Lemma counting non-intersecting lattice paths, [11,
+8], which was instrumental in proving the conjecture counting the number of
+n × n alternating sign matrices, the story of which is told in [1].
+To determine the number of such partial permutations, we develop an al-
+gorithm that works recursively to verify whether growing partial permutations
+are possible. Our implementation is similar to Miller and Peterson’s geomet-
+ric approach to solving questions about More Sums Than Differences sets, [13,
+Lemma 2.1], and also similar to Laaksonen’s approach to enumerating OEIS
+sequence A237749, [10]. Both of these problems plus the problem we study here
+could be phrased in terms of enumerating chambers in a hyperplane arrange-
+ment, potentially including a restriction to a specific cone within the hyperplane
+arrangement. Enumerating chambers is a stream of research on its own (e.g.
+[16, 6]), and there are existing theorems counting chambers by using M¨obius
+inversion, [20, 19]. However, here we do not need to enumerate every chamber
+without restrictions because this is again equivalent to counting the possible
+total orderings of {xi + xj}0≤i<j≤n.
+Lemma 4.2. Consider antisymmetric block colorings with endpoints (x0, x1, . . . , xn)
+for n even, so that xk + xn−k = 1 for 0 ≤ k ≤ n. Then, the number of ways to
+insert the pairs {xi + xj}i̸=j into the ordering 0 = 2x0 < 2x1 < · · · < 2xn = 2
+grows as follows, starting with n = 0 and with n increasing by twos:
+1, 1, 3, 23, 357, 9391, 371219, . . .
+Proof. The key computational tool in our proof is linear programming: with
+existing linear programming packages like the GNU Linear Programming Kit
+(GLPK, [9]) we can easily check whether a single system of inequalities has a
+valid solution. Thus, we create a running list of partial systems of inequalities
+that have valid solutions, and count in how many ways it is possible to extend
+each system with a single additional inequality. Below, we give pseudocode for
+the algorithm we use, followed by a brief explanation of some of the technicalities
+required to make this code run correctly and efficiently. The full code is posted
+online at https://cocalc.com/TorinGreenwood/MonochromeSequences/Mo
+nochromaticProgressions.
+Pseudocode to Enumerate BCG Diagrams
+1
+\\ Initialize a running list of partial systems of inequalities
+2
+PartialInequalitiesOld =
+�
+{0 = x0, xk + xn−k = 1 for 0 ≤ k ≤ n,
+3
+xk ≤ xk+1 for 0 ≤ k ≤ n − 1}
+�
+4
+5
+\\ For each partial system of inequalities (i.e. for each set of
+6
+partial constraints), find all ways to add a new inequality
+17
+
+7
+2xk ≤ xi + xj ≤ 2xk+1 by deciding where xi + xj fits between
+8
+successive 2xk
+9
+for (i, j) with i ̸= j:
+10
+PartialInequalitiesNew = {}
+11
+for k from i to j − 1:
+12
+for constraints in PartialInequalitiesOld:
+13
+if constraints ∪ {2xk < xi + xj < 2xk+1} is valid:
+14
+PartialInequalitiesNew + =
+�
+constraints
+15
+∪ {2xk < xi + xj < 2xk+1}
+�
+16
+PartialInequalitiesOld = PartialInequalitiesNew
+17
+return PartialInequalitiesOld
+We now discuss some important aspects of our implementation with GLPK
+that ensured the code ran efficiently and correctly. The uninterested reader may
+skip the rest of this proof without a loss of continuity. First, mixed integer linear
+programs typically search to optimize a linear objective function in the variables
+x over a region of linear inequalities written in terms of x. Here, our goal was
+simply to check whether a system of linear inequalities was feasible, meaning
+that a solution exists. This can be achieved with linear programming by setting
+the objective function to be any constant, C, because the linear program will
+return a certificate x∗ where the maximum is achieved. When the objective
+function is constant, this is simply any feasible solution.
+As an added layer of complexity, linear programming typically only al-
+lows for inequalities that are not strict.
+However, exponentially many ar-
+rangements of the pairs {xi + xj}i̸=j can be achieved trivially by the solution
+x∗ = (0, 1/2, 1/2, . . . , 1/2, 1), since any sum of distinct endpoints xi + xj would
+equal 1/2, 1, or 3/2. In fact, many such arrangements can only be achieved
+by these trivial solutions. If we allow such solutions, it is not possible for the
+program to finish due to an explosion in the number of possible systems of
+inequalities. To avoid this scenario, we force all inequalities in every system
+to be strict.
+Thus, we introduce a single auxiliary variable ϵ that converts
+strict inequalities into weak inequalities. For example, the strict inequalities
+2xk < xi + xj < 2xk+1 become a pair of weak inequalities 2xk + ϵ ≤ xi + xj and
+xi + xj + ϵ ≤ 2xk+1. After adding ϵ to every inequality, we change the objective
+function from a constant C to the variable ϵ, and search for the maximum value
+of ϵ within the region where the inequality system is true. As long as a value of
+ϵ > 0 is found, the set of inequalities is feasible.
+Generally, linear programming implementations work with floating point
+arithmetic, leading to rounding errors. Because there is no way to bound how
+small a feasible region could be, we used the version of GLPK that works using
+rational arithmetic.
+Even still, GLPK returns its solutions as floating point
+numbers, occasionally with roundoff errors. Thus, we set the threshold for ϵ to
+be near the limits of floating point arithmetic at 5 × 10−15. We found that the
+smallest ϵ value above this threshold was on the order of 10−3, illustrating that
+any value below 5 × 10−15 was due to precision error.
+Unfortunately, rational solvers tend to be much slower than their floating
+18
+
+point counterparts. To address this, we needed to optimize our code. One factor
+that impacted runtime significantly was the order in which pairs (i, j) were
+checked in the for loop in Line 9 of the pseudocode above. After experimenting
+with different orderings, we found that checking the pairs in decreasing order of
+j − i was several times faster than checking the pairs in lexicographic order.
+Additionally, the rational solver became stuck in an infinite loop for 26 of
+the millions of feasibility checks it ran on systems of inequalities for the n = 12
+case. This issue was resolved by changing the order of the inequalities within
+these problematic systems of inequalities before they were input into GLPK. We
+did not find a single ordering that avoided infinite loops for all of the feasibility
+checks. Instead, we found that for any specific set of inequalities, there always
+existed some ordering where GLPK would halt rapidly.
+4.3
+Optimizing over all BCG diagrams
+Now that we have found the number of possible BCG diagrams for f(xn) for
+each n ≤ 12 and xn that are antisymmetric, we can finally leverage the power
+of calculus.
+Despite being a piecewise function, we soon find that f(xn) is
+continuous with continuous partial derivatives.
+This implies that its global
+maximum happens either at a critical point, or at a boundary point of the
+domain of the function. In Lemma 4.3, we prove that f(xn) has continuous
+partial derivatives for any fixed n, after which we can finish the proof of Theorem
+1.2.
+Lemma 4.3. Consider all block colorings with n blocks and endpoints x =
+(x0, x1, . . . , xn).
+In the region 0 = x0 < x1 < . . . < xn = 1, f(xn) is a
+continuous function with continuous partial derivatives in each variable xj.
+Proof. From the diagram representation in Figure 1, it is clear that f is a
+continuous function of the endpoints, xn. To verify that the partial derivatives
+are continuous, we give a geometric argument: consider a single region R in the
+diagram, like those drawn in Figure 3. Let fR(xn) be the area of this single
+region as a function of the endpoints. The region has up to 6 sides, and each
+side is a line whose position is determined by some single endpoint xi. Thus,
+∂
+∂xi fR(xn) is equal to the total length of the boundaries of R determined by the
+variable xi. (Indeed, moving a single xi by a small ∆xi changes the area of the
+polygon R by ∆xi · ℓi + O(∆xi)2 as ∆xi → 0, where ℓi is the total length of the
+boundaries of R determined by xi.)
+Now, we consider several cases. As xn varies, R may do any of the following:
+stay the same type of polygon, change polygon types, or enter or leave the
+diagram altogether. It is clear that when R stays the same type of polygon, its
+side lengths change continuously in xn, so fR(xn) has continuous partials in this
+case. When R changes polygon type, the change must occur when a diagonal
+line crosses over a corner of the box formed by the horizontal and vertical strips
+shown in Figure 3. This means that any time a region changes polygon type, the
+side that enters or leaves the region does so with initial length 0, again implying
+that the partials are continuous. Finally, we consider when R enters or leaves
+19
+
+the diagram. There are two ways this can happen: either a horizontal, vertical,
+or diagonal strip collapses to width 0, or a diagonal line crosses over the corner
+of the box described above. When a strip collapses to width 0, this means that
+there are two consecutive endpoints xi and xi+1 where (xi+1 −xi) tends to zero.
+Thus, although the partial derivative is not continuous in this case, it is on the
+boundary of the region of xn values we consider. On the other hand, when a
+diagonal line crosses over the corner of a box, all the side lengths of the polygon
+approach zero, so the partials are again continuous.
+The diagram representation of f(xn) makes it clear that f(xn) has a bounded
+number of regions: at most one for each intersection of a horizontal, vertical, and
+diagonal strip. Since fR(xn) is continuous with continuous partial derivatives
+for every region R, f(xn) is too.
+Now that we have shown that f(xn) is continuous with continuous partial
+derivatives for a fixed n, we are ready to complete the proof of Theorem 1.2
+with the following lemma.
+Lemma 4.4. Let x12 = (x0, . . . , x12) with 0 ≤ x0 ≤ · · · ≤ x12 = 1 and x12
+antisymmetric. The global minimum of f(x12) over all such x12 is 117/548,
+occurring uniquely at the coloring from Equation 1.
+Proof. Because f(x12) is a C1 function on the polytope 0 = x0 < x1 < . . . <
+x12 = 1, its global minimum occurs on the boundary of the polytope or at a
+critical point within the interior of the polytope. The boundary of this polytope
+is the union of polytopes of the same form with fewer variables. For this reason,
+we find the critical points for f(xn) for each even value of n between 0 and 12.
+Lemma 4.1 implies that f(xn) is a piecewise-quadratic function for each
+n. Fix n, and consider any piece of this function, which can be extended to a
+function f ∗(xn) on all of Rn/2−1 (since x1 through xn/2−1 determine the coloring
+because it is anti-symmetric). The partial derivatives of f ∗(xn) are piecewise
+linear functions. The critical points of this everywhere-defined quadratic are the
+solution to a linear system of equations. Therefore, there are either no critical
+points, or a vector space of critical points. In the case that the vector space
+has positive dimension, the value of f ∗(xn) must be constant among all of its
+critical points. Thus, when f ∗(xn) has critical points, it suffices to check the
+value of f ∗(xn) at a single critical point when checking for the values of local
+optima.
+This leads us to the following pseudocode to search for the global minimum
+of f(x12) on the polytope 0 = x0 ≤ x1 ≤ · · · ≤ x12 = 1. (The full version of the
+code is posted at https://cocalc.com/TorinGreenwood/MonochromeSequen
+ces/MonochromaticProgressions.)
+Pseudocode to find the minimum value of f(x12)
+18
+\\ We search the interior of f(xn) for n = 2, 4, 6, 8, 10, and 12.
+19
+>> for n from 0 to 12 by twos:
+20
+20
+
+21
+>> for each piece f ∗(xn) of the piecewise function f(xn)
+22
+(identified by Lemma 4.2):
+23
+24
+>> calculate the quadratic polynomial corresponding to
+25
+f ∗(xn) (by using Lemma 4.1)
+26
+27
+\\In the next line, we can feed into GLPK all of the
+28
+inequalities defining the configuration for f ∗(xn) plus
+29
+the equalities that set each of the partial derivatives
+30
+of f ∗(xn) to zero.
+31
+>> use GLPK to check the existence of a critical point
+32
+of f ∗(xn) within the region of xn-values where
+33
+f(xn) ≡ f ∗(xn)
+34
+35
+>> if critical points exist:
+36
+>> evaluate f ∗(xn) at any critical point cn
+37
+>> store cn and f ∗(cn) if this is a new record minimum
+38
+39
+>> return the minimum cn and f ∗(cn) values
+This code verifies that the global minimum of f(xn) when n is at most 12 is
+117/548, which is attained only at the coloring with endpoints given in Equation
+(1) (without the N in each coordinate).
+The number of pieces of the function f(xn) for n ≥ 14 grows very rapidly,
+making an analysis of its critical points increasingly challenging. However, we
+can guarantee that the optimal is always rational:
+Corollary 4.5. For each n ∈ Z+, the minimum value of f(xn) is rational,
+regardless of whether xn is restricted to be anti-symmetric or not.
+Proof. This is nearly immediate from our proof structure: the minimum of
+f(xn) occurs at some critical point of f(xℓ) with xℓ in the interior of where
+f(xℓ) is defined, for an ℓ ≤ n. These critical points are defined by a system
+of linear equations with rational coefficients. Whenever there are only finitely
+many critical points, they all must have rational coordinates. On the other hand,
+if there is a piece f ∗(xℓ) of the piecewise function f(xℓ) that has infinitely many
+critical points, all of the critical points of f ∗(xℓ) attain the same constant value.
+This implies that there still exists a critical point with rational coordinates where
+the minimum is attained. Finally, because each piece of f(xℓ) is a quadratic
+with rational coefficients, the minimum is thus also rational.
+5
+Circle colorings
+As a variation on the theme of enumerating monochromatic progressions within
+colorings of [N], some authors have also investigated properties of arithmetic
+21
+
+progressions within colorings of the cyclic group ZN. For example, given a fixed
+red and blue 2-coloring of Zp for p prime, the fraction of monochromatic 3-term
+progressions that are red or blue depends only on the proportion of elements
+colored red, and not on the exact positioning of the red and blue elements, [5, 12].
+Even when N is not prime, the fraction of monochromatic 3-term progressions
+in ZN is bounded below by the quantity given if N were prime, [12].
+Inspired by these results, we now explore a continuous analogue to the enu-
+meration of monochromatic progressions within 2-colorings of ZN. Color each of
+the numbers in the unit circle S1 = {e2πiθ : θ ∈ [0, 1)} with red or blue, and con-
+sider 3-term arithmetic progressions of the form (e2πix1, e2πi(x1+d), e2πi(x1+2d))
+for x1, d ∈ [0, 1). To properly discuss the “fraction” of these that are monochro-
+matic for a given coloring, we introduce the uniform probability measure µ on
+[0, 1) and randomly sample arithmetic progressions by independently choosing
+x1, d ∈ [0, 1) according to µ. Using this framework, the probability of selecting a
+monochromatic progression depends only on the Lebesgue measure of the set of
+points colored red (i.e. the likelihood that, say, e2πix1 is red) and not on which
+points were colored red, which is an analogous result to the one for 2-colorings
+of the discrete group Zp.
+Lemma 5.1. Let C : S1 → {0, 1} be any measurable coloring of S1 with
+p := µ
+��
+θ ∈ [0, 1) : C
+�
+e2πiθ�
+= 0
+��
+defined as the proportion of points colored red, and let m(C) be the set containing
+all pairs (x1, d) ∈ [0, 1) × [0, 1) such that (e2πix1, e2πi(x1+d), e2πi(x1+2d)) are
+monochromatic. Then,
+(µ × µ)(m(C)) = 1 − 3p + 3p2.
+In particular, if we randomly select a starting point x1 and an increment d
+independently from each other according to the uniform distribution on S1, then
+the probability that the associated 3-AP is monochrome depends only on the
+proportion p of red points and not on how these points are distributed around
+S1.
+Proof. We take a probabilistic approach that follows the proof structure of The-
+orem 6 from [12]. To that end, let x1 and d be independent draws from µ and
+for i = 1, 2, 3 let Ai (respectively, Bi) be the event that the ith term in the
+progression (e2πix1, e2πi(x1+d), e2πi(x1+2d)) is red (respectively, blue). Then, via
+inclusion/exclusion, we have
+P(A1 ∪ A2 ∪ A3) =
+� 3
+�
+i=1
+P(Ai)
+�
+−
+�
+�
+�
+1≤i<j≤3
+P(Ai ∩ Aj)
+�
+� + P(A1 ∩ A2 ∩ A3).
+We note that P(A1 ∪ A2 ∪ A3) = 1 − P(B1 ∩ B2 ∩ B3).
+Since P(m(C)) =
+P(A1 ∩ A2 ∩ A3) + P(B1 ∩ B2 ∩ B3), we can rearrange the above to obtain
+P(m(C)) = 1 −
+3
+�
+i=1
+P(Ai) +
+�
+1≤i<j≤3
+P(Ai ∩ Aj).
+(9)
+22
+
+The random variables e2πix1, e2πi(x1+d), and e2πi(x1+2d) are pairwise indepen-
+dent and uniformly distributed on S1.
+This is true based on the rotational
+invariance of the uniform distribution on S1 and the fact that e2πix1 and e2πid
+are independent and uniformly distributed on S1. It follows that P(Ai) = p and
+P(Ai ∩ Aj) = p2 for i, j = 1, 2, 3. Substituting these into (9) yields
+(µ × µ)(m(C)) = P(m(C)) = 1 − 3p + 3p2.
+6
+Future Work
+In Sections 3 and 4 above, we outlined an approach to identifying the optimal
+coloring of [N] and the interval [0, 1] that minimizes the fraction of monochro-
+matic 3-APs for any fixed upper bound on the number of blocks n. A natural
+question is whether we can show that the optimal coloring for any n > 12 is the
+same as the optimal coloring for n = 12. One possibility is to prove that the
+colorings are no better for n = 14, and then argue that adding arbitrarily more
+intervals is no better than adding just two more intervals.
+Besides investigating how colorings of [N], ZN, [0, 1], and S1 affect the preva-
+lence of monochromatic arithmetic progressions of length 3, there are other re-
+lated problems that have yet to be explored. Perhaps the most natural question
+to ask is how the analysis changes if we consider longer arithmetic progressions,
+and the articles [18, 2, 12, 3] make partial progress in this direction for several
+different lengths of progressions. A slightly less obvious question is to ask what
+happens when we consider arithmetic progressions of color-dependent lengths.
+For example, we could attempt to color [0, 1] or [N] in a way that simultane-
+ously minimizes the fractions of monochromatic blue progressions of length 3
+and monochromatic red progressions of length 4.
+Another natural generalization is to add more colors. What do the 3- and 4-
+colorings of [0, 1] that minimize monochrome arithmetic progressions of length
+3 look like?
+Can anything be said about the rate at which the fraction of
+monochrome progressions decays as the number of colors increases? All of these
+questions have natural analogues in the setting of Ramsey theory as applied
+to graphs, and of course these generalizations might interact in any number of
+ways.
+When the problems studied in this paper were first posed, it was unclear
+whether or not colorings could perform better than random. Although they
+can perform better than random in the cases we present in detail above, is this
+also true for related problems? Recent work in [4] gives interesting insights into
+some classes of problems where solutions must be better than random.
+In addition to changing the number of colors or length of the progressions
+we study, we could also consider colorings in other geometries. For example,
+we wonder how to color an interval that has a gap in the middle in order
+to minimize monochromatic APs therein. By varying the length of the gap,
+we might gain insight into why antisymmetry is seemingly important in the
+23
+
+optimal block colorings of [0, 1] that we discuss above. Furthermore, we have
+already seen that in the contexts of Zp for p prime and the continuous circle, the
+performance of colorings with respect to 3-term progressions depends only on the
+ratios of the colors present. What other algebraic and geometric settings exhibit
+similar behavior? Alternatively, what would happen if we were to consider S1
+as in Section 5 but sample 3-APs by choosing the start point and increment
+according to a different distribution than uniform?
+7
+Acknowledgments
+Computations were performed using High Performance Computing infrastruc-
+ture provided by the Mathematical Sciences Support unit at the University of
+the Witwatersrand, and for this the authors are thankful.
+Additionally, the
+authors are grateful for invaluable tips from Professor Antti Laaksonen on how
+to optimize the code in Lemma 4.2.
+References
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+https://doi.org/10.1090/memo/0154.
+A
+Appendix: 20 Polygonal Regions
+The 20 possible regions from Lemma 4.1 are given below.
+1
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj ≤ xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(xi+1 − xi)(xj+1 − xj)
+2
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj ≤ 2xk ≤ min(xi + xj+1, xi+1 + xj),
+xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(xi+1 − xi)(xj+1 − xj) − (2xk − xi − xj)2/2
+3
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi+1 + xj ≤ 2xk ≤ xi + xj+1 ≤ xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(xi+1 − xi)(xj+1 + xi/2 + xi+1/2 − 2xk)
+4
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj+1 ≤ 2xk ≤ xi+1 + xj ≤ xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(xj+1 − xj)(xi+1 + xj/2 + xj+1/2 − 2xk)
+26
+
+5
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+max(xi + xj+1, xi+1 + xj) ≤ 2xk ≤ xi+1 + xj+1 ≤ 2xk+1
+Region Area:
+(2xk − xi+1 − xj+1)2/2
+6
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj,
+max(xi + xj+1, xi+1 + xj) ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(xi+1 − xi)(xj+1 − xj) − (2xk+1 − xj+1 − xi+1)2/2
+7
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj ≤ 2xk ≤ min(xi + xj+1, xi+1 + xj),
+max(xi + xj+1, xi+1 + xj) ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(xi+1 − xi)(xj+1 − xj) − (2xk − xi − xj)2/2
+− (2xk+1 − xi+1 − xj+1)2/2
+8
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi+1 + xj ≤ 2xk ≤ xi + xj+1 ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(xi2 − xi1)(xj2 + xi1/2 + xi2/2 − 2xk1)
+− (2xk2 − xi2 − xj2)2/2
+9
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj+1 ≤ 2xk ≤ xi+1 + xj ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(xj+1 − xj) · (xi+1 + xj/2 + xj+1/2 − 2xk)
+− (2xk+1 − xi+1 − xj+1)2/2
+10
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+max(xi + xj+1, xi+1 + xj) ≤ 2xk ≤ 2xk+1 ≤ xi+1 + xj+1
+Region Area:
+(2xk − xi+1 − xj+1)2/2 − (2xk+1 − xi+1 − xj+1)2/2
+27
+
+11
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj ≤ xi+1 + xj ≤ 2xk+1 ≤ xi + xj+1
+Region Area:
+(xi+1 − xi)(2xk+1 − xj − xi/2 − xi+1/2)
+12
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj ≤ 2xk ≤ xi+1 + xj ≤ 2xk+1 ≤ xi + xj+1
+Region Area:
+(xi+1 − xi)(2xk+1 − xj − xi/2 − xi+1/2)
+− (2xk − xi − xj)2/2
+13
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi+1 + xj ≤ 2xk ≤ 2xk+1 ≤ xi + xj+1
+Region Area:
+(xi+1 − xi)(2xk+1 − 2xk)
+14
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj ≤ xi + xj+1 ≤ 2xk+1 ≤ xi+1 + xj
+Region Area:
+(xj+1 − xj)(2xk+1 − xj/2 − xj+1/2 − xi)
+15
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj ≤ 2xk ≤ xi + xj+1 ≤ 2xk+1 ≤ xi+1 + xj
+Region Area:
+(xj+1 − xj)(2xk+1 − xj/2 − xj+1/2 − xi)
+− (2xk − xi − xj)2/2
+16
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj+1 ≤ 2xk ≤ 2xk+1 ≤ xi+1 + xj
+Region Area:
+(xj+1 − xj)(2xk+1 − 2xk)
+28
+
+17
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk ≤ xi + xj ≤ 2xk+1 ≤ min(xi + xj+1, xi+1 + xj)
+Region Area:
+(2xk+1 − xi − xj)2/2
+18
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi + xj ≤ 2xk ≤ 2xk+1 ≤ min(xi + xj+1, xi+1 + xj)
+Region Area:
+(2xk+1 − xi − xj)2/2 − (2xk − xi − xj)2/2
+19
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+2xk+1 ≤ xi + xj
+Region Area:
+0
+20
+xi
+i+1
+j
+j+1
+x
+x
+x
+k+1
+x
+k
+x
+Characterizing Inequalities:
+xi+1 + xj+1 ≤ 2xk
+Region Area:
+0
+29
+
diff --git a/GtAyT4oBgHgl3EQffPh0/content/tmp_files/load_file.txt b/GtAyT4oBgHgl3EQffPh0/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..626bfda84602930b60ef421772dcc1302caa6993
--- /dev/null
+++ b/GtAyT4oBgHgl3EQffPh0/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf,len=1060
+page_content='From discrete to continuous: Monochromatic  3-term arithmetic progressions  Torin Greenwood∗, Jonathan Kariv†, Noah Williams‡  December 31, 2022  Abstract  We prove a known 2-coloring of the integers [N] := {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=', N}  minimizes the number of monochromatic arithmetic 3-progressions under  certain restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' A monochromatic arithmetic progression is a set of  equally-spaced integers that are all the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Previous work by  Parrilo, Robertson and Saracino conjectured an optimal coloring for large  N that involves 12 colored blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Here, we prove that the conjecture is  optimal among anti-symmetric colorings with 12 or fewer colored blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  We leverage a connection to the coloring of the continuous interval [0, 1]  used by Parrilo, Robertson, and Saracino as well as by Butler, Costello  and Graham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Our proof identifies classes of colorings with permutations,  then counts the permutations using mixed integer linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  1  Introduction  Consider coloring each of the integers in [N] with one of r colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' A κ-term  arithmetic progression is any subset of κ equally-spaced integers, denoted a κ-  AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' An arithmetic progression is monochromatic if every term is colored the  same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Can we color [N] in a way that avoids all monochromatic κ-APs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  A classic result is van der Waerden’s Theorem:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1 (van der Waerden, [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For any integers r, κ ≥ 1, there exists  a number N such that every r-coloring of [N] has a monochromatic κ-AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Given that monochromatic κ-APs are guaranteed to exist when enough num-  bers are colored, we ask a refined question: what is the minimum number of  monochromatic κ-APs that could exist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' To be more precise, define Cr(N) to  be the set of r-colorings of [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For any c ∈ Cr(N), let mκ(c) be the number of  ∗Department  of  Mathematics,  North  Dakota  State  University,  Fargo,  ND  USA,  torin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='greenwood@ndsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='edu  †Isazi Consulting, Johannesburg, South Africa, jkariv@isaziconsulting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='za  ‡Department of Mathematical Sciences, Appalachian State University, Boone, NC USA,  williamsnn@appstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='edu  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='00336v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='CO]  1 Jan 2023  monochromatic κ-APs induced by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Finally, let APκ(N) be the total number  of κ-APs in [N], regardless of whether they are monochromatic or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then,  we look at  Pr,κ(N) :=  min  c∈Cr(N)  mκ(c)  APκ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  The focus of this paper is to examine the minimum for monochromatic 3-APs  within 2-colorings, P(N) := P2,3(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In 1999, Ron Graham proposed that  limn→∞ P(n) = β for some constant, β, and offered a $100 prize for finding  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Originally, it was not clear whether colorings could perform better than  random in the long run: for large values of N, is it possible to color [N] so  the probability that a randomly selected 3-AP is monochromatic is less than  (1/2)3 + (1/2)3 = 1/4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' It is notable that the analogous question for 2-colorings  of Zp is answered negatively for p prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Indeed, Lu and Peng [12] show that for  a given 2-coloring of Zp, the fraction of 3-APs that are monochromatic depends  only on the fraction of each color present in the coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  For our question concerning 2-colorings of [N], Parrilo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' [14] and Butler  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' [2] verified independently but nearly simultaneously that it is possible  to do better than random, and they found upper and lower bounds for the  minimum monochromatic APs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The upper bound was attained through simu-  lating good colorings and finding one that performed well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' They landed on the  following 12-block coloring:  Explicitly, when coloring [N], the blocks would be approximately of the following  sizes:  �28N  548 , 6N  548, 28N  548 , 37N  548 , 59N  548 , 116N  548 , 116N  548 , 59N  548 , 37N  548 , 28N  548 , 6N  548, 28N  548  �  (1)  Due to this coloring, P(N) ≤  117  548 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Note that this coloring is anti-  symmetric: the left half of the coloring is a mirror image of the right half  but uses opposite colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In [2], Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' performed many computer sim-  ulations using genetic algorithms to find the optimal coloring, and noted that  this same 12-block coloring consistently appeared regardless of the seed coloring  with which they started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' They noted that a remaining challenge would be to  analyze the case of rapidly alternating colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  The goal of this paper is to show that as N → ∞, the 2-coloring of [N] that  has alternating color blocks with sizes given in Equation (1) is globally optimal  among anti-symmetric colorings with at most 12 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' As far as the authors  are aware, this is the first result of optimality under any restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Here, we  let ˜C2(N) be the 2-colorings of [N] that are anti-symmetric and have at most  12 contiguous segments of red or blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then, define  ˜P(N) =  min  c∈ ˜C2(N)  m3(c)  AP3(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Our main result is as follows:  2  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Consider coloring each integer in [N] with either red or blue  such that the coloring is anti-symmetric and has at most 12 contiguous blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Then, as N increases the minimum possible fraction of arithmetic progressions  approaches 117  548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' That is, limN→∞ ˜P(N) = 117  548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Below, we provide a proof sketch that outlines the sections in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Sketch of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' First, we will convert from discrete colorings of [N] to continu-  ous colorings of [0, 1] with at most 12 contiguous segments, referred to as block  colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' After restricting the number of color changes that can occur within  a coloring, it turns out that optimizing the discrete colorings is the same as  optimizing the continuous colorings, as described rigorously in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  When switching to the continuous realm, we let a continuous coloring be a  function c : [0, 1] → {0, 1}, where 0 and 1 (in the range) represent red and blue,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then, we let f[0,1](c) be the fraction of arithmetic progressions in  the coloring c that are monochromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We can represent this fraction geomet-  rically by a BCG diagram, described by Butler, Costello, and Graham in [2] and  illustrated in Figure 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' When c consists of 12 contiguous segments, we  label the endpoints of the coloring as (x0 = 0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , x12 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' As we allow the  coloring c to vary, f[0,1](c) is a piecewise quadratic function in the xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Moreover,  each piece of f[0,1](c) is determined completely by the relative ordering of the  pairs of sums {xi + xj}, as described in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Next, we aim to identify every piece of the quadratic function over all color-  ings c of [0, 1] with 12 intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Using the GNU Linear Programming Kit [9],  we count 371, 219 possible arrangements of {xi + xj} that could give distinct  quadratics in f[0,1], as proved in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2 with the help of our code available  online at https://cocalc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='com/TorinGreenwood/MonochromeSequences/Mo  nochromaticProgressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Finally, once we have identified the 371, 219 possible pieces in the quadratic  function, we search for the global minimum of f[0,1] among all these pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Fortunately, from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='3, it turns out that f[0,1] is a continuous function  with continuous partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Thus, we can minimize f[0,1] by search-  ing for all critical points within each piece of the quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Because f[0,1] is  piecewise quadratic, its critical points are determined by systems of linear in-  equalities (defining the domain of a piece of f[0,1]) and equalities (setting the  partial derivatives of f[0,1] to zero), allowing us again to use linear programming  to identify the critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We describe our search for these critical points  in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  A byproduct of our proof structure is that among colorings with a fixed  number of contiguous blocks, there exist optimal colorings with rational end-  points, as described in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In Section 5, we show that with respect  to 2-colorings of the continuous unit circle S1, the fraction of monochromatic  APs depends only on the measure of points colored red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This is analogous to  the results in [5, 12] that concern colorings of Zp for p prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  3  2  Background  When searching for bounds on the number of monochromatic arithmetic pro-  gressions in [N], Frankl, Graham, and R¨odl developed the following theorem:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1 (Frankl, Graham, R¨odl, [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For fixed r and κ, there exists ℓ > 0  so that the number of monochromatic κ-APs in any r-coloring of {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , N}  is at least ℓN 2 + o(N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  This proved that a positive fraction of APs must be monochromatic in the  long run, but gave no indication of how small ℓ could be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Datskovsky made progress on a related problem in [5], analyzing the minimal  number of monochromatic Schur triples in [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' A Schur triple (a, b, c) from  [N] is any triple of integers where a + b = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Datskovsky investigated the  minimum possible number of monochromatic Schur triples when coloring each  integer red or blue, and proved that asymptotically, the minimum is N 2/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The  proof relied on using a discrete Fourier transform, which yielded a combinatorial  identity that broke down counts of Schur triples into a few easier to analyze  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Although our proof does not use the discrete Fourier transform, it also  will transform a discrete problem into a continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In [14], Parrilo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' applied some of the tools from Datskovsky’s work  to arithmetic progressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Again, the authors found a combinatorial identity  breaking down sets of arithmetic progressions into simpler sets, but it was no  longer possible to enumerate these sets exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Instead, the authors ended up  with bounds on the minimum number of monochromatic progressions possible in  [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' They also identified the coloring shown in Equation (1) in the introduction  above, and verified it was locally optimal among colorings with 12 intervals  that are antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Our paper aims to prove that this coloring is optimal  globally among the same set of colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Constellations are a generalization of APs studied in [2], where instead of  all points being equally spaced like in an AP, the consecutive differences of  terms must satisfy some fixed proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' analyzed constellations  by representing sets of monochromatic constellations using integrals of indicator  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This led them to represent monochromatic regions in two-dimensional  diagrams which we refer to as BCG diagrams, as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Visu-  alizing progressions via these diagrams is crucial to our proof, and provides the  connection we need between discrete and continuous realms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  One important aspect of our proof is enumerating the number of ways  pairwise sums {xi + xj} can be ordered for a list of positive real numbers  x0 ≤ x1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' ≤ xn with n even and xi + xn−i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  This problem could  be framed as counting the number of chambers in a hyperplane arrangement,  and there already exists a rich set of tools for counting chambers, as seen for ex-  ample in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' However, in this paper, we use mixed integer linear programming,  which is well-suited to determining whether a system of linear inequalities has  a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This coding approach was also employed by Miller and Peterson in  [13] when they counted more sums than differences sets, and also by Laaksonen  4  in [10] when he counted closely-related arrangements of sums of pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' More  details on this approach are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  The current best known bounds on the minimum number of monochromatic  κ-APs in the general (non-antisymmetric) case for κ > 3 are found using an  “unrolling” strategy, described in [12] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Here, an optimal coloring of  some interval {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , ℓ} for ℓ ≪ N is found explicitly, and then repeated to fill  the interval [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Although this strategy works well for κ > 3, when κ = 3, the  colorings do no better than random in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  3  Relationship between discrete and continuous  case  In this section, we define a precise connection between discrete 2-colorings of  [N], and a natural continuous analogue of 2-coloring [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' First, we pause to  define a 3-AP in [N] formally: a 3-AP is any set of 3 terms (a, a+d, a+2d) each  in [N] where d is any integer including negative values or zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' It is convenient  for us to include the case where d ≤ 0 in our arguments, although this choice  ultimately does not change which colorings minimize monochromatic APs nor  the minimum they attain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  For the interval [0, 1], we identify any 3-AP (a, a + d, a + 2d) by its first and  last term (a, a + 2d) in [0, 1] × [0, 1], now allowing d to be any real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  We obtain a measure on the set of 3-APs in [0, 1] by choosing the starting and  ending point of the progressions uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' A coloring of the interval is defined  to be a function c : [0, 1] → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In this section, we begin by discussing measurable colorings of [0, 1], which  can be approximated in a standard way by bead colorings, defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then,  we show that minimizing monochromatic APs over all measurable colorings of  [0, 1] is the same as minimizing all APs over just bead colorings, as formalized  in Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Next, we justify that every discrete coloring of [N] has a corresponding con-  tinuous coloring of [0, 1], and that the fraction of monochromatic APs in the  continuous coloring is a function of both the monochromatic APs and monochro-  matic off-by-1 APs in the discrete coloring, as explained above and in Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Using this connection, we find that when the number of blocks of contiguous  runs of colors in a coloring is bounded by n, the fraction of APs in a continuous  coloring versus its discrete analogue is small as N grows large, formalized in  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Finally, this allows us to prove our main result of the section: that  minimizing over discrete colorings with a fixed number of blocks is the same as  minimizing over continuous colorings with the same number of blocks, stated  rigorously in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Now, we begin stating our results formally, starting with the definition of a  Lebesgue-measurable coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' A coloring of [0, 1] is Lebesgue-measurable if c−1(0) is Lebesgue-  measurable (or equivalently c−1(1) is Lebesgue-measurable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  5  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' A bead coloring of [0, 1] is a coloring where for some ℓ, each  of the intervals ( i  ℓ, i+1  ℓ ) is monochrome for i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Each interval  ( i  ℓ, i+1  ℓ ) is called a bead, and we sometimes refer to such a coloring as an ℓ-bead  coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  We introduce bead colorings because they are the continuous analogue of  coloring the integers [N] obtained by fattening each integer into an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Our goal is to show that when optimizing colorings over the interval [0, 1], we  may restrict our attention to bead colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We call the set of bead colorings B  and the set of Lebesgue-measurable colorings M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Observe that B ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Finally  we define a difference between two colorings as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For two colorings ca ∈ M and cb ∈ M we define d(ca, cb) :=  µ({x | ca(x) ̸= cb(x)}), where µ is the usual Lebesgue measure on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Recall that we identify an arithmetic progression in [0, 1] by the pair of  starting and ending points in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For a coloring c on [N], we define f[N](c) to  be the fraction of arithmetic progressions that are monochromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Analogously,  when c is a coloring of [0, 1], we have the following definition:  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For a coloring c : [0, 1] → {0, 1}, let f[0,1](c) be the Lebesgue  measure of the set of monochromatic arithmetic 3-term progressions (viewed as  a subset of [0, 1]2) induced by the coloring c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  We justify our restriction to bead colorings with the following standard  measure-theoretic lemmas (proved for completeness momentarily):  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For any two measurable colorings c1 and c2 of [0, 1], if d(c1, c2) <  ϵ, then |f[0,1](c1) − f[0,1](c2)| < 4ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For any measurable coloring cm of [0, 1] and any ϵ > 0 there exists  a bead coloring cb such that cm and cb disagree on a set of measure at most ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  As B ⊂ M, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2 immediately implies the following:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Optimizing monochromatic 3-APs over bead colorings is the same  as optimizing over all measurable colorings in the following sense:  inf  cb∈B f[0,1](cb) =  inf  cm∈M f[0,1](cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  We begin with the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Let A ⊂ [0, 1] be a set of measure ϵ, and consider flipping  the colors of all elements in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' There are three classes of monochromatic 3-APs  that could be created or destroyed: the APs where the first, middle or last  element is flipped (where some APs may belong to more than one class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We  consider the measure of each of these three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' As the first and last elements  of a progression are chosen uniformly, the corresponding classes have measure  ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The middle element is the average of two uniform random variables, and so  6  has a triangular distribution on [0, 1] with maximum density 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Therefore the  set of monochrome progressions whose middle term is in A would have measure  at most 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Summing the measures of these three classes yields an upper bound  for their union of 4ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  We now justify Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2, whose proof is a standard measure-theoretic  argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' By hypothesis, the set Xbl := c−1  m (0) of blue-colored el-  ements of [0, 1] is measurable with finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' So, a standard result from  measure theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  [15, Theorem 12]) establishes the existence of a finite  disjoint collection of open intervals I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , Iℓ ⊂ [0, 1] satisfying  µ  �� ℓ�  i=1  Ii  �  \\ Xbl  �  + µ  �  Xbl \\  ℓ�  i=1  Ii  �  < ϵ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Since the rationals are dense in [0, 1], we can perturb the 2ℓ endpoints of the  intervals {Ii}, each by some amount less than  ϵ  4ℓ, to find a disjoint collection  I′  1, I′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , I′  ℓ of open intervals with rational endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Let Ubl be the union of  these intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then, Ubl and Xbl have a symmetric difference of measure at  most ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' It follows that the coloring cb defined by coloring each interval of Ubl  blue is a bead coloring for which d(cb, cm) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Call a progression an off-by-1 AP if it is of the form (a, a + d, a + 2d ± 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  We will show that we can easily compute f[0,1](cb) for a bead coloring cb with N  beads by considering the colored beads as an integer coloring of [N], computing  the number of 3-term APs in this sequence, and adding half of the off-by-1 APs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Recall that for a discrete coloring c, m3(c) is the number of monochromatic  3-APs induced by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Let m′  3(c) be the number of monochromatic off-by-1 APs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Then, we have the following comparison between colorings of [0, 1] with exactly  N beads (of not necessarily alternating colors) and corresponding colorings of  [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Let cb be an N-bead coloring of [0, 1], and let c∗  b be the discrete  coloring of [N] corresponding to cb, where the number i is colored blue if and  only if the ith bead in cb is colored blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then,  f[0,1](cb) = m3(c∗  b) + m′  3(c∗  b)/2  N 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Consider a randomly chosen progression in [0, 1] identified by its end-  points (a, b), and a fixed N-bead coloring cb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We use a probabilistic proof, so  we rewrite  (µ × µ)((a, b) ∈ [0, 1]2 : (a, b) is monochromatic) =: P((a, b) monochromatic),  where µ × µ is the usual Lebesgue measure on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We will condition on which  beads S and E contain a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Let M be the bead containing the middle  7  element of the progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Given a bead coloring of [0, 1], it is useful to define  the distance between two beads A and B, db(A, B) as 0 when A = B and as one  more than the number of other beads strictly between A and B otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Note  that when db(S, E) is even, then S, M, and E must form a 3-AP of beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' On the  other hand, when db(S, E) is odd, S, M, and E form an off-by-one progression  and M could be two possible beads depending on the internal positioning of a  and b within S and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Formally, letting {Bi}N  i=1 be the set of beads,  f[0,1](cb) =  �  i,j  P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)  =  �  d(Bi,Bj) even  P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)  +  �  d(Bi,Bj) odd  P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)  (2)  Now, P(a ∈ Bi, b ∈ Bj) = 1/N 2 for each i and j since a and b are indepen-  dently and uniformly distributed among the beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Also, because our coloring  is fixed, when db(Bi, Bj) is even P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) is 0 or  1 depending on whether or not the beads S, M, and E form a monochromatic  3-term AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Thus,  �  d(Bi,Bj) even  P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)  = m3(c∗  b) · 1  N 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  (3)  When db(Bi, Bj) is odd, there are two choices for M: B(i+j−1)/2 or B(i+j+1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Thus, we can condition on these two choices:  P((a, b) monochromatic|a ∈ Bi, b ∈ Bj)  = P((a, b) mono.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='|a ∈ Bi, b ∈ Bj, M = B(i+j−1)/2) · P(M = B(i+j−1)/2|a ∈ Bi, b ∈ Bj)  + P((a, b) mono.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='|a ∈ Bi, b ∈ Bj, M = B(i+j+1)/2) · P(M = B(i+j+1)/2|a ∈ Bi, b ∈ Bj)  Here, P(M = B(i+j−1)/2|a ∈ Bi, b ∈ Bj) = P(M = B(i+j+1)/2|a ∈ Bi, b ∈ Bj) =  1/2 because a and b are positioned uniformly within S and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Additionally,  P((a, b) monochromatic|a ∈ Bi, b ∈ Bj, M = B(i+j−1)/2)  is 0 or 1 depending on whether the off-by-1 progression in c∗  b is monochromatic  or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Hence, these two terms combined simplify to  �  d(Bi,Bj) odd  P((a, b) monochromatic|a ∈ Bi, b ∈ Bj) · P(a ∈ Bi, b ∈ Bj)  =  1  2N 2 m′  3(c∗  b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  (4)  8  Plugging in Equations (3) and (4) into Equation (2) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Much of the rest of this paper will deal with a particular class of colorings  called “block colorings” which we now define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Informally, they are partitions of  I into disjoint intervals which are alternately colored red and blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For a finite collection of endpoints {xi} such that 0 = x0 <  x1 < x2 < · · · < xn−1 < xn = 1, we define the associated “block” coloring as  the coloring where the n intervals Ji = (xi−1, xi) (for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , n}) are all  monochrome and alternate in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Note that the colors assigned to the endpoints {xi} (or indeed to any points  within a measure zero set) do not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' With these definitions, we can now  compare the performance of discrete colorings with their continuous analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Let C(N, n) be the set of 2-colorings of [N] with at most n contigu-  ous blocks of colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For any coloring c ∈ C(N, n), let c∗ be the corresponding  block coloring of [0, 1] where the interval [(i − 1)/N, i/N) is colored blue by c∗ if  and only if i ∈ [N] is colored blue by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then,  max  c∈C(N,n)  ��f[0,1](c∗) − f[N](c)  �� = O  � n  N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Here, there exists a C > 0 independent of N and n such that |O(n/N)| < Cn/N  for all positive integers n and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Our proof will use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4 to rewrite f[0,1](c∗) in terms of f[N](c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Before proceeding with this, we will interpret the number of off-by-1 monochro-  matic APs induced by c, m′  3(c), in terms of the regular monochromatic APs,  m3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We claim the following:  m′  3(c) = 2m3(c) + O(nN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  (5)  To verify this, note that each AP (a, a + d, a + 2d) in [N] corresponds almost  bijectively to a pair of off-by-1 APs by moving the first or last endpoint inwards  by one: (a + 1, a + d, a + 2d) or (a, a + d, a + 2d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' (When N is odd, this  misses exactly two off-by-1 APs: (1, (N −1)/2, N) and (1, (N +1)/2, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' When  N is even, this is truly a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=')  Using this near bijection, we now compare when APs and off-by-1 APs are  monochromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Under this contraction action, the only time an AP (a, a +  d, a + 2d) is monochromatic while one of its corresponding off-by-1 APs is not  monochromatic is when a or a + 2d is adjacent to a number of the opposite  color, and the same could be said if the original AP is not monochromatic but  the off-by-1 AP is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' If our coloring only has n intervals total, there are only n−1  ways to position a immediately before a color change, and similarly only n − 1  ways to position a + 2d immediately after a color change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Since d can still be  chosen freely, there are O(nN) possible off-by-1 APs that disagree with their  corresponding APs on being monochromatic, verifying our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  9  Now, in the notation of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4, we see that (c∗)∗ = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Thus,  f[0,1](c∗) = m3(c)  N 2  + m′  3(c)/2  N 2  = 2m3(c) + O(nN)  N 2  = m3(c)  N 2/2 + O  � n  N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  The proof of the lemma will be complete if we can verify the following:  m3(c)  N 2/2 = f[N](c) + O  � 1  N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  To see this, recall that by definition f[N](c) = m3(c)/ AP3(N), so that  m3(c)  N 2/2 − f[N](c) =  m3(c)  AP3(N) · AP3(N) − N 2/2  N 2/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  (6)  We have m3(c) ≤ AP3(N), so that the first fraction on the right in Equation (6)  is at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Next, note that AP3(N) = N 2/2+O(N): it is easy to compute this  explicitly for when N is even or odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' But, intuitively, if we pick two numbers x  and y from [N] at random, there are N 2 ways to do this, and about half the time  x − y is even and these correspond to the start and end of a 3-AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Therefore,  AP3(N) − N 2/2 = O(N), and plugging this into Equation (6) completes the  proof with  m3(c)  N 2/2 − f[N](c) = O  � 1  N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Finally, we end this section with the result rigorously justifying our conver-  sion between discrete and continuous colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Let Sn be the block 2-colorings of [0, 1] with at most n blocks, and  let C(N, n) be the 2-colorings of [N] with at most n contiguous blocks, where  n = o(N) as N approaches infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Then, minimizing monochromatic APs  over Sn is the same as minimizing monochromatic APs over C(N, n) in the  following sense:  lim  N→∞  ���� inf  c∈Sn f[0,1](c) −  min  c∈C(N,n) f[N](c)  ���� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Here, we consider block colorings of [0, 1] where the edge of a block is at a  possibly irrational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' However, as we will see later, all optimal colorings  of [0, 1] with a fixed number of blocks must have rational endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  10  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This is mostly a standard ϵ argument, so let ϵ > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We aim to  show for all N sufficiently large,  ���� inf  c∈Sn f[0,1](c) −  min  c∈C(N,n) f[N](c)  ���� ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  We prove this in two halves, first proving the infimum is nearly bounded above  by the minimum, and then arguing the reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Consider any coloring ˜c ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1, for every N sufficiently large, we can find a n-block  coloring ˜cN of [0, 1] with endpoints of the form r/N for r an integer such that  ��f[0,1](˜c) − f[0,1](˜cN)  �� < ϵ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  (7)  This is true because we can round each endpoint to the nearest 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then,  we define ˜c∗  N to be the coloring of [N] where i is colored blue if and only if the  ith block of ˜cN is blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Note that ˜c∗  N still only has at most n blocks, and that  using the notation from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='5, (˜c∗  N)∗ = ˜cN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' So, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='5, for N  sufficiently large (independent of the colorings ˜c, ˜cN, ˜c∗  N),  |f[0,1](˜cN) − f[N](˜c∗  N)| = O(n/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  (8)  By choosing N sufficiently large (independent of the colorings ˜c, ˜cN, ˜c∗  N), Equa-  tions (7) and (8) imply  min  c∈C(N,n) f[N](c) ≤ f[N](˜c∗  N) < f[0,1](˜c) + ϵ  where this bound holds for all N sufficiently large and for all ˜c ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Therefore,  for all N sufficiently large,  min  c∈C(N,n) f[N](c) ≤ inf  c∈Sn f[0,1](c) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Now, we prove the reverse inequality: consider any coloring ˆc of [N], and  let ˆc∗ be the coloring of [0, 1] induced by ˆc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Again, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='5, for N  sufficiently large,  ��f[N](ˆc) − f[0,1](ˆc∗)  �� < ϵ,  and since this is true for any coloring ˆc ∈ C(N, n), this proves that for N  sufficiently large,  inf  c∈Sn f[0,1](c) ≤  min  c∈C(N,n) f[N](c) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Combining this with the complementary inequality above completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  At this point, we have justified that once bounding the number of blocks  in our coloring, optimizing colorings of [N] is the same as optimizing colorings  of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We only make use of this result when the number of blocks n = 12  because that is the conjectured global optimal number of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' But, the same  proof shows that switching to the continuous realm works whenever n = o(N)  as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  11  first term in progression  third term in progression  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='8  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='8  1  coloring  of [0, 1]  Figure 1: Below the horizontal axis a coloring, c, is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The horizontal  axis represents the first term a in an arithmetic progression, and the vertical  axis represents the third term a+2d in the progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Whenever a point in the  diagram is colored red (blue), this corresponds to the progression (a, a+d, a+2d)  being colored red (blue) by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  4  Proofs for the continuous case  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1  Colorings can be represented by BCG diagrams  Consider any block coloring c : [0, 1] → {0, 1} of the interval with endpoints  of the blocks given by {x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , xn} with x0 = 0 and xn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Without loss  of generality, assume that the first block (x0, x1) is colored blue, and alternate  colors for each remaining interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Recall that the colors of the endpoints of the  blocks can be assigned in any way, since this does not change the probability of  selecting a monochromatic progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In [2], Butler, Costello, and Graham proposed a method of visualizing the  monochromatic arithmetic progressions associated to a coloring in terms of di-  agrams like in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Any arithmetic progression (a, a + d, a + 2d) can be  identified uniquely by its first and last coordinates, which are represented by  the horizontal and vertical axes of such a diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Note that the diagram is  divided into vertical strips, horizontal strips, and northwest/southeast diagonal  strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Consider any region identified as the intersection of one horizontal, one  vertical, and one diagonal strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For a block coloring, this region corresponds  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='8  xto a collection of monochromatic arithmetic progressions if and only if the in-  dices of the vertical, horizontal, and diagonal strips defining the region all have  matching parities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Because the total area of the square in any diagram like Figure 1 is one, the  measure of the set of monochromatic sequences is equal to the sum of the areas  of the red and blue regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1 of [2], Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' express the  total colored area as the sum of two integrals involving an indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Their work applied to constellations, a generalization of arithmetic progressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Here, we instead derive explicit polynomial equations for the areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Consider  any one colored region in such a diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' As the endpoints xi are perturbed  slightly, the region remains the same type of polygon although its dimensions  may change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  This implies that the area of each region can be represented  locally as a quadratic in the variables {xi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Denote a block coloring c by its list  of endpoints x := (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then, summing over all monochromatic regions  shows that the measure of the monochromatic progressions, f[0,1](x), is locally  quadratic in the {xi}, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We now denote f(x) := f[0,1](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' When we restrict  f to act on colorings with exactly n blocks, we will write f(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  As x varies, the regions in the diagram change polygon type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Thus, for each  n, f(xn) is a piecewise function that is locally quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In order to minimize  f globally, we wish to identify the boundaries of these pieces in terms of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The  following lemma describes how to identify the polygons in such a diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The region that is the intersection of the ith vertical strip, jth  horizontal strip, and kth diagonal strip of a diagram is empty or forms a closed  polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The type of polygon is determined by testing whether each of the four  values {xi + xj, xi + xj+1, xi+1 + xj, xi+1 + xj+1} is greater than or less than  the two values {2xk, 2xk+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  If this ordering is known, the area of the cor-  responding region can be expressed as a quadratic polynomial in the variables  {xi, xi+1, xj, xj+1, xk, xk+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In the diagrams like in Figure 1, the horizontal lines all are given by  {y = xi}n  i=0 and the vertical lines by {x = xi}n  i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  At any point (x, y) in  the diagram, the middle value in the corresponding arithmetic progression is  (x+y)/2, and setting this equal to any endpoint in our coloring implies that the  diagonal lines are given by {y = 2xi − x}n  i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' As described above, for any triple  (i, j, k) where i, j, k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , 12} all have matching parities, the intersection of  the ith vertical strip, jth horizontal strip, and kth diagonal strip corresponds  to a region of monochromatic arithmetic progressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  To determine the shape of the region of the monochromatic progressions,  first consider the rectangle formed by the intersection of the ith vertical strip  and jth horizontal strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The corners of this rectangle have coordinates (xi, xj),  (xi+1, xj), (xi, xj+1), and (xi+1, xj+1), as labelled in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In order for the  intersection of this rectangle with the kth diagonal strip to be non-empty, we  need the upper diagonal line y = 2xk+1 − x to be above the lower left corner of  the rectangle, (xi, xj), and the lower diagonal line y = 2xk − x to be below the  upper right corner of the rectangle, (xi+1, xj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This is the same as requiring  the inequalities 2xk+1 ≥ xi + xj and 2xk ≤ xi+1 + xj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  13  ith vertical strip  jth  horizontal  strip  kth diagonal strip  (x    , x    )  i+1  j+1  (x    , x )  i+1  j  (x , x )  i  j  (x , x    )  i  j+1  y = 2x  - x  k  y = 2x     - x  k+1  Figure 2: Above is the intersection of the ith vertical strip, jth horizontal  strip, and kth diagonal strip determined by a block coloring with endpoints  x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Whether the intersection is empty can be determined by  comparing the diagonal lines {y = 2xk − x, y = 2xk+1 − x} to the corners of  the box {(xi, xj), (xi+1, xj+1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  xi  i+1  j  j+1  x  x  x  k+1  x  k  x  Characterizing Inequalities:  xi+1 + xj ≤ 2xk ≤ xi + xj+1 ≤ xi+1 + xj+1 ≤ 2xk+1  Region Area:  (xi+1 − xi)(xj+1 + xi/2 + xi+1/2 − 2xk)  xi  i+1  j  j+1  x  x  x  k+1  x  k  x  Characterizing Inequalities:  2xk ≤ xi + xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  max(xi + xj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' xi+1 + xj) ≤ 2xk+1 ≤ xi+1 + xj+1  Region Area:  (xi+1 − xi)(xj+1 − xj) − (2xk+1 − xj+1 − xi+1)2/2  Figure 3: Illustrated here are two different ways that the kth diagonal strip  can intersect with the ith horizontal and jth vertical strip in a coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The  resulting monochromatic region is shaded in gray, and the area of the region is  given as a quadratic in x below the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The type of polygon is determined  by the partial permutation given below each diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The other 18 possibilities  are enumerated in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  14  Additionally, the type of polygon formed by the intersection of the strips is  determined by whether the two diagonal lines y = 2xk − x and y = 2xk+1 − x  are above or below each of the four corners of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For any specific relation-  ship between the lines and the four corners, some basic geometric arguments  allow us to find the area of the polygon enclosed by the strips in terms of  {xi, xi+1, xj, xj+1, xk, xk+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' It turns out that there are 20 possible arrange-  ments of the lines that yield distinct polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In Figure 3, two possibilities are  given, along with the corresponding quadratic equations for their areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The  full list of 20 polygons is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2  Enumerating BCG diagrams  Now that we have identified criteria that allow us to determine the shape of each  monochromatic region in a diagram, we wish to compute how many collections  of shapes are possible between all diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In other words, we now know  that for each fixed n the function f(xn) is a piecewise quadratic function in  the endpoints xn, but we would like to identify how many pieces it has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' From  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1, we have that the ordering of the pairwise sums {xi + xj}0≤i<j≤n  completely determines the shapes in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This is sequence A237749  in the On-Line Encyclopedia of Integer Sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Currently only 9 elements  in the sequence are known, ending with 771, 505, 180 possible orderings for the  pairwise sums with n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Thus, this sequence grows much too quickly to be  useful in checking every piece of f(xn) for n = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Note that if f(xn) were everywhere concave up, it could only have a single  local minimum, which would necessarily be the global minimum as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Since  the conjectured optimum solution is a local minimum, the proof would be com-  plete for any coloring with a finite number of intervals regardless of whether  the coloring is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Additionally, a gradient descent algorithm would  quickly lead to the global minimum even if it were unknown in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Unfor-  tunately, through computational search, it is easy to find pieces of f(xn) that  are not concave up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For this reason, we must search for local minima on each  piece of f(xn) individually in order to guarantee the conjectured coloring is  globally minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Counting pairwise orderings of {xi + xj}0≤i<j≤n is closely related to other  combinatorial problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In Figure 4, we find that any ordering of {xi + xj}  could be encoded within a standard Young tableau of inverse staircase shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Here, the filling of the (r, s) entry of the tableau (where the top of the tableau  is the (0, 0) entry) is equal to the position of xr + xs when all pairs {xi + xj}  are placed in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  To simplify our computations, we make two restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' First, because of  numerical simulations in [2] and our own, we consider only colorings that are  antisymmetric: when reflected about the middle of the unit interval, almost  every point in the coloring is swapped to the opposite color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Equivalently, we  require that xk +xn−k = 1 for 0 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Additionally, we do not need to know  all of the relations in the total ordering of {xi+xj}0≤i<j≤n in order to determine  a configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Instead, for each pair (i, j) with i ̸= j, we search for the value of  15  x0  x1  x2  x3  x4  x5  x0  x1  x2  x3  x4  x5  1  2  4  5  3  6  11  9  7  8 10  14  13  15  12 17  16  19 20  18  21  Figure 4: Above is pictured a standard Young diagram corresponding to the  choice of endpoints x0 = 0, x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='19, x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='9, x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='6, x4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='65, and x5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  The entry labelled 3 tells us that x0 + x2 is the third smallest in the ordering  of pairs {xi + xj}5  i,j=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Additionally, the non-crossing lattice paths from the  line y = −x to the lower left of the Young diagram partition the diagram into  regions where all the corresponding pairwise sums are between two consecutive  values 2xk−1 and 2xk for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  16  k where 2xk−1 ≤ xi + xj ≤ 2xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Thus, we look for the number of ways to insert  the pairwise sums {xi + xj}i̸=j into the line, 0 = 2x0 ≤ 2x1 ≤ · · · ≤ 2xn = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In Figure 4, this reframing corresponds to not needing to know the entire filling  of the diagram, but instead having a family of non-crossing lattice paths each  starting at different points down the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This set-up is very similar to the  Lindstr¨om-Gessel-Viennot Lemma counting non-intersecting lattice paths, [11,  8], which was instrumental in proving the conjecture counting the number of  n × n alternating sign matrices, the story of which is told in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  To determine the number of such partial permutations, we develop an al-  gorithm that works recursively to verify whether growing partial permutations  are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Our implementation is similar to Miller and Peterson’s geomet-  ric approach to solving questions about More Sums Than Differences sets, [13,  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1], and also similar to Laaksonen’s approach to enumerating OEIS  sequence A237749, [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Both of these problems plus the problem we study here  could be phrased in terms of enumerating chambers in a hyperplane arrange-  ment, potentially including a restriction to a specific cone within the hyperplane  arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Enumerating chambers is a stream of research on its own (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  [16, 6]), and there are existing theorems counting chambers by using M¨obius  inversion, [20, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' However, here we do not need to enumerate every chamber  without restrictions because this is again equivalent to counting the possible  total orderings of {xi + xj}0≤i<j≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Consider antisymmetric block colorings with endpoints (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , xn)  for n even, so that xk + xn−k = 1 for 0 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then, the number of ways to  insert the pairs {xi + xj}i̸=j into the ordering 0 = 2x0 < 2x1 < · · · < 2xn = 2  grows as follows, starting with n = 0 and with n increasing by twos:  1, 1, 3, 23, 357, 9391, 371219, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The key computational tool in our proof is linear programming: with  existing linear programming packages like the GNU Linear Programming Kit  (GLPK, [9]) we can easily check whether a single system of inequalities has a  valid solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Thus, we create a running list of partial systems of inequalities  that have valid solutions, and count in how many ways it is possible to extend  each system with a single additional inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Below, we give pseudocode for  the algorithm we use, followed by a brief explanation of some of the technicalities  required to make this code run correctly and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The full code is posted  online at https://cocalc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='com/TorinGreenwood/MonochromeSequences/Mo  nochromaticProgressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Pseudocode to Enumerate BCG Diagrams  1  \\\\ Initialize a running list of partial systems of inequalities  2  PartialInequalitiesOld =  �  {0 = x0, xk + xn−k = 1 for 0 ≤ k ≤ n,  3  xk ≤ xk+1 for 0 ≤ k ≤ n − 1}  �  4  5  \\\\ For each partial system of inequalities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' for each set of  6  partial constraints),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' find all ways to add a new inequality  17  7  2xk ≤ xi + xj ≤ 2xk+1 by deciding where xi + xj fits between  8  successive 2xk  9  for (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' j) with i ̸= j:  10  PartialInequalitiesNew = {}  11  for k from i to j − 1:  12  for constraints in PartialInequalitiesOld:  13  if constraints ∪ {2xk < xi + xj < 2xk+1} is valid:  14  PartialInequalitiesNew + =  �  constraints  15  ∪ {2xk < xi + xj < 2xk+1}  �  16  PartialInequalitiesOld = PartialInequalitiesNew  17  return PartialInequalitiesOld  We now discuss some important aspects of our implementation with GLPK  that ensured the code ran efficiently and correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The uninterested reader may  skip the rest of this proof without a loss of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' First, mixed integer linear  programs typically search to optimize a linear objective function in the variables  x over a region of linear inequalities written in terms of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Here, our goal was  simply to check whether a system of linear inequalities was feasible, meaning  that a solution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This can be achieved with linear programming by setting  the objective function to be any constant, C, because the linear program will  return a certificate x∗ where the maximum is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' When the objective  function is constant, this is simply any feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  As an added layer of complexity, linear programming typically only al-  lows for inequalities that are not strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  However, exponentially many ar-  rangements of the pairs {xi + xj}i̸=j can be achieved trivially by the solution  x∗ = (0, 1/2, 1/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , 1/2, 1), since any sum of distinct endpoints xi + xj would  equal 1/2, 1, or 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In fact, many such arrangements can only be achieved  by these trivial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' If we allow such solutions, it is not possible for the  program to finish due to an explosion in the number of possible systems of  inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' To avoid this scenario, we force all inequalities in every system  to be strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Thus, we introduce a single auxiliary variable ϵ that converts  strict inequalities into weak inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For example, the strict inequalities  2xk < xi + xj < 2xk+1 become a pair of weak inequalities 2xk + ϵ ≤ xi + xj and  xi + xj + ϵ ≤ 2xk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' After adding ϵ to every inequality, we change the objective  function from a constant C to the variable ϵ, and search for the maximum value  of ϵ within the region where the inequality system is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' As long as a value of  ϵ > 0 is found, the set of inequalities is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Generally, linear programming implementations work with floating point  arithmetic, leading to rounding errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Because there is no way to bound how  small a feasible region could be, we used the version of GLPK that works using  rational arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Even still, GLPK returns its solutions as floating point  numbers, occasionally with roundoff errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Thus, we set the threshold for ϵ to  be near the limits of floating point arithmetic at 5 × 10−15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We found that the  smallest ϵ value above this threshold was on the order of 10−3, illustrating that  any value below 5 × 10−15 was due to precision error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Unfortunately, rational solvers tend to be much slower than their floating  18  point counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' To address this, we needed to optimize our code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' One factor  that impacted runtime significantly was the order in which pairs (i, j) were  checked in the for loop in Line 9 of the pseudocode above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' After experimenting  with different orderings, we found that checking the pairs in decreasing order of  j − i was several times faster than checking the pairs in lexicographic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Additionally, the rational solver became stuck in an infinite loop for 26 of  the millions of feasibility checks it ran on systems of inequalities for the n = 12  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This issue was resolved by changing the order of the inequalities within  these problematic systems of inequalities before they were input into GLPK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We  did not find a single ordering that avoided infinite loops for all of the feasibility  checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Instead, we found that for any specific set of inequalities, there always  existed some ordering where GLPK would halt rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='3  Optimizing over all BCG diagrams  Now that we have found the number of possible BCG diagrams for f(xn) for  each n ≤ 12 and xn that are antisymmetric, we can finally leverage the power  of calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Despite being a piecewise function, we soon find that f(xn) is  continuous with continuous partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  This implies that its global  maximum happens either at a critical point, or at a boundary point of the  domain of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='3, we prove that f(xn) has continuous  partial derivatives for any fixed n, after which we can finish the proof of Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Consider all block colorings with n blocks and endpoints x =  (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In the region 0 = x0 < x1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' < xn = 1, f(xn) is a  continuous function with continuous partial derivatives in each variable xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' From the diagram representation in Figure 1, it is clear that f is a  continuous function of the endpoints, xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' To verify that the partial derivatives  are continuous, we give a geometric argument: consider a single region R in the  diagram, like those drawn in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Let fR(xn) be the area of this single  region as a function of the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The region has up to 6 sides, and each  side is a line whose position is determined by some single endpoint xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Thus,  ∂  ∂xi fR(xn) is equal to the total length of the boundaries of R determined by the  variable xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' (Indeed, moving a single xi by a small ∆xi changes the area of the  polygon R by ∆xi · ℓi + O(∆xi)2 as ∆xi → 0, where ℓi is the total length of the  boundaries of R determined by xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=')  Now, we consider several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' As xn varies, R may do any of the following:  stay the same type of polygon, change polygon types, or enter or leave the  diagram altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' It is clear that when R stays the same type of polygon, its  side lengths change continuously in xn, so fR(xn) has continuous partials in this  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' When R changes polygon type, the change must occur when a diagonal  line crosses over a corner of the box formed by the horizontal and vertical strips  shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This means that any time a region changes polygon type, the  side that enters or leaves the region does so with initial length 0, again implying  that the partials are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Finally, we consider when R enters or leaves  19  the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' There are two ways this can happen: either a horizontal, vertical,  or diagonal strip collapses to width 0, or a diagonal line crosses over the corner  of the box described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' When a strip collapses to width 0, this means that  there are two consecutive endpoints xi and xi+1 where (xi+1 −xi) tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Thus, although the partial derivative is not continuous in this case, it is on the  boundary of the region of xn values we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' On the other hand, when a  diagonal line crosses over the corner of a box, all the side lengths of the polygon  approach zero, so the partials are again continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  The diagram representation of f(xn) makes it clear that f(xn) has a bounded  number of regions: at most one for each intersection of a horizontal, vertical, and  diagonal strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Since fR(xn) is continuous with continuous partial derivatives  for every region R, f(xn) is too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Now that we have shown that f(xn) is continuous with continuous partial  derivatives for a fixed n, we are ready to complete the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2  with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Let x12 = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' , x12) with 0 ≤ x0 ≤ · · · ≤ x12 = 1 and x12  antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The global minimum of f(x12) over all such x12 is 117/548,  occurring uniquely at the coloring from Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Because f(x12) is a C1 function on the polytope 0 = x0 < x1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' <  x12 = 1, its global minimum occurs on the boundary of the polytope or at a  critical point within the interior of the polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The boundary of this polytope  is the union of polytopes of the same form with fewer variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For this reason,  we find the critical points for f(xn) for each even value of n between 0 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1 implies that f(xn) is a piecewise-quadratic function for each  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Fix n, and consider any piece of this function, which can be extended to a  function f ∗(xn) on all of Rn/2−1 (since x1 through xn/2−1 determine the coloring  because it is anti-symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The partial derivatives of f ∗(xn) are piecewise  linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' The critical points of this everywhere-defined quadratic are the  solution to a linear system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Therefore, there are either no critical  points, or a vector space of critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In the case that the vector space  has positive dimension, the value of f ∗(xn) must be constant among all of its  critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Thus, when f ∗(xn) has critical points, it suffices to check the  value of f ∗(xn) at a single critical point when checking for the values of local  optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  This leads us to the following pseudocode to search for the global minimum  of f(x12) on the polytope 0 = x0 ≤ x1 ≤ · · · ≤ x12 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' (The full version of the  code is posted at https://cocalc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='com/TorinGreenwood/MonochromeSequen  ces/MonochromaticProgressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=')  Pseudocode to find the minimum value of f(x12)  18  \\\\ We search the interior of f(xn) for n = 2, 4, 6, 8, 10, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  19  >> for n from 0 to 12 by twos:  20  20  21  >> for each piece f ∗(xn) of the piecewise function f(xn)  22  (identified by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2):  23  24  >> calculate the quadratic polynomial corresponding to  25  f ∗(xn) (by using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1)  26  27  \\\\In the next line, we can feed into GLPK all of the  28  inequalities defining the configuration for f ∗(xn) plus  29  the equalities that set each of the partial derivatives  30  of f ∗(xn) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  31  >> use GLPK to check the existence of a critical point  32  of f ∗(xn) within the region of xn-values where  33  f(xn) ≡ f ∗(xn)  34  35  >> if critical points exist:  36  >> evaluate f ∗(xn) at any critical point cn  37  >> store cn and f ∗(cn) if this is a new record minimum  38  39  >> return the minimum cn and f ∗(cn) values  This code verifies that the global minimum of f(xn) when n is at most 12 is  117/548,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' which is attained only at the coloring with endpoints given in Equation  (1) (without the N in each coordinate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  The number of pieces of the function f(xn) for n ≥ 14 grows very rapidly,  making an analysis of its critical points increasingly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' However, we  can guarantee that the optimal is always rational:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For each n ∈ Z+, the minimum value of f(xn) is rational,  regardless of whether xn is restricted to be anti-symmetric or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' This is nearly immediate from our proof structure: the minimum of  f(xn) occurs at some critical point of f(xℓ) with xℓ in the interior of where  f(xℓ) is defined, for an ℓ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' These critical points are defined by a system  of linear equations with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Whenever there are only finitely  many critical points, they all must have rational coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' On the other hand,  if there is a piece f ∗(xℓ) of the piecewise function f(xℓ) that has infinitely many  critical points, all of the critical points of f ∗(xℓ) attain the same constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  This implies that there still exists a critical point with rational coordinates where  the minimum is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Finally, because each piece of f(xℓ) is a quadratic  with rational coefficients, the minimum is thus also rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  5  Circle colorings  As a variation on the theme of enumerating monochromatic progressions within  colorings of [N], some authors have also investigated properties of arithmetic  21  progressions within colorings of the cyclic group ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For example, given a fixed  red and blue 2-coloring of Zp for p prime, the fraction of monochromatic 3-term  progressions that are red or blue depends only on the proportion of elements  colored red, and not on the exact positioning of the red and blue elements, [5, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Even when N is not prime, the fraction of monochromatic 3-term progressions  in ZN is bounded below by the quantity given if N were prime, [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Inspired by these results, we now explore a continuous analogue to the enu-  meration of monochromatic progressions within 2-colorings of ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Color each of  the numbers in the unit circle S1 = {e2πiθ : θ ∈ [0, 1)} with red or blue, and con-  sider 3-term arithmetic progressions of the form (e2πix1, e2πi(x1+d), e2πi(x1+2d))  for x1, d ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' To properly discuss the “fraction” of these that are monochro-  matic for a given coloring, we introduce the uniform probability measure µ on  [0, 1) and randomly sample arithmetic progressions by independently choosing  x1, d ∈ [0, 1) according to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Using this framework, the probability of selecting a  monochromatic progression depends only on the Lebesgue measure of the set of  points colored red (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' the likelihood that, say, e2πix1 is red) and not on which  points were colored red, which is an analogous result to the one for 2-colorings  of the discrete group Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Let C : S1 → {0, 1} be any measurable coloring of S1 with  p := µ  ��  θ ∈ [0, 1) : C  �  e2πiθ�  = 0  ��  defined as the proportion of points colored red, and let m(C) be the set containing  all pairs (x1, d) ∈ [0, 1) × [0, 1) such that (e2πix1, e2πi(x1+d), e2πi(x1+2d)) are  monochromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then,  (µ × µ)(m(C)) = 1 − 3p + 3p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In particular, if we randomly select a starting point x1 and an increment d  independently from each other according to the uniform distribution on S1, then  the probability that the associated 3-AP is monochrome depends only on the  proportion p of red points and not on how these points are distributed around  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' We take a probabilistic approach that follows the proof structure of The-  orem 6 from [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' To that end, let x1 and d be independent draws from µ and  for i = 1, 2, 3 let Ai (respectively, Bi) be the event that the ith term in the  progression (e2πix1, e2πi(x1+d), e2πi(x1+2d)) is red (respectively, blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Then, via  inclusion/exclusion, we have  P(A1 ∪ A2 ∪ A3) =  � 3  �  i=1  P(Ai)  �  −  �  �  �  1≤i<j≤3  P(Ai ∩ Aj)  �  � + P(A1 ∩ A2 ∩ A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  We note that P(A1 ∪ A2 ∪ A3) = 1 − P(B1 ∩ B2 ∩ B3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Since P(m(C)) =  P(A1 ∩ A2 ∩ A3) + P(B1 ∩ B2 ∩ B3), we can rearrange the above to obtain  P(m(C)) = 1 −  3  �  i=1  P(Ai) +  �  1≤i<j≤3  P(Ai ∩ Aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  (9)  22  The random variables e2πix1, e2πi(x1+d), and e2πi(x1+2d) are pairwise indepen-  dent and uniformly distributed on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  This is true based on the rotational  invariance of the uniform distribution on S1 and the fact that e2πix1 and e2πid  are independent and uniformly distributed on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' It follows that P(Ai) = p and  P(Ai ∩ Aj) = p2 for i, j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Substituting these into (9) yields  (µ × µ)(m(C)) = P(m(C)) = 1 − 3p + 3p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  6  Future Work  In Sections 3 and 4 above, we outlined an approach to identifying the optimal  coloring of [N] and the interval [0, 1] that minimizes the fraction of monochro-  matic 3-APs for any fixed upper bound on the number of blocks n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' A natural  question is whether we can show that the optimal coloring for any n > 12 is the  same as the optimal coloring for n = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' One possibility is to prove that the  colorings are no better for n = 14, and then argue that adding arbitrarily more  intervals is no better than adding just two more intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Besides investigating how colorings of [N], ZN, [0, 1], and S1 affect the preva-  lence of monochromatic arithmetic progressions of length 3, there are other re-  lated problems that have yet to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Perhaps the most natural question  to ask is how the analysis changes if we consider longer arithmetic progressions,  and the articles [18, 2, 12, 3] make partial progress in this direction for several  different lengths of progressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' A slightly less obvious question is to ask what  happens when we consider arithmetic progressions of color-dependent lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  For example, we could attempt to color [0, 1] or [N] in a way that simultane-  ously minimizes the fractions of monochromatic blue progressions of length 3  and monochromatic red progressions of length 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Another natural generalization is to add more colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' What do the 3- and 4-  colorings of [0, 1] that minimize monochrome arithmetic progressions of length  3 look like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Can anything be said about the rate at which the fraction of  monochrome progressions decays as the number of colors increases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' All of these  questions have natural analogues in the setting of Ramsey theory as applied  to graphs, and of course these generalizations might interact in any number of  ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  When the problems studied in this paper were first posed, it was unclear  whether or not colorings could perform better than random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Although they  can perform better than random in the cases we present in detail above, is this  also true for related problems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Recent work in [4] gives interesting insights into  some classes of problems where solutions must be better than random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In addition to changing the number of colors or length of the progressions  we study, we could also consider colorings in other geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' For example,  we wonder how to color an interval that has a gap in the middle in order  to minimize monochromatic APs therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' By varying the length of the gap,  we might gain insight into why antisymmetry is seemingly important in the  23  optimal block colorings of [0, 1] that we discuss above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Furthermore, we have  already seen that in the contexts of Zp for p prime and the continuous circle, the  performance of colorings with respect to 3-term progressions depends only on the  ratios of the colors present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' What other algebraic and geometric settings exhibit  similar behavior?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Alternatively, what would happen if we were to consider S1  as in Section 5 but sample 3-APs by choosing the start point and increment  according to a different distribution than uniform?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  7  Acknowledgments  Computations were performed using High Performance Computing infrastruc-  ture provided by the Mathematical Sciences Support unit at the University of  the Witwatersrand, and for this the authors are thankful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  Additionally, the  authors are grateful for invaluable tips from Professor Antti Laaksonen on how  to optimize the code in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  References  [1]  David M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Bressoud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Proofs and Confirmations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Cambridge University Press,  Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' isbn: 9780511613449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1017/cbo9780511613449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' url:  http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1017/CBO9780511613449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  [2]  Steve Butler, Kevin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Costello, and Ron Graham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' “Finding Patterns  Avoiding Many Monochromatic Constellations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In: Experimental Math-  ematics 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='4 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1080/10586458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='10  390631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1080%2F10586458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='10390631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  [3]  Steve Butler, Ron Graham, and Linyuan Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' “Unrolling Residues to Avoid  Progressions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In: Mathematics Magazine 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 83–94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' doi: 10  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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+page_content='  [4]  Kevin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Costello and Gabriel Elvin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Avoiding Monochromatic Solutions  to 3-term Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='03350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' url:  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='org/abs/2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='03350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  [5]  Boris A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Datskovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' “On the number of monochromatic Schur triples”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In: Advances in Applied Mathematics 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1 (2003), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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+page_content=' Real analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Fourth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Prentice Hall,  Boston, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' isbn: 0-13-143747-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  [16]  Richard P Stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' An introduction to hyperplane arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  accessed 21-July-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='cis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='edu/~cis6  10/sp06stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  [17]  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' van der Waerden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' “Beweis einer Baudetschen Vermutung”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In: Nieuw  Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Wisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 15 (1927), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 212–216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  [18]  Julia Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' “The minimum number of monochromatic 4-term progressions  in Zp”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In: Journal of Combinatorics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 53–68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  25  [19]  Thomas Zaslavsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' “A combinatorial analysis of topological dissections”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  In: Advances in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='3 (1977), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 267–285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' issn: 0001-8708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1016/0001-8708(77)90076-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1016/000  1-8708(77)90076-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  [20]  Thomas Zaslavsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' “Facing up to arrangements: face-count formulas for  partitions of space by hyperplanes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' In: Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='issue 1,  154 (1975), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' vii+102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' issn: 0065-9266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1090/memo/0154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' url:  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1090/memo/0154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  A  Appendix: 20 Polygonal Regions  The 20 possible regions from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='1 are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  1  xi  i+1  j  j+1  x  x  x  k+1  x  k  x  Characterizing Inequalities:  2xk ≤ xi + xj ≤ xi+1 + xj+1 ≤ 2xk+1  Region Area:  (xi+1 − xi)(xj+1 − xj)  2  xi  i+1  j  j+1  x  x  x  k+1  x  k  x  Characterizing Inequalities:  xi + xj ≤ 2xk ≤ min(xi + xj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' xi+1 + xj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='xi+1 + xj+1 ≤ 2xk+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='Region Area:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='(xi+1 − xi)(xj+1 − xj) − (2xk − xi − xj)2/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='xi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='Characterizing Inequalities:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='xi+1 + xj ≤ 2xk ≤ xi + xj+1 ≤ xi+1 + xj+1 ≤ 2xk+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='Region Area:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='(xi+1 − xi)(xj+1 + xi/2 + xi+1/2 − 2xk)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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+page_content='xi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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+page_content='max(xi + xj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' xi+1 + xj) ≤ 2xk ≤ xi+1 + xj+1 ≤ 2xk+1  Region Area:  (2xk − xi+1 − xj+1)2/2  6  xi  i+1  j  j+1  x  x  x  k+1  x  k  x  Characterizing Inequalities:  2xk ≤ xi + xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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+page_content=' xi+1 + xj) ≤ 2xk+1 ≤ xi+1 + xj+1  Region Area:  (xi+1 − xi)(xj+1 − xj) − (2xk+1 − xj+1 − xi+1)2/2  7  xi  i+1  j  j+1  x  x  x  k+1  x  k  x  Characterizing Inequalities:  xi + xj ≤ 2xk ≤ min(xi + xj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
+page_content=' xi+1 + xj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAyT4oBgHgl3EQffPh0/content/2301.00336v1.pdf'}
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diff --git a/GtE3T4oBgHgl3EQfWQqL/content/tmp_files/2301.04467v1.pdf.txt b/GtE3T4oBgHgl3EQfWQqL/content/tmp_files/2301.04467v1.pdf.txt
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+FrustumFormer: Adaptive Instance-aware Resampling for Multi-view 3D
+Detection
+Yuqi Wang1,2
+Yuntao Chen3
+Zhaoxiang Zhang1,2,3
+1Institute of Automation, Chinese Academy of Sciences (CASIA)
+2 School of Artificial Intelligence, University of Chinese Academy of Sciences
+3 Centre for Artificial Intelligence and Robotics, HKISI CAS
+{wangyuqi2020,zhaoxiang.zhang}@ia.ac.cn
+chenyuntao08@gmail.com
+Abstract
+The transformation of features from 2D perspective
+space to 3D space is essential to multi-view 3D object de-
+tection. Recent approaches mainly focus on the design of
+view transformation, either pixel-wisely lifting perspective
+view features into 3D space with estimated depth or grid-
+wisely constructing BEV features via 3D projection, treat-
+ing all pixels or grids equally. However, choosing what to
+transform is also important but has rarely been discussed
+before. The pixels of a moving car are more informative
+than the pixels of the sky.
+To fully utilize the informa-
+tion contained in images, the view transformation should
+be able to adapt to different image regions according to
+their contents. In this paper, we propose a novel frame-
+work named FrustumFormer, which pays more attention to
+the features in instance regions via adaptive instance-aware
+resampling. Specifically, the model obtains instance frus-
+tums on the bird’s eye view by leveraging image view object
+proposals. An adaptive occupancy mask within the instance
+frustum is learned to refine the instance location. Moreover,
+the temporal frustum intersection could further reduce the
+localization uncertainty of objects. Comprehensive exper-
+iments on the nuScenes dataset demonstrate the effective-
+ness of FrustumFormer, and we achieve a new state-of-the-
+art performance on the benchmark. Codes will be released
+soon.
+1. Introduction
+Perception in 3D space has gained increasing attention in
+both academia and industry. Despite the success of LiDAR-
+based methods [14,33,41,44], camera-based 3D object de-
+tection [19, 35, 36, 43] has earned a wide audience, due to
+the low cost for deployment and advantages for long-range
+detection. Recently, multi-view 3D detection in Bird’s-Eye-
+View (BEV) has made fast progresses. Due to the unified
+representation in 3D space, multi-view features and tem-
+poral information can be fused conveniently, which leads
+to significant performance improvement over monocular
+methods [5,28,35,39].
+Transforming perspective view features into the bird’s-
+eye view is the key to the success of modern BEV 3D de-
+tectors [12,18,19,22]. As shown in Fig. 1, we categorize the
+existing methods into lifting-based ones like LSS [30] and
+BEVDet [12] and query-based ones like BEVFormer [19]
+and Ego3RT [25]. However, these methods mainly focus
+on the design of view transformation strategies while over-
+looking the significance of choosing the right features to
+transform during view transformation. Regions containing
+objects like vehicles and pedestrians are apparently more in-
+formative than the empty background like sky and ground.
+But all previous methods treat them with equal importance.
+We suggest that the view transformation should be adaptive
+with respect to the image content. Therefore, we propose
+Adaptive Instance-aware Resampling (AIR), an instance-
+aware view transformation, as shown in Fig. 1c. The core
+idea of AIR is to reduce instance localization uncertainty by
+focusing on a selective part of BEV queries. Localizing in-
+stance regions is difficult directly on the BEV plane but rel-
+atively easy in the image view. Therefore, the instance frus-
+tum, lifting from instance proposals in image views, gives
+geometrical hints of the possible locations of objects in the
+3D space. Though the instance frustum has provided initial
+prior locations, it is still a large uncertain area. We propose
+an occupancy mask predictor and a temporal frustum fusion
+module to further reduce the localization uncertainty. Our
+model learns an occupancy mask for frustum queries on the
+1
+arXiv:2301.04467v1  [cs.CV]  10 Jan 2023
+
+(a) Grid Sampling in Image.
+(b) Grid Sampling in BEV.
+(c) Instance-aware Sampling in Frustum.
+Figure 1. Comparison of different sampling strategies for the feature transformation from image view to bird’s eye view. (a)
+represents the sampling in image view and lift features [12] to BEV with pixel-wise depth estimation. (b) shows the grid sampling in BEV
+and queries back [19] to obtain image features via camera projection. (c) illustrates our proposed strategy: instance-aware sampling in the
+frustum, which is adaptive sampling according to the view content. More attention will focus on instance regions.
+BEV plane, predicting the possibility that a region might
+contain objects. We also fuse instance frustums across dif-
+ferent time steps, where the intersection area poses geomet-
+ric constraints for actual locations of objects.
+We propose a novel framework called FrustumFormer
+based on the insights mentioned previously, which effec-
+tively enhances the learning of instance-aware BEV fea-
+tures via Adaptive Instance-aware Resampling. Frustum-
+Former utilizes the instance frustum to establish the con-
+nection between perspective and bird’s eye view regions,
+which contains two key designs: (1) frustum encoder to en-
+hance instance-aware features via adaptive instance-aware
+resampling within the instance frustum. (2) frustum fusion
+module to aggregate history instance frustum features for
+accurate localization and velocity prediction. In conclusion,
+the contributions of this work are as follows:
+• We propose FrustumFormer, a novel framework that
+exploits the geometric constraints behind perspective
+view and birds’ eye view by instance frustum.
+• We propose that choosing what to transform is also im-
+portant during view transformation. The view transfor-
+mation should adapt to the view content. Instance re-
+gions should gain more attention rather than be treated
+equally.
+Therefore, we design Adaptive Instance-
+aware Resampling (AIR) to focus more on the in-
+stance regions, enhancing the learning of instance-
+aware BEV features.
+• We evaluate the proposed FrustumFormer on the
+nuScenes dataset. We achieve improved performance
+compared to prior arts. FrustumFormer achieves 58.9
+NDS and 51.6 mAP on nuScenes test set without bells
+and whistles.
+2. Related Work
+2.1. Frustum-based 3D Object Detection
+Frustum indicates the possible locations of 3D objects in
+a 3D space by projecting 2D interested regions. The frustum
+is commonly used to aid fusion [8,27,31,37,42] in 3D ob-
+ject detection when RGB images and LiDAR data are avail-
+able. Frustum PointNets [31] takes advantage of mature 2D
+object detectors and performs 3D object instance segmenta-
+tion within the trimmed 3D frustums. Frustum Fusion [26]
+leverages the intersection volume of the two frustums in-
+duced by the 2D detection on stereo images. To deal with
+LiDAR sparsity, Faraway-Frustum [42] proposes a novel fu-
+sion strategy for detecting faraway objects. In this paper, we
+introduce the idea of frustum into camera-only 3D detection
+for enhancing instance-aware BEV features.
+2.2. Multi-view 3D Object Detection
+Multi-view 3D object detection aims to predict the 3D
+bounding boxes and categories of the objects with multi-
+view images as input. Current methods can be divided into
+two schemes: lifting 2D to 3D and Querying 2D from 3D.
+Lifting 2D to 3D. Following the spirit of LSS [30],
+BEVDet [12] lifts multi-view 2D image features into a
+depth-aware frustum and splats into a unified bird’s-eye-
+view (BEV) representation and applies to the detection task.
+BEVDepth [18] utilizes LiDAR points as supervision to
+learn reliable depth estimation. BEVDet4D [11] incorpo-
+rates the temporal information and extends the BEVDet to
+the spatial-temporal 4D working space. Recently, STS [38],
+BEVStereo [16] and SOLOFusion [29] further attempt to
+improve the depth learning by combining temporal geomet-
+ric constraints.
+2
+
+T
+T-1
+Backbone
+Frustum Fusion
+Frustum Encoder
+Backbone
+BEV Feature
+Detection Head
+AIR
+Cross 
+Attention
+Cross 
+Attention
++
+Instance Feature
+Scene Feature
+Frustum Encoder
+AIR
+Cross 
+Attention
+Cross 
+Attention
++
+Instance Feature
+Scene Feature
+AIR
+Instance Frustum
+Occupancy Mask
+BEV Feature
+(a) Overall architecture of FrustumFormer.
+T"
+T#
+(b) Temporal Frustum Fusion.
+Figure 2. Illustration of our proposed FrustumFormer. (a) shows the overall pipeline. The image backbone first extracts the multi-view
+image features. The frustum encoder transforms the multi-view image features into a unified BEV feature by integrating temporal infor-
+mation from frustum fusion. Then the detection head decodes the BEV feature to the final outputs. Adaptive Instance-aware Resampling
+(AIR) is utilized to adaptively adjust the sampling area and points according to the view content. It consists of two parts: instance frustum
+query generation and frustum occupancy mask prediction. (b) illustrates the hints for object locations during temporal frustum fusion.
+Querying
+2D
+from
+3D.
+Following
+DETR
+[2],
+DETR3D [36] predicts learnable queries in 3D space
+and projects back to query the corresponding 2D im-
+age features.
+PETR [22, 23] proposes to query directly
+with 3D position-aware features, which are generated
+by encoding the 3D position embedding into 2D image
+features.
+Ego3RT [25] introduces the polarized grid of
+dense imaginary eyes and sends rays backward to 2D visual
+representation.
+BEVFormer [19] learns spatiotemporal
+BEV features via deformable attention, which explicitly
+constructs the BEV grid samples in 3D space and queries
+back to aggregate multi-view image features.
+Polar-
+Former [13] further generates polar queries in the Polar
+coordinate and encodes BEV features in Polar space.
+3. Method
+In multi-view 3D object detection task, N monocular
+views of images I = {Ii ∈ R3×H×W }N
+i=1, together with
+camera intrinsics K = {Ki ∈ R3×3}N
+i=1 and camera extrin-
+sics T = {Ti ∈ R4×4}N
+i=1 are given. The objective of the
+model is to output the 3D attributes (locations, size and ve-
+locity) and the corresponding category of objects contained
+in multi-view images.
+As shown in Fig. 2a, FrustumFormer mainly focuses
+on the feature transformation process and is composed of
+four components: an image backbone, a frustum encoder,
+a frustum fusion module, and a detection head. The im-
+age backbone first extracts the image features of multi-view
+images. Aiding by the frustum fusion module, the image
+features transform into a unified BEV feature via the frus-
+tum encoder. Finally, a query-based detection head decodes
+the BEV feature to the 3D outputs of the detection task.
+3.1. Frustum Encoder
+Frustum encoder transforms the multi-scale multi-view
+image features F to a unified BEV feature B. Instead of
+treating all regions equally during the feature view trans-
+formation, our frustum encoder adaptively transforms the
+image features according to the view content. As shown
+in Fig. 2a, we use two types of BEV queries to construct
+the final BEV features, the scene queries Qs and the in-
+stance queries Qi. The learning process of the scene query
+is similar to BEVFormer [19]. However, the instance query
+is only learned inside instance regions, and the learned in-
+stance features are further combined with the scene feature
+to form the final instance-aware BEV features.
+Specifi-
+cally, the selection of instance regions is made via adaptive
+instance-aware resampling, which consists of (1) instance
+frustum query generation and (2) frustum occupancy mask
+prediction. Finally, the instance feature is learned by (3)
+instance frustum cross attention computed in selected in-
+stance regions. We will introduce these three parts in the
+following.
+Instance Frustum Query Generation. This section in-
+troduces the query generation for a single instance frus-
+tum Qf, which is a subset of instance query Qi.
+The
+core insight is to leverage the instance mask from perspec-
+tive views and select the corresponding region on the BEV
+plane. Following the query-based [13,19] view transforma-
+tion, we define a group of grid-shape learnable parameters
+Qi ∈ RH×W ×C as the instance queries. H, W are the spa-
+tial shape of BEV queries, and C is the channel dimension.
+We first generate sampling points {pk
+i = (xi, yi, zk), i ∈
+H × W, k ∈ K} corresponding to a single BEV query Qpi
+at grid region center pi = (xi, yi), and then project these
+3
+
+points to different image views. K is the number of sam-
+pling points in the vertical direction of pillars. The projec-
+tion between sampling points pk
+i and its corresponding 2D
+reference point (uk
+ij, vk
+ij) on the j-th image view is formu-
+lated as:
+πj(pk
+i ) = (uk
+ij, vk
+ij)
+(1)
+dk
+ij · [uk
+ij, vk
+ij, 1]T = Tj · [xk
+i , yk
+i , zk
+i , 1]T
+(2)
+where πj(pk
+i ) denotes the projection of the k-th sampling
+point at location pi on the j-th camera view. Tj ∈ R3×4
+is the projection matrix of the j-th camera. xk
+i , yk
+i , zk
+i rep-
+resents the 3D location of the sampling point in the vehicle
+frame. uk
+ij, vk
+ij denotes corresponding 2D reference point
+on j-th view projected from 3D sampling point pk
+i . dk
+ij is
+the depth in the camera frame.
+Predicting the instance frustum region directly in bird’s
+eye view is challenging, but detecting the objects in per-
+spective view [9, 32] is relatively mature. Inspired by this,
+we take advantage of object masks on the image plane and
+leverage its geometric clues for the BEV plane. The in-
+stance frustum queries Qf of a specific 2D instance could
+be defined as all instance BEV queries Qi with image plane
+projection points inside the object mask S Eq. (3):
+Qf = {Qpi ∈ Qi|
+∃
+π(pk
+i ) ∈ S}
+(3)
+Qpi ∈ R1×C is the query located at pi = (xi, yi). pk
+i rep-
+resents the k-th sampling points in the pillars at pi. π(pk
+i )
+denotes the projection points of pk
+i to the image plane.
+Frustum Occupancy Mask Prediction. Although instance
+frustum provides potential locations for objects, it still cor-
+responds to a large area on the BEV plane due to depth un-
+certainty. Therefore, we propose to predict an occupancy
+mask for all frustum to further reduce the localization un-
+certainty of instances within the instance frustum. Besides
+perspective constraints from instance frustum, another con-
+straint we could utilize is the supervision directly on the
+BEV plane. Specifically, given the union of all instance
+frustum queries ∪Qf, we design a AdaMask module to pre-
+dict a binary occupancy mask Obev ∈ RH×W ×1 on the
+BEV plane for all instance frustum queries in a single shot.
+The occupancy mask reflects the probability of a grid-wise
+region containing the objects, and is computed by Eq. (4):
+Obev = AdaMask(∪Qf)
+(4)
+AdaMask is a learned module composed of 2D convolu-
+tions. The supervision comes from the projection of the
+ground truth 3D bounding boxes on the BEV plane. We
+choose the focal loss [21] for learning occupancy mask in
+Eq. (5):
+Lm = Focal Loss(Obev, Ω)
+(5)
+where Ω denotes the projection mask of 3D bounding boxes
+on the BEV plane.
+We choose the minimum projecting
+bounding box on the BEV plane as the supervise signal.
+The minimum projecting bounding box is composed of the
+outermost corners of the objects, considering the rotation.
+Furthermore, the supervision for Obev adds to each layer of
+the frustum encoder for iterative refinement, together with
+the BEV instance feature learning. We utilize the last out-
+put instance frustum query for the current layer to predict
+the current occupancy mask. Therefore, the sampling areas
+adapt to the previous layer output.
+Instance Frustum Cross Attention.
+Instance Frustum
+Cross Attention (IFCA) is designed for the feature interac-
+tion between instance queries Qi and image view features
+F. The instance queries Qi is selected by Eq. (6):
+Qi = {Qpi ∈ ∪Qf|Obev(i) = 1}
+(6)
+Instance queries are selected from the instance frustum
+queries Qf. Obev(i) denotes the occupancy value at po-
+sition pi on the BEV plane. For each query Qpi in Qf, if
+Obev(i) predicts the occupancy value is 1, then the query
+Qpi is marked as instance query. The process of instance
+frustum cross-attention (IFCA) can be formulated as:
+IFCA(Qpi
+i , Fj) =
+M
+�
+m=1
+DeformAttn(Qpi
+i , πj(pm
+i ), Fj)
+(7)
+where Qpi
+i
+is an instance query at location pi, πj(pm
+i ) is
+the projection to get the m-th 2D reference point on the j-
+th camera view. M is the total number of sampling points
+in an instance query. Fj is the image features of the j-th
+camera view.
+3.2. Frustum Fusion Module
+Temporal information plays an important role in camera-
+based 3D object detection, especially in inferring the mo-
+tion state of objects and recognizing objects under heavy
+occlusions. Beyond learning occupancy mask on the BEV
+plane, another solution for eliminating the location uncer-
+tainty in the instance frustum is to fuse the temporal infor-
+mation.
+Temporal Frustum Intersection. As shown in Fig. 2b, the
+intersection area of the instance frustum at different times-
+tamps leaves hints for the accurate location of 3D objects.
+Inspired by this, we constrain the query interaction within
+instance frustum regions, implicitly learning features from
+interaction areas. Given instance frustum queries Qf at cur-
+rent timestamp t and history instance frustum queries Hf
+preserved at timestamp t′. For a query ∪Qpi
+f at position
+pi, we use the information from ego-motion (∆x, ∆y, ∆θ)
+to compute the corresponding position p′
+i at timestamp t′.
+The cross attention for query Qpi
+f only compute the his-
+tory queries around position p′
+i of Hf. Following [19], we
+adopt an RNN-like [6] way to fuse the historical instance
+4
+
+frustum queries sequentially. In this way, the long-range
+hints for the intersection area can be aggregated.
+Temporal Frustum Cross Attention. Temporal Frustum
+Cross Attention (TFCA) aggregates the information of his-
+tory instance frustum queries Hf into the current instance
+frustum queries Qf. Since the objects might be movable
+in the scene, causing the misalignment if only computing
+the query at p′
+i. Deformable attention [46] is utilized to
+reduce the influence of object movement. The process of
+temporal frustum cross attention (TFCA) can be formulated
+as follows:
+TFCA(Qpi
+f , Hf) =
+M
+�
+m=1
+DeformAttn(Qpi
+f , p′m
+i , Hf) (8)
+where Qpi
+f denotes the instance frustum query located at
+pi = (xi, yi). Hf represents the history instance frustum
+query. p′
+i is the aligned position by ego-motion. For each
+query at location p′
+i, we sample M points p′m
+i to query the
+history instance frustum feature.
+4. Experiment
+4.1. Datasets
+We conduct experiments on the challenging public au-
+tonomous driving datasets, namely nuScenes dataset [1].
+nuScenes dataset [1]. The nuScenes dataset provides 1000
+sequences of different scenes collected in Boston and Singa-
+pore. These sequences are officially split into 700/150/150
+ones for training, validation, and testing. Each sequence is
+roughly about 20s duration, and the key samples are anno-
+tated at 2Hz, contributing to a total of 1.4M objects bound-
+ing boxes.
+Each sample consists of RGB images from
+6 cameras covering the 360-degree horizontal FOV: front,
+front left, front right, back, back left, and back right. The
+image resolution is 1600×900 pixels in all views. 10 classes
+are annotated for the object-detecting task: car, truck, bus,
+trailer, construction vehicle, pedestrian, motorcycle, bicy-
+cle, barrier, and traffic cone.
+Evaluation metrics. For the official evaluation protocol
+in the nuScenes dataset, the metrics include mean Average
+Precision (mAP) and a set of True Positive (TP) metrics,
+which contains the average translation error (ATE), average
+scale error (ASE), average orientation error (AOE), average
+velocity error (AVE), and average attribute error (AAE). Fi-
+nally, the nuScenes detection score (NDS) is defined to con-
+sider the above metrics as in Eq. (9):
+NDS = 1
+10
+�
+5mAP +
+�
+mTP∈T P
+(1 − min (1, mTP))
+�
+(9)
+4.2. Experimental Settings
+Implementation Details. Following previous methods [19,
+35, 36], we utilize two types of backbone: ResNet101-
+DCN [7, 10] that initialized from FCOS3D [35], and
+VoVnet-99 [15] that initialized from DD3D [28]. We uti-
+lize the output multi-scale features from FPN [20] with
+sizes 1/8, 1/16, 1/32, and 1/64, and the feature dimension
+is 256. The frustum encoder has 6 layers, and we imple-
+ment it based on BEVFormer [19]. The default size of BEV
+queries is 200 × 200, and the perception ranges are [-51.2m,
+51.2m] for the X and Y axis and [-3m, 5m] for the Z axis.
+We sample K = 8 points for each pillar-like region of the
+BEV query. We adopt learnable position embedding for
+BEV queries. For the 2D instance proposals, we utilized the
+Mask R-CNN [9] pre-trained on the nuImages [1]. We use
+the output bounding boxes to generate object mask regions,
+and the score threshold is set to 0.5. The loss weight for
+Lm is set to 5. For the frustum fusion module, the tempo-
+ral window size W is set to 8, and we randomly sampled 4
+keyframes in the training phase. We utilized a query-based
+detection head [36] to decode the BEV features. The num of
+the object query is set to 600 and has 3 groups of queries [4]
+during training.
+Training. We train the model on 8 NVIDIA A100 GPUs
+with batch size 1 per GPU. We train our model with
+AdamW [24] optimizer for 24 epochs, an initial learning
+rate of 2 × 10−4 with a cosine annealing schedule. The
+input of the images is cropped to 1600 × 640. We adopt
+data augmentations like image scaling, flipping, color dis-
+tortion, and GridMask [3]. For the ablation study, we train
+the model with a total batch size of 8 for 24 epochs with-
+out data augmentation. We use the ResNet-50 [10] as the
+backbone. The image resolution is resized at a scale of 0.8,
+which is 1280 × 512.
+Inference. During inference, the previous BEV features
+are saved and used for the next, corresponding to the infi-
+nite temporal window of a sequence. This online inference
+strategy is time-efficient. Since we adopted three groups of
+queries during training, only one group is utilized at infer-
+ence time. We do not adopt model-agnostic tricks such as
+model ensemble and test time augmentation when evaluat-
+ing our model on both val and test sets.
+4.3. 3D Object Detection Results
+We compare our method with the state of the art on both
+val and test sets of nuScenes.
+nuScenes test set.
+Table 1 compares the results on the
+nuScenes test set. We achieved 51.6 mAP and 58.9 NDS
+without utilizing extra depth supervision from LiDAR. Un-
+der the setting without utilizing LiDAR as supervision, our
+method outperforms the previous state of the art. We eval-
+uate our model in two types of backbone mentioned in
+the implementation details.
+With R101-DCN [7] as the
+backbone, we could achieve 47.8 mAP and 56.1 NDS, a
+significant improvement (+2.1 mAP and 1.8 NDS) over
+previous methods.
+For the final performance, we train
+5
+
+Methods
+Backbone
+CBGS
+LiDAR
+mAP↑
+NDS↑
+mATE↓
+mASE↓
+mAOE↓
+mAVE↓
+mAAE↓
+FCOS3D‡ [35]
+R101†
+0.358
+0.428
+0.690
+0.249
+0.452
+1.434
+0.124
+PGD [34]
+R101†
+0.386
+0.448
+0.626
+0.245
+0.451
+1.509
+0.127
+BEVFormer [19]
+R101†
+0.445
+0.535
+0.631
+0.257
+0.405
+0.435
+0.143
+PolarFormer [13]
+R101†
+0.457
+0.543
+0.612
+0.257
+0.392
+0.467
+0.129
+FrustumFormer
+R101†
+0.478
+0.561
+0.575
+0.257
+0.402
+0.411
+0.132
+DD3D [28]‡
+V2-99*
+0.418
+0.477
+0.572
+0.249
+0.368
+1.014
+0.124
+DETR3D‡ [36]
+V2-99*
+✓
+0.412
+0.479
+0.641
+0.255
+0.394
+0.845
+0.133
+Ego3RT [25]
+V2-99*
+0.425
+0.473
+0.549
+0.264
+0.433
+1.014
+0.145
+M2BEV [40]
+X-101
+0.429
+0.474
+0.583
+0.254
+0.376
+1.053
+0.190
+BEVDet4D‡ [11]
+Swin-B
+✓
+0.451
+0.569
+0.511
+0.241
+0.386
+0.301
+0.121
+UVTR [17]
+V2-99*
+0.472
+0.551
+0.577
+0.253
+0.391
+0.508
+0.123
+BEVFormer [19]
+V2-99*
+0.481
+0.569
+0.582
+0.256
+0.375
+0.378
+0.126
+PolarFormer [13]
+V2-99*
+0.493
+0.572
+0.556
+0.256
+0.364
+0.440
+0.127
+PETRv2 [23]
+V2-99*
+0.490
+0.582
+0.561
+0.243
+0.361
+0.343
+0.120
+BEVDepth‡ [18]
+V2-99*
+✓
+✓
+0.503
+0.600
+0.445
+0.245
+0.378
+0.320
+0.126
+BEVStereo [16]
+V2-99*
+✓
+✓
+0.525
+0.610
+0.431
+0.246
+0.358
+0.357
+0.138
+FrustumFormer
+V2-99*
+0.516
+0.589
+0.555
+0.249
+0.372
+0.389
+0.126
+Table 1. Comparison to state-of-art on the nuScenes test set. * notes that VoVNet-99(V2-99) [15] was pre-trained on the depth
+estimation task with extra data [28]. †Initialized from FCOS3D [35] backbone. ‡ means utilizing test-time augmentation during inference.
+The commonly used scheme for training is 24 epochs, and CBGS [45] would increase the training epochs by nearly 4.5×. LiDAR means
+training depth branch utilizing extra modality supervision from LiDAR.
+Methods
+Backbone
+CBGS
+LiDAR
+mAP↑
+NDS↑
+mATE↓
+mASE↓
+mAOE↓
+mAVE↓
+mAAE↓
+FCOS3D [35]
+R101†
+0.295
+0.372
+0.806
+0.268
+0.511
+1.315
+0.170
+DETR3D [36]
+R101†
+✓
+0.349
+0.434
+0.716
+0.268
+0.379
+0.842
+0.200
+PGD [34]
+R101†
+0.358
+0.425
+0.667
+0.264
+0.435
+1.276
+0.177
+PETR [22]
+R101†
+✓
+0.370
+0.442
+0.711
+0.267
+0.383
+0.865
+0.201
+UVTR [17]
+R101†
+0.379
+0.483
+0.731
+0.267
+0.350
+0.510
+0.200
+BEVFormer [19]
+R101†
+0.416
+0.517
+0.673
+0.274
+0.372
+0.394
+0.198
+PolarFormer [13]
+R101†
+0.432
+0.528
+0.648
+0.270
+0.348
+0.409
+0.201
+BEVDepth [18]
+R101
+✓
+✓
+0.412
+0.535
+0.565
+0.266
+0.358
+0.331
+0.190
+STS [38]
+R101
+✓
+✓
+0.431
+0.542
+0.525
+0.262
+0.380
+0.369
+0.204
+FrustumFormer
+R101†
+0.457
+0.546
+0.624
+0.265
+0.362
+0.380
+0.191
+Table 2. Comparison to state-of-art on the nuScenes val set. †Initialized from FCOS3D [35] backbone. Our model is trained for 24
+epochs without CBGS [45]. LiDAR means training depth branch utilizing extra modality supervision from LiDAR.
+FrustumFormer on the trainval split for 24 epochs without
+CBGS [45], with VoVNet(V2-99) as backbone architecture
+with a pre-trained checkpoint from DD3D [28].
+nuScenes validation set. Table 2 shows that our method
+achieves leading performance on the nuScenes val set. We
+achieved 45.7 mAP and 54.6 NDS without bells and whis-
+tles. Unlike the evaluation on test set, all the methods are
+compared with a fair backbone here. Since BEVDepth [18]
+and STS [38] utilized extra modality supervision in training,
+our NDS metric only improved slightly compared to them,
+but our mAP improved significantly. The translation error
+would be reduced with LiDAR supervision for the depth es-
+timation, but this required extra modality data from LiDAR.
+Besides, our model is trained for 24 epochs, while they actu-
+ally trained 90 epochs if using CBGS [45]. More qualitative
+results are shown in supplementary materials.
+4.4. Ablation Study
+We
+conduct
+several
+ablation
+experiments
+on
+the
+nuScenes val set to validate the design of FrustumFormer.
+As mentioned in the implementation details, we use the
+ResNet-50 [10] as the backbone, and the image resolution
+is resized to 0.8 scales for all ablation experiments.
+Ablation of Components in FrustumFormer. Table 3 ab-
+lates the components designed in FrustumFormer. (a) is the
+baseline setting of our method. (b) is the baseline with the
+6
+
+instance frustum query, which resamples the points in the
+whole instance frustum region. (c) is the baseline with the
+occupancy mask, which gets supervision on the BEV plane.
+(d) is the baseline with adaptive instance-aware resampling,
+which consists of instance frustum query and occupancy
+mask. By utilizing the adaptive instance-aware resampling
+to enhance the instance-aware BEV feature, both mAP and
+NDS can be significantly improved. (e) is based on (d) and
+further adds the history frustum information to incorporate
+temporal clues. Above all, our FrustumFormer could im-
+prove 4.2 mAP and 9.7 NDS compared to the baseline.
+IF
+OM
+FF
+mAP↑
+NDS↑
+mATE↓
+(a)
+0.318
+0.366
+0.771
+(b)
+✓
+0.326
+0.373
+0.765
+(c)
+✓
+0.328
+0.381
+0.759
+(d)
+✓
+✓
+0.337
+0.383
+0.749
+(e)
+✓
+✓
+✓
+0.360
+0.463
+0.719
+Table 3. Ablation of components in FrustumFormer. IF denotes
+instance frustum, OM denotes occupancy mask, and FF means
+temporal frustum fusion. Adaptive instance-aware resampling is
+the combination of IF and OM, shown in (d).
+Ablation of Instance-aware Sampling. Table 4 proves the
+effectiveness of instance-aware sampling. (a) represents the
+baseline setting, which treats all regions equally and sam-
+ples 1x points for a cell region. (b) increases the sampling
+points to 2x for all regions. (c) selectively resamples the
+points inside instance regions. Compared with (b) and (c),
+we found that instance-aware sampling is more effective
+since simply increasing the sampling points for all regions
+has no gain.
+Total
+Scene
+Instance
+mAP↑
+NDS↑
+(a)
+1×
+1×
+-
+0.318
+0.366
+(b)
+2×
+2×
+-
+0.318
+0.362
+(c)
+2×
+1×
+1×
+0.326
+0.373
+Table 4. Ablation of instance-aware sampling. In our method,
+1× means sampling 8 points for a cell region on the BEV plane.
+Ablation of Occupancy Mask Learning. Table 5 com-
+pares different supervision for learning occupancy masks
+on the BEV plane. (a) is the baseline without explicit su-
+pervision. (b) adds the supervision on the BEV plane, in
+which the instance area can be obtained by projecting an-
+notated bounding boxes. To ease the learning, we slightly
+enlarge the bounding boxes by 1.0 meters in this setting. (c)
+uses the strict projection area (without enlargement) from
+the bounding boxes. (d) increases the loss weight for Lm
+from 5.0 to 10.0. We choose the (c) as our default setting.
+Ablation for Temporal Frustum Fusion. In Table 6, we
+first demonstrate the effectiveness of utilizing frustum in
+Supervision
+α
+mAP↑
+NDS↑
+mATE↓
+(a)
+w/o
+-
+0.318
+0.366
+0.771
+(b)
+w/ BEV box*
+5.0
+0.324
+0.374
+0.756
+(c)
+w/ BEV box
+5.0
+0.328
+0.381
+0.759
+(d)
+w/ BEV box
+10.0
+0.322
+0.381
+0.749
+Table 5. Ablation of occupancy mask learning. BEV box means
+utilizing the ground truth bounding boxes’ projection on the BEV
+plane as supervision. * denotes enlarged bounding box. α is the
+loss weight for learning occupancy mask.
+temporal fusion, and then we ablate the parameters for win-
+dow size W and keyframes K. (a) is the baseline for com-
+parison, with window size 4 and 2 keyframes. (b) adds the
+temporal frustum information.
+(c) enlarges the temporal
+window size to 8 and uses 4 keyframes in the temporal win-
+dow, which achieves the best performance. Both mAP and
+NDS would improve with a longer temporal window. We
+use the parameters in (d) as the default setting.
+W
+K
+Frustum
+mAP↑
+NDS↑
+mAVE↓
+(a)
+4
+2
+0.353
+0.454
+0.497
+(b)
+4
+2
+✓
+0.355
+0.457
+0.479
+(c)
+8
+4
+✓
+0.360
+0.463
+0.463
+Table 6. Ablation for temporal frustum fusion. W means the
+history window size. K determines the key frames sampled in tem-
+poral window during model training.
+0.35
+0.45
+0.55
+0.65
+0.75
+0.85
+car
+traffic cone
+barrier
+pedestrian motorcycle
+truck
+bicycle
+trailer
+bus
+construction 
+vehicle
+Baseline
+w/ AIR
+Figure 3. Improvement of Recall Under Low Visibility. We
+compute the recall under the visibility at 0-40% for all categories
+on the nuScenes val set. The recall for bus, bicycle, trailer, and
+construction vehicle categories improved significantly.
+Improvement of Recall Under Low Visibility.
+The
+nuScenes dataset provides the visibility labels of objects
+in four subsets {0-40%, 40-60%, 60-80%, 80-100%}. As
+shown in Fig. 3, we compare the recall between baseline
+and baseline with adaptive instance-aware resampling un-
+der low visibility (0-40%). We found that the recall for cat-
+egories of bicycle, trailer, bus, and construction vehicle im-
+proved a lot under the low visibility. Since nearly 29% of
+7
+
+objects belong to the visibility of 0-40%, such improvement
+is crucial for a better 3D object detector. More quantitative
+results and visualizations under heavy occlusions are in sup-
+plementary materials.
+4.5. Qualitative Analysis
+Visualization of Recall Improvement. As shown in Fig. 4,
+we illustrate the recall improvement with AIR from the pre-
+diction on the nuScenes val set. The prediction boxes are
+marked in blue, while the ground truth boxes are marked in
+green. In the red circle region, the objects would be discov-
+ered by utilizing AIR to enhance the learning of instance
+features.
+v
+Figure 4. Visualization of Recall Improvement. The left side is
+the baseline, and the right is the baseline with AIR. Examples are
+selected from the prediction on the nuScenes val set. The predic-
+tion boxes are marked in blue, while the ground truth boxes are
+marked in green. In the red circle region, more objects can be
+discovered by our methods.
+Visualization of Instance-aware Sampling. As shown in
+Fig. 5, we illustrate the instance-aware sampling points of
+our method on both perspective view and bird’s eye view.
+The sampling points are highly related to the instance re-
+gions, enhancing the learning of the instance-aware feature.
+Examples are selected from the nuScenes dataset.
+Visualization of Instance-aware BEV Feature. As shown
+in Fig. 6, we visualize the BEV feature output by our frus-
+tum encoder. Examples are selected from the prediction
+on the nuScenes val set. The BEV feature learned by our
+frustum encoder is instance-aware and has strong relations
+to the real positions. Here we visualize the corresponding
+ground truth boxes on the right side. Furthermore, we com-
+pare the BEV feature between the baseline and the base-
+line with adaptive instance-aware resampling. When utiliz-
+ing AIR, more instance regions would be discovered (cor-
+Figure 5. Visualization of Instance-aware Sampling. We visual-
+ize the instance-aware sampling on the perspective view and bird’s
+eye view. Ground truth bounding boxes are marked in green color.
+Examples are selected from the nuScenes dataset.
+responding to the recall improvement), and the features are
+more instance discriminative in the dense areas.
+Figure 6.
+Visualization of the instance-aware BEV feature.
+From left to right, we compare the feature heatmaps output by
+frustum encoder (w/o AIR), with AIR, and ground truth boxes in
+green shown on the right. The colors for the feature heatmap corre-
+spond to the norm value. The BEV feature learned by our frustum
+encoder is instance-aware and has strong relations to the actual po-
+sitions in 3D space. By using AIR, more instance regions would
+be discovered, and the features are more instance discriminative in
+the dense areas. Examples are selected from the prediction on the
+nuScenes val set.
+5. Conclusion
+In this paper, we propose FrustumFormer, a novel
+framework for multi-view 3D object detection.
+The
+core insight of FrustumFormer is to transform adaptively
+according to the view contents. To this end, we designed
+adaptive instance-aware resampling, paying more attention
+to the instance regions during feature view transformation.
+By utilizing adaptive instance-aware resampling in the
+frustum encoder and temporal frustum fusion module, the
+model can better locate the instance regions while learning
+the instance-aware BEV features.
+Experimental results
+on the nuScenes dataset demonstrate the effectiveness
+of our method for multi-view 3D object detection.
+Our
+method significantly improved on mAP over the previous
+methods due to the focus on the instance regions.
+We
+hope our framework can serve as a new baseline for
+8
+
+XI厂l:
+.:80
+口:following 3D perception work, and let more people pay
+attention to the view content during feature transformation.
+References
+[1] Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora,
+Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Gi-
+ancarlo Baldan, and Oscar Beijbom.
+nuscenes: A multi-
+modal dataset for autonomous driving. In Proceedings of
+the IEEE/CVF conference on computer vision and pattern
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diff --git a/GtE3T4oBgHgl3EQfWQqL/content/tmp_files/load_file.txt b/GtE3T4oBgHgl3EQfWQqL/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf,len=782
+page_content='FrustumFormer: Adaptive Instance-aware Resampling for Multi-view 3D  Detection  Yuqi Wang1,2  Yuntao Chen3  Zhaoxiang Zhang1,2,3  1Institute of Automation, Chinese Academy of Sciences (CASIA)  2 School of Artificial Intelligence, University of Chinese Academy of Sciences  3 Centre for Artificial Intelligence and Robotics, HKISI CAS  {wangyuqi2020,zhaoxiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='zhang}@ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='cn  chenyuntao08@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='com  Abstract  The transformation of features from 2D perspective  space to 3D space is essential to multi-view 3D object de-  tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Recent approaches mainly focus on the design of  view transformation, either pixel-wisely lifting perspective  view features into 3D space with estimated depth or grid-  wisely constructing BEV features via 3D projection, treat-  ing all pixels or grids equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' However, choosing what to  transform is also important but has rarely been discussed  before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The pixels of a moving car are more informative  than the pixels of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  To fully utilize the informa-  tion contained in images, the view transformation should  be able to adapt to different image regions according to  their contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' In this paper, we propose a novel frame-  work named FrustumFormer, which pays more attention to  the features in instance regions via adaptive instance-aware  resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Specifically, the model obtains instance frus-  tums on the bird’s eye view by leveraging image view object  proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' An adaptive occupancy mask within the instance  frustum is learned to refine the instance location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Moreover,  the temporal frustum intersection could further reduce the  localization uncertainty of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Comprehensive exper-  iments on the nuScenes dataset demonstrate the effective-  ness of FrustumFormer, and we achieve a new state-of-the-  art performance on the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Codes will be released  soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Introduction  Perception in 3D space has gained increasing attention in  both academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Despite the success of LiDAR-  based methods [14,33,41,44], camera-based 3D object de-  tection [19, 35, 36, 43] has earned a wide audience, due to  the low cost for deployment and advantages for long-range  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Recently, multi-view 3D detection in Bird’s-Eye-  View (BEV) has made fast progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Due to the unified  representation in 3D space, multi-view features and tem-  poral information can be fused conveniently, which leads  to significant performance improvement over monocular  methods [5,28,35,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Transforming perspective view features into the bird’s-  eye view is the key to the success of modern BEV 3D de-  tectors [12,18,19,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 1, we categorize the  existing methods into lifting-based ones like LSS [30] and  BEVDet [12] and query-based ones like BEVFormer [19]  and Ego3RT [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' However, these methods mainly focus  on the design of view transformation strategies while over-  looking the significance of choosing the right features to  transform during view transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Regions containing  objects like vehicles and pedestrians are apparently more in-  formative than the empty background like sky and ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  But all previous methods treat them with equal importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  We suggest that the view transformation should be adaptive  with respect to the image content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Therefore, we propose  Adaptive Instance-aware Resampling (AIR), an instance-  aware view transformation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The core  idea of AIR is to reduce instance localization uncertainty by  focusing on a selective part of BEV queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Localizing in-  stance regions is difficult directly on the BEV plane but rel-  atively easy in the image view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Therefore, the instance frus-  tum, lifting from instance proposals in image views, gives  geometrical hints of the possible locations of objects in the  3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Though the instance frustum has provided initial  prior locations, it is still a large uncertain area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We propose  an occupancy mask predictor and a temporal frustum fusion  module to further reduce the localization uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Our  model learns an occupancy mask for frustum queries on the  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='04467v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='CV]  10 Jan 2023  (a) Grid Sampling in Image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  (b) Grid Sampling in BEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  (c) Instance-aware Sampling in Frustum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Comparison of different sampling strategies for the feature transformation from image view to bird’s eye view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (a)  represents the sampling in image view and lift features [12] to BEV with pixel-wise depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (b) shows the grid sampling in BEV  and queries back [19] to obtain image features via camera projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (c) illustrates our proposed strategy: instance-aware sampling in the  frustum, which is adaptive sampling according to the view content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' More attention will focus on instance regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  BEV plane, predicting the possibility that a region might  contain objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We also fuse instance frustums across dif-  ferent time steps, where the intersection area poses geomet-  ric constraints for actual locations of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  We propose a novel framework called FrustumFormer  based on the insights mentioned previously, which effec-  tively enhances the learning of instance-aware BEV fea-  tures via Adaptive Instance-aware Resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Frustum-  Former utilizes the instance frustum to establish the con-  nection between perspective and bird’s eye view regions,  which contains two key designs: (1) frustum encoder to en-  hance instance-aware features via adaptive instance-aware  resampling within the instance frustum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (2) frustum fusion  module to aggregate history instance frustum features for  accurate localization and velocity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' In conclusion,  the contributions of this work are as follows:  We propose FrustumFormer, a novel framework that  exploits the geometric constraints behind perspective  view and birds’ eye view by instance frustum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  We propose that choosing what to transform is also im-  portant during view transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The view transfor-  mation should adapt to the view content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Instance re-  gions should gain more attention rather than be treated  equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Therefore, we design Adaptive Instance-  aware Resampling (AIR) to focus more on the in-  stance regions, enhancing the learning of instance-  aware BEV features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  We evaluate the proposed FrustumFormer on the  nuScenes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We achieve improved performance  compared to prior arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' FrustumFormer achieves 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='9  NDS and 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='6 mAP on nuScenes test set without bells  and whistles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Related Work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Frustum-based 3D Object Detection  Frustum indicates the possible locations of 3D objects in  a 3D space by projecting 2D interested regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The frustum  is commonly used to aid fusion [8,27,31,37,42] in 3D ob-  ject detection when RGB images and LiDAR data are avail-  able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Frustum PointNets [31] takes advantage of mature 2D  object detectors and performs 3D object instance segmenta-  tion within the trimmed 3D frustums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Frustum Fusion [26]  leverages the intersection volume of the two frustums in-  duced by the 2D detection on stereo images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' To deal with  LiDAR sparsity, Faraway-Frustum [42] proposes a novel fu-  sion strategy for detecting faraway objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' In this paper, we  introduce the idea of frustum into camera-only 3D detection  for enhancing instance-aware BEV features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Multi-view 3D Object Detection  Multi-view 3D object detection aims to predict the 3D  bounding boxes and categories of the objects with multi-  view images as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Current methods can be divided into  two schemes: lifting 2D to 3D and Querying 2D from 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Lifting 2D to 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Following the spirit of LSS [30],  BEVDet [12] lifts multi-view 2D image features into a  depth-aware frustum and splats into a unified bird’s-eye-  view (BEV) representation and applies to the detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  BEVDepth [18] utilizes LiDAR points as supervision to  learn reliable depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' BEVDet4D [11] incorpo-  rates the temporal information and extends the BEVDet to  the spatial-temporal 4D working space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Recently, STS [38],  BEVStereo [16] and SOLOFusion [29] further attempt to  improve the depth learning by combining temporal geomet-  ric constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  2  T  T-1  Backbone  Frustum Fusion  Frustum Encoder  Backbone  BEV Feature  Detection Head  AIR  Cross  Attention  Cross  Attention  +  Instance Feature  Scene Feature  Frustum Encoder  AIR  Cross  Attention  Cross  Attention  +  Instance Feature  Scene Feature  AIR  Instance Frustum  Occupancy Mask  BEV Feature  (a) Overall architecture of FrustumFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  T"  T#  (b) Temporal Frustum Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Illustration of our proposed FrustumFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (a) shows the overall pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The image backbone first extracts the multi-view  image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The frustum encoder transforms the multi-view image features into a unified BEV feature by integrating temporal infor-  mation from frustum fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Then the detection head decodes the BEV feature to the final outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Adaptive Instance-aware Resampling  (AIR) is utilized to adaptively adjust the sampling area and points according to the view content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' It consists of two parts: instance frustum  query generation and frustum occupancy mask prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (b) illustrates the hints for object locations during temporal frustum fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Querying  2D  from  3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Following  DETR  [2],  DETR3D [36] predicts learnable queries in 3D space  and projects back to query the corresponding 2D im-  age features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  PETR [22, 23] proposes to query directly  with 3D position-aware features, which are generated  by encoding the 3D position embedding into 2D image  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Ego3RT [25] introduces the polarized grid of  dense imaginary eyes and sends rays backward to 2D visual  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  BEVFormer [19] learns spatiotemporal  BEV features via deformable attention, which explicitly  constructs the BEV grid samples in 3D space and queries  back to aggregate multi-view image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Polar-  Former [13] further generates polar queries in the Polar  coordinate and encodes BEV features in Polar space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Method  In multi-view 3D object detection task, N monocular  views of images I = {Ii ∈ R3×H×W }N  i=1, together with  camera intrinsics K = {Ki ∈ R3×3}N  i=1 and camera extrin-  sics T = {Ti ∈ R4×4}N  i=1 are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The objective of the  model is to output the 3D attributes (locations, size and ve-  locity) and the corresponding category of objects contained  in multi-view images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 2a, FrustumFormer mainly focuses  on the feature transformation process and is composed of  four components: an image backbone, a frustum encoder,  a frustum fusion module, and a detection head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The im-  age backbone first extracts the image features of multi-view  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Aiding by the frustum fusion module, the image  features transform into a unified BEV feature via the frus-  tum encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Finally, a query-based detection head decodes  the BEV feature to the 3D outputs of the detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Frustum Encoder  Frustum encoder transforms the multi-scale multi-view  image features F to a unified BEV feature B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Instead of  treating all regions equally during the feature view trans-  formation, our frustum encoder adaptively transforms the  image features according to the view content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' As shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 2a, we use two types of BEV queries to construct  the final BEV features, the scene queries Qs and the in-  stance queries Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The learning process of the scene query  is similar to BEVFormer [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' However, the instance query  is only learned inside instance regions, and the learned in-  stance features are further combined with the scene feature  to form the final instance-aware BEV features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Specifi-  cally, the selection of instance regions is made via adaptive  instance-aware resampling, which consists of (1) instance  frustum query generation and (2) frustum occupancy mask  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Finally, the instance feature is learned by (3)  instance frustum cross attention computed in selected in-  stance regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We will introduce these three parts in the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Instance Frustum Query Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' This section in-  troduces the query generation for a single instance frus-  tum Qf, which is a subset of instance query Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  The  core insight is to leverage the instance mask from perspec-  tive views and select the corresponding region on the BEV  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Following the query-based [13,19] view transforma-  tion, we define a group of grid-shape learnable parameters  Qi ∈ RH×W ×C as the instance queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' H, W are the spa-  tial shape of BEV queries, and C is the channel dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  We first generate sampling points {pk  i = (xi, yi, zk), i ∈  H × W, k ∈ K} corresponding to a single BEV query Qpi  at grid region center pi = (xi, yi), and then project these  3  points to different image views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' K is the number of sam-  pling points in the vertical direction of pillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The projec-  tion between sampling points pk  i and its corresponding 2D  reference point (uk  ij, vk  ij) on the j-th image view is formu-  lated as:  πj(pk  i ) = (uk  ij, vk  ij)  (1)  dk  ij · [uk  ij, vk  ij, 1]T = Tj · [xk  i , yk  i , zk  i , 1]T  (2)  where πj(pk  i ) denotes the projection of the k-th sampling  point at location pi on the j-th camera view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Tj ∈ R3×4  is the projection matrix of the j-th camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' xk  i , yk  i , zk  i rep-  resents the 3D location of the sampling point in the vehicle  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' uk  ij, vk  ij denotes corresponding 2D reference point  on j-th view projected from 3D sampling point pk  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' dk  ij is  the depth in the camera frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Predicting the instance frustum region directly in bird’s  eye view is challenging, but detecting the objects in per-  spective view [9, 32] is relatively mature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Inspired by this,  we take advantage of object masks on the image plane and  leverage its geometric clues for the BEV plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The in-  stance frustum queries Qf of a specific 2D instance could  be defined as all instance BEV queries Qi with image plane  projection points inside the object mask S Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (3):  Qf = {Qpi ∈ Qi|  ∃  π(pk  i ) ∈ S}  (3)  Qpi ∈ R1×C is the query located at pi = (xi, yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' pk  i rep-  resents the k-th sampling points in the pillars at pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' π(pk  i )  denotes the projection points of pk  i to the image plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Frustum Occupancy Mask Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Although instance  frustum provides potential locations for objects, it still cor-  responds to a large area on the BEV plane due to depth un-  certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Therefore, we propose to predict an occupancy  mask for all frustum to further reduce the localization un-  certainty of instances within the instance frustum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Besides  perspective constraints from instance frustum, another con-  straint we could utilize is the supervision directly on the  BEV plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Specifically, given the union of all instance  frustum queries ∪Qf, we design a AdaMask module to pre-  dict a binary occupancy mask Obev ∈ RH×W ×1 on the  BEV plane for all instance frustum queries in a single shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  The occupancy mask reflects the probability of a grid-wise  region containing the objects, and is computed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (4):  Obev = AdaMask(∪Qf)  (4)  AdaMask is a learned module composed of 2D convolu-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The supervision comes from the projection of the  ground truth 3D bounding boxes on the BEV plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We  choose the focal loss [21] for learning occupancy mask in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (5):  Lm = Focal Loss(Obev, Ω)  (5)  where Ω denotes the projection mask of 3D bounding boxes  on the BEV plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  We choose the minimum projecting  bounding box on the BEV plane as the supervise signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  The minimum projecting bounding box is composed of the  outermost corners of the objects, considering the rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Furthermore, the supervision for Obev adds to each layer of  the frustum encoder for iterative refinement, together with  the BEV instance feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We utilize the last out-  put instance frustum query for the current layer to predict  the current occupancy mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Therefore, the sampling areas  adapt to the previous layer output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Instance Frustum Cross Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Instance Frustum  Cross Attention (IFCA) is designed for the feature interac-  tion between instance queries Qi and image view features  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The instance queries Qi is selected by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (6):  Qi = {Qpi ∈ ∪Qf|Obev(i) = 1}  (6)  Instance queries are selected from the instance frustum  queries Qf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Obev(i) denotes the occupancy value at po-  sition pi on the BEV plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' For each query Qpi in Qf, if  Obev(i) predicts the occupancy value is 1, then the query  Qpi is marked as instance query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The process of instance  frustum cross-attention (IFCA) can be formulated as:  IFCA(Qpi  i , Fj) =  M  �  m=1  DeformAttn(Qpi  i , πj(pm  i ), Fj)  (7)  where Qpi  i  is an instance query at location pi, πj(pm  i ) is  the projection to get the m-th 2D reference point on the j-  th camera view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' M is the total number of sampling points  in an instance query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Fj is the image features of the j-th  camera view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Frustum Fusion Module  Temporal information plays an important role in camera-  based 3D object detection, especially in inferring the mo-  tion state of objects and recognizing objects under heavy  occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Beyond learning occupancy mask on the BEV  plane, another solution for eliminating the location uncer-  tainty in the instance frustum is to fuse the temporal infor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Temporal Frustum Intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 2b, the  intersection area of the instance frustum at different times-  tamps leaves hints for the accurate location of 3D objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Inspired by this, we constrain the query interaction within  instance frustum regions, implicitly learning features from  interaction areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Given instance frustum queries Qf at cur-  rent timestamp t and history instance frustum queries Hf  preserved at timestamp t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' For a query ∪Qpi  f at position  pi, we use the information from ego-motion (∆x, ∆y, ∆θ)  to compute the corresponding position p′  i at timestamp t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  The cross attention for query Qpi  f only compute the his-  tory queries around position p′  i of Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Following [19], we  adopt an RNN-like [6] way to fuse the historical instance  4  frustum queries sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' In this way, the long-range  hints for the intersection area can be aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Temporal Frustum Cross Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Temporal Frustum  Cross Attention (TFCA) aggregates the information of his-  tory instance frustum queries Hf into the current instance  frustum queries Qf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Since the objects might be movable  in the scene, causing the misalignment if only computing  the query at p′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Deformable attention [46] is utilized to  reduce the influence of object movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The process of  temporal frustum cross attention (TFCA) can be formulated  as follows:  TFCA(Qpi  f , Hf) =  M  �  m=1  DeformAttn(Qpi  f , p′m  i , Hf) (8)  where Qpi  f denotes the instance frustum query located at  pi = (xi, yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Hf represents the history instance frustum  query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' p′  i is the aligned position by ego-motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' For each  query at location p′  i, we sample M points p′m  i to query the  history instance frustum feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Experiment  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Datasets  We conduct experiments on the challenging public au-  tonomous driving datasets, namely nuScenes dataset [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  nuScenes dataset [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The nuScenes dataset provides 1000  sequences of different scenes collected in Boston and Singa-  pore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' These sequences are officially split into 700/150/150  ones for training, validation, and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Each sequence is  roughly about 20s duration, and the key samples are anno-  tated at 2Hz, contributing to a total of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='4M objects bound-  ing boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Each sample consists of RGB images from  6 cameras covering the 360-degree horizontal FOV: front,  front left, front right, back, back left, and back right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The  image resolution is 1600×900 pixels in all views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 10 classes  are annotated for the object-detecting task: car, truck, bus,  trailer, construction vehicle, pedestrian, motorcycle, bicy-  cle, barrier, and traffic cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' For the official evaluation protocol  in the nuScenes dataset, the metrics include mean Average  Precision (mAP) and a set of True Positive (TP) metrics,  which contains the average translation error (ATE), average  scale error (ASE), average orientation error (AOE), average  velocity error (AVE), and average attribute error (AAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Fi-  nally, the nuScenes detection score (NDS) is defined to con-  sider the above metrics as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (9):  NDS = 1  10  �  5mAP +  �  mTP∈T P  (1 − min (1, mTP))  �  (9)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Experimental Settings  Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Following previous methods [19,  35, 36], we utilize two types of backbone: ResNet101-  DCN [7, 10] that initialized from FCOS3D [35], and  VoVnet-99 [15] that initialized from DD3D [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We uti-  lize the output multi-scale features from FPN [20] with  sizes 1/8, 1/16, 1/32, and 1/64, and the feature dimension  is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The frustum encoder has 6 layers, and we imple-  ment it based on BEVFormer [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The default size of BEV  queries is 200 × 200, and the perception ranges are [-51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='2m,  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='2m] for the X and Y axis and [-3m, 5m] for the Z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  We sample K = 8 points for each pillar-like region of the  BEV query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We adopt learnable position embedding for  BEV queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' For the 2D instance proposals, we utilized the  Mask R-CNN [9] pre-trained on the nuImages [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We use  the output bounding boxes to generate object mask regions,  and the score threshold is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The loss weight for  Lm is set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' For the frustum fusion module, the tempo-  ral window size W is set to 8, and we randomly sampled 4  keyframes in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We utilized a query-based  detection head [36] to decode the BEV features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The num of  the object query is set to 600 and has 3 groups of queries [4]  during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We train the model on 8 NVIDIA A100 GPUs  with batch size 1 per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We train our model with  AdamW [24] optimizer for 24 epochs, an initial learning  rate of 2 × 10−4 with a cosine annealing schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The  input of the images is cropped to 1600 × 640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We adopt  data augmentations like image scaling, flipping, color dis-  tortion, and GridMask [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' For the ablation study, we train  the model with a total batch size of 8 for 24 epochs with-  out data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We use the ResNet-50 [10] as the  backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The image resolution is resized at a scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='8,  which is 1280 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' During inference, the previous BEV features  are saved and used for the next, corresponding to the infi-  nite temporal window of a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' This online inference  strategy is time-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Since we adopted three groups of  queries during training, only one group is utilized at infer-  ence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We do not adopt model-agnostic tricks such as  model ensemble and test time augmentation when evaluat-  ing our model on both val and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 3D Object Detection Results  We compare our method with the state of the art on both  val and test sets of nuScenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  nuScenes test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Table 1 compares the results on the  nuScenes test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We achieved 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='6 mAP and 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content=' Un-  der the setting without utilizing LiDAR as supervision, our  method outperforms the previous state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We eval-  uate our model in two types of backbone mentioned in  the implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='126  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Comparison to state-of-art on the nuScenes test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' * notes that VoVNet-99(V2-99) [15] was pre-trained on the depth  estimation task with extra data [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' †Initialized from FCOS3D [35] backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' ‡ means utilizing test-time augmentation during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  The commonly used scheme for training is 24 epochs, and CBGS [45] would increase the training epochs by nearly 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='5×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' LiDAR means  training depth branch utilizing extra modality supervision from LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Methods  Backbone  CBGS  LiDAR  mAP↑  NDS↑  mATE↓  mASE↓  mAOE↓  mAVE↓  mAAE↓  FCOS3D [35]  R101†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='170  DETR3D [36]  R101†  ✓  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='200  PGD [34]  R101†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='177  PETR [22]  R101†  ✓  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='201  UVTR [17]  R101†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='379  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='200  BEVFormer [19]  R101†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='416  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='198  PolarFormer [13]  R101†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='432  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='412  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='190  STS [38]  R101  ✓  ✓  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='431  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='204  FrustumFormer  R101†  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='457  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='546  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='624  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='265  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='362  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='191  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Comparison to state-of-art on the nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' †Initialized from FCOS3D [35] backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Our model is trained for 24  epochs without CBGS [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' LiDAR means training depth branch utilizing extra modality supervision from LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  FrustumFormer on the trainval split for 24 epochs without  CBGS [45], with VoVNet(V2-99) as backbone architecture  with a pre-trained checkpoint from DD3D [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  nuScenes validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Table 2 shows that our method  achieves leading performance on the nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We  achieved 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='7 mAP and 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='6 NDS without bells and whis-  tles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Unlike the evaluation on test set, all the methods are  compared with a fair backbone here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Since BEVDepth [18]  and STS [38] utilized extra modality supervision in training,  our NDS metric only improved slightly compared to them,  but our mAP improved significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The translation error  would be reduced with LiDAR supervision for the depth es-  timation, but this required extra modality data from LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Besides, our model is trained for 24 epochs, while they actu-  ally trained 90 epochs if using CBGS [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' More qualitative  results are shown in supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Ablation Study  We  conduct  several  ablation  experiments  on  the  nuScenes val set to validate the design of FrustumFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  As mentioned in the implementation details, we use the  ResNet-50 [10] as the backbone, and the image resolution  is resized to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='8 scales for all ablation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Ablation of Components in FrustumFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Table 3 ab-  lates the components designed in FrustumFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (a) is the  baseline setting of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (b) is the baseline with the  6  instance frustum query, which resamples the points in the  whole instance frustum region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (c) is the baseline with the  occupancy mask, which gets supervision on the BEV plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  (d) is the baseline with adaptive instance-aware resampling,  which consists of instance frustum query and occupancy  mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' By utilizing the adaptive instance-aware resampling  to enhance the instance-aware BEV feature, both mAP and  NDS can be significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (e) is based on (d) and  further adds the history frustum information to incorporate  temporal clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Above all, our FrustumFormer could im-  prove 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='2 mAP and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='7 NDS compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  IF  OM  FF  mAP↑  NDS↑  mATE↓  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='318  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='366  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='771  (b)  ✓  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='326  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='373  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='765  (c)  ✓  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='328  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+page_content='759  (d)  ✓  ✓  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='337  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='383  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='749  (e)  ✓  ✓  ✓  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='360  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='463  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='719  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Ablation of components in FrustumFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' IF denotes  instance frustum, OM denotes occupancy mask, and FF means  temporal frustum fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Adaptive instance-aware resampling is  the combination of IF and OM, shown in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Ablation of Instance-aware Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Table 4 proves the  effectiveness of instance-aware sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (a) represents the  baseline setting, which treats all regions equally and sam-  ples 1x points for a cell region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (b) increases the sampling  points to 2x for all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (c) selectively resamples the  points inside instance regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Compared with (b) and (c),  we found that instance-aware sampling is more effective  since simply increasing the sampling points for all regions  has no gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Total  Scene  Instance  mAP↑  NDS↑  (a)  1×  1× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='318  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='366  (b)  2×  2× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='318  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='362  (c)  2×  1×  1×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='326  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='373  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Ablation of instance-aware sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' In our method,  1× means sampling 8 points for a cell region on the BEV plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Ablation of Occupancy Mask Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Table 5 com-  pares different supervision for learning occupancy masks  on the BEV plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (a) is the baseline without explicit su-  pervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (b) adds the supervision on the BEV plane, in  which the instance area can be obtained by projecting an-  notated bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' To ease the learning, we slightly  enlarge the bounding boxes by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='0 meters in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (c)  uses the strict projection area (without enlargement) from  the bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (d) increases the loss weight for Lm  from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='0 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We choose the (c) as our default setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Ablation for Temporal Frustum Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' In Table 6, we  first demonstrate the effectiveness of utilizing frustum in  Supervision  α  mAP↑  NDS↑  mATE↓  (a)  w/o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='318  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='366  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='771  (b)  w/ BEV box*  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='324  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='374  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='756  (c)  w/ BEV box  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='328  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='381  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='759  (d)  w/ BEV box  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='322  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='381  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='749  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Ablation of occupancy mask learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' BEV box means  utilizing the ground truth bounding boxes’ projection on the BEV  plane as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' * denotes enlarged bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' α is the  loss weight for learning occupancy mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  temporal fusion, and then we ablate the parameters for win-  dow size W and keyframes K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (a) is the baseline for com-  parison, with window size 4 and 2 keyframes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' (b) adds the  temporal frustum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  (c) enlarges the temporal  window size to 8 and uses 4 keyframes in the temporal win-  dow, which achieves the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Both mAP and  NDS would improve with a longer temporal window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We  use the parameters in (d) as the default setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  W  K  Frustum  mAP↑  NDS↑  mAVE↓  (a)  4  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='353  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='454  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='497  (b)  4  2  ✓  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='355  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='457  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='479  (c)  8  4  ✓  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='360  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='463  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='463  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Ablation for temporal frustum fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' W means the  history window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' K determines the key frames sampled in tem-  poral window during model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='85  car  traffic cone  barrier  pedestrian motorcycle  truck  bicycle  trailer  bus  construction  vehicle  Baseline  w/ AIR  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Improvement of Recall Under Low Visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We  compute the recall under the visibility at 0-40% for all categories  on the nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The recall for bus, bicycle, trailer, and  construction vehicle categories improved significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Improvement of Recall Under Low Visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  The  nuScenes dataset provides the visibility labels of objects  in four subsets {0-40%, 40-60%, 60-80%, 80-100%}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' As  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 3, we compare the recall between baseline  and baseline with adaptive instance-aware resampling un-  der low visibility (0-40%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We found that the recall for cat-  egories of bicycle, trailer, bus, and construction vehicle im-  proved a lot under the low visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Since nearly 29% of  7  objects belong to the visibility of 0-40%, such improvement  is crucial for a better 3D object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' More quantitative  results and visualizations under heavy occlusions are in sup-  plementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Qualitative Analysis  Visualization of Recall Improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 4,  we illustrate the recall improvement with AIR from the pre-  diction on the nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The prediction boxes are  marked in blue, while the ground truth boxes are marked in  green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' In the red circle region, the objects would be discov-  ered by utilizing AIR to enhance the learning of instance  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  v  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Visualization of Recall Improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The left side is  the baseline, and the right is the baseline with AIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Examples are  selected from the prediction on the nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The predic-  tion boxes are marked in blue, while the ground truth boxes are  marked in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' In the red circle region, more objects can be  discovered by our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Visualization of Instance-aware Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' As shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 5, we illustrate the instance-aware sampling points of  our method on both perspective view and bird’s eye view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  The sampling points are highly related to the instance re-  gions, enhancing the learning of the instance-aware feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Examples are selected from the nuScenes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Visualization of Instance-aware BEV Feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' As shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' 6, we visualize the BEV feature output by our frus-  tum encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Examples are selected from the prediction  on the nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The BEV feature learned by our  frustum encoder is instance-aware and has strong relations  to the real positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Here we visualize the corresponding  ground truth boxes on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Furthermore, we com-  pare the BEV feature between the baseline and the base-  line with adaptive instance-aware resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' When utiliz-  ing AIR, more instance regions would be discovered (cor-  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Visualization of Instance-aware Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' We visual-  ize the instance-aware sampling on the perspective view and bird’s  eye view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Ground truth bounding boxes are marked in green color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Examples are selected from the nuScenes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  responding to the recall improvement), and the features are  more instance discriminative in the dense areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Visualization of the instance-aware BEV feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  From left to right, we compare the feature heatmaps output by  frustum encoder (w/o AIR), with AIR, and ground truth boxes in  green shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The colors for the feature heatmap corre-  spond to the norm value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' The BEV feature learned by our frustum  encoder is instance-aware and has strong relations to the actual po-  sitions in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' By using AIR, more instance regions would  be discovered, and the features are more instance discriminative in  the dense areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Examples are selected from the prediction on the  nuScenes val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' Conclusion  In this paper, we propose FrustumFormer, a novel  framework for multi-view 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  The  core insight of FrustumFormer is to transform adaptively  according to the view contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' To this end, we designed  adaptive instance-aware resampling, paying more attention  to the instance regions during feature view transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  By utilizing adaptive instance-aware resampling in the  frustum encoder and temporal frustum fusion module, the  model can better locate the instance regions while learning  the instance-aware BEV features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Experimental results  on the nuScenes dataset demonstrate the effectiveness  of our method for multi-view 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  Our  method significantly improved on mAP over the previous  methods due to the focus on the instance regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  We  hope our framework can serve as a new baseline for  8  XI厂l:  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content=' :80  口:following 3D perception work, and let more people pay  attention to the view content during feature transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  References  [1] Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora,  Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Gi-  ancarlo Baldan, and Oscar Beijbom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
+page_content='  nuscenes: A multi-  modal dataset for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE3T4oBgHgl3EQfWQqL/content/2301.04467v1.pdf'}
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+ITA-ELECTION-2022: A multi-platform dataset of social media conversations
+around the 2022 Italian general election
+Francesco Pierri, Geng Liu, Stefano Ceri
+Dipartimento di Elettronica, Informazione e Bioingegneria,
+Politecnico di Milano, Milano, Italy
+E-mail: {name.surname}@polimi.it
+Abstract
+Online social media play a major role in shaping public dis-
+course and opinion, especially during political events. We
+present the first public multi-platform dataset of Italian-
+language political conversations, focused on the 2022 Italian
+general election taking place on September 25th. Leveraging
+public APIs and a keyword-based search, we collected mil-
+lions of posts published by users, pages and groups on Face-
+book, Instagram and Twitter, along with metadata of TikTok
+and YouTube videos shared on these platforms, over a period
+of four months. We augmented the dataset with a collection
+of political ads sponsored on Meta platforms, and a list of so-
+cial media handles associated with political representatives.
+Our data resource will allow researchers and academics to
+further our understanding of the role of social media in the
+democratic process.
+Introduction
+Online social media provide researchers and academics with
+unprecedented opportunities to observe a wide range of po-
+litical and societal phenomena (Rossi, Righetti, and Marino
+2021). They also play a critical role in shaping public opin-
+ion during political events (Vitak et al. 2011), and represent
+a rich source of data to study the interplay between politi-
+cal actors’ campaigns (Sahly, Shao, and Kwon 2019), media
+outlets’ agenda settings (Kim et al. 2016), and users’ news
+consumption (Allcott and Gentzkow 2017).
+In Italy, as of 20221, YouTube is the platform used by the
+largest amount of internet users (88%), followed by Meta
+platforms (64%) and TikTok (54%), whereas Twitter only
+accounts for approximately 7%2. However, previous studies
+of online social media during Italian elections and referen-
+dum mostly focused on Twitter (Rossi, Righetti, and Marino
+2021), due to the large availability of data via its APIs. In
+this work, instead, we present a public data resource of polit-
+ical conversations and user-generated content shared around
+the 2022 Italian general election, which allows researchers
+and academics to study multiple social platforms simultane-
+ously.
+Copyright © 2022, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+1www.statista.com/statistics/1311549/top-social-platforms-
+italy/
+2datareportal.com/reports/digital-2022-italy
+The 2022 Italian general election was the first ever to take
+place in autumn, as a consequence of the fall of the govern-
+ment of national unity led by Mario Draghi in July3. The
+election had a record-low voter turnout and it was won by
+the right-wing coalition of Giorgia Meloni with over 43% of
+the vote share. Among the opponents, the Centre-left coali-
+tion led by Enrico Letta obtained approximately 25% of
+the voters, the populist Movimento 5 Stelle led by former
+PM Giuseppe Conte reached less than 16%, and the liberal
+and centrist Third Pole, which included former PM Matteo
+Renzi, obtained almost 8% of the vote share.
+We present ITA-ELECTION-2022, the first public
+multi-platform dataset of Italian-language political conver-
+sations taking place on online social media, with a focus
+on the 2022 Italian general election. We collected millions
+of social media posts from Facebook, Instagram and Twit-
+ter, as well as advertisements sponsored on Meta platforms
+and metadata for TikTok and YouTube videos shared on the
+aforementioned platforms. We finally augment the dataset
+with a collection of social media handles associated with
+Italian political representatives. To collect the data, we em-
+ployed a snowball sampling procedure and curated a list
+of relevant terms to accordingly perform a keyword-based
+search during a period of four months (July 2022 - October
+2022). We provide public access to the data via GitHub and
+DataVerse repositories, as detailed next.
+The outline of this paper is the following: in the next sec-
+tion we review existing public data resources related to the
+present work; then, we describe the data collection proce-
+dure(s) carried out to build the dataset; next, we describe a
+few potential applications of the collected data; finally, we
+discuss limitations, draw conclusions and provide some eth-
+ical remarks.
+Related Work
+There are several public datasets that allow to study social
+media conversations around political issues. We focus our
+literature review on the Italian context, and then describe a
+few datasets related to other countries. We also refer the in-
+terested reader to (Rossi, Righetti, and Marino 2021) for an
+overview of studies that describe the interplay between so-
+cial media and Italian politics.
+3en.wikipedia.org/wiki/2022 Italian general election
+arXiv:2301.05119v1  [cs.SI]  12 Jan 2023
+
+(Basile, Lai, and Sanguinetti 2018) collect tweets in the
+Italian language continuously from 2012 to 2018, extracting
+a number of smaller datasets enriched with different kinds
+of annotations for linguistic purposes. They provide access
+to tweet IDs and annotations in a public repository.
+(Pierri, Artoni, and Ceri 2020) analyze the prevalence
+of Italian disinformation spreading on Twitter in the five
+months preceding the 2019 European Parliament election.
+They collect over 300 k tweets sharing thousands of news
+articles originating from websites flagged as unreliable by
+journalists and fact-checkers, providing public access to
+tweet IDs and lists of websites. The same authors provide
+a similar dataset collected in a different period of 2019, and
+that contains tweets sharing links to mainstream and tradi-
+tional news websites, both in the Italian and French language
+(Pierri 2020).
+(Di Giovanni and Brambilla 2021) study the polarization
+around the 2020 Italian constitutional referendum. They col-
+lect a dataset of 1.2 M tweets discussing the event – and
+provide access to their IDs –, with the goal of designing
+a hashtag-based semi-automatic approach to label Twitter
+users’ stance towards the referendum.
+Following the COVID-19 pandemic, several researchers
+collected social media data to study conversations around
+the crisis, with a particular focus on the impact of vaccine
+misinformation. (Crupi et al. 2022) study the evolution of
+Italian Twitter conversations around vaccines during the pe-
+riod 2019-2021, whereas (Di Giovanni et al. 2022) collect
+tweets in multiple languages (French, German and Italian)
+during the first year of world vaccination programs. Both
+contributions give public access to tweet IDs, with the lat-
+ter providing also a set of labeled pro/anti-vaccines tweets
+and hashtags that can be used for training machine learning
+classifiers.
+(Calisir and Brambilla 2020) provide a dataset of tweets
+discussing Brexit for a period of 45 months, from January
+2016 until September 2019. The data, which comprises 50.8
+million tweets and 3.97 million users, is enriched with meta-
+data such as the bot score of users, sentiment score of tweets,
+and political stance labels predicted by a classifier developed
+by the authors.
+There is a large number of datasets that focus on the U.S.
+elections (both presidential and midterms), and we provide
+here a non-exhaustive list of available resources. (Hanna
+et al. 2011) mapped candidates from the 2010 U.S. Midterm
+election with their Twitter accounts and a random sample
+of their followers. (Bovet and Makse 2019) collected over
+171 M tweets in the English language, mentioning Donald
+Trump and Hillary Clinton during the 2016 U.S. Presiden-
+tial election. (Deb et al. 2019) and (Yang, Hui, and Menczer
+2022) collected tweets discussing the 2018 U.S. Midterm
+election, both using a hashtag-based search (e.g. tweets shar-
+ing the hashtag ”#ivoted” on election day) and querying
+Twitter APIs with general keywords related to the midterm
+election. (Chen, Deb, and Ferrara 2022) provide a longitudi-
+nal dataset of over 1.2 billion U.S. politics- and election-
+related tweets shared around the period of the 2020 U.S.
+Presidential election. Related to the same election, (Abilov
+et al. 2021) released a multi-modal dataset of 7.6 M tweets
+Figure 1: Example of an ad run on Meta platforms along
+with the information provided by Meta Ad Library API.
+and 25.6 M retweets from 2.6 M users related to voter fraud
+claims. They augmented the data with cluster labels, users’
+suspension status, and perceptual hashes of tweeted images
+as well as aggregate data from external links and YouTube
+videos shared on Twitter.
+Data Collection
+This section describes the data collection procedure(s) car-
+ried out to gather data from different social media platforms.
+We remark that we employed the same list of keywords
+related to the Italian election, which we obtained through
+a snowball sampling approach using Twitter data only, to
+query different APIs. Our dataset conforms with FAIR prin-
+ciples: it is Findable, Accessible and Reusable as it is pub-
+licly accessible in an online Github4 and DataVerse reposi-
+tory5, where we provide the means to recreate it almost com-
+pletely (see limitations discussed next). It is also Interopera-
+ble as the data files are released in “.csv” and “.txt” formats.
+We summarize some statistics of the dataset in Table 1.
+Twitter
+We collected all tweets in the Italian language related to the
+election by using tweepy Python library to query Twitter
+v1.1 Filter streaming API endpoint6 in the period September
+4github.com/frapierri/ita-election-2022
+5doi.org/10.7910/DVN/EALXH2
+6developer.twitter.com/en/docs/twitter-api/v1/tweets/filter-
+realtime/overview
+
+Inactive
+8Sep2022-23Sep2022
+Platforms 
+Categories
+:Estimatedaudiencesize:100K-500kpeople
+自Amount spent (EUR):<E100
+@Impressions:4K-5k
+ID:376439398025966
+See ad details
+Sebastiano Valenti
+Sponsored·PaidforbySebastianoMarioValenti
+SonoufficialmentecandidatoalleelezioniregionalicolMoVimento5
+Stelle ☆★
+LaureatoinInformaticaehosemprelavoratoinquestoambito
+Nonfacciopartedinessunacordatapolitica,
+hosempremessoimpegnonelportareavantiproposteeleggiche
+migliorinolavitadellapersone,checomeme,tuttiigiornisialzano..
+COMESIVOTA
+0050
+NELLASCHEDA VERDE
+SICILIA
+EGNAAL SIMBOLO
+SCRIVI WALENTI
+SEGNADIPAOL
+2022
+DIPAOLA
+VALENTI
+NUCCIO
+SEBASTIANG
+VALENTI
+DIPAOLA
+Sebastiano Valenti
+Send Messa...
+Personal blogFigure 2: Daily number of social media posts and ads col-
+lected in our dataset, for different platforms. Solid lines
+show 3-day moving averages.
+2nd, 2022 - October 20th, 2022. We also leveraged Twit-
+ter’s historical Search API v2 endpoint7 to collect tweets
+retrospectively in the period July 1st, 2022 - September 2nd,
+2022. To query Twitter’s APIs we employed a snowball sam-
+pling approach, following existing work (Di Giovanni et al.
+2022; DeVerna et al. 2021), and generated a list of relevant
+keywords starting with seed terms such as “elezioni2022”
+and “elezioni”8; the final list contains 62 keywords and it
+is available in the repository associated with this paper. A
+sample is provided in Table 2. The total collection of tweets
+contains 19,087,594 tweets shared by 618,089 unique users.
+We remark that to abide by Twitter’s terms of service we
+only share tweet IDs publicly. These can be “re-hydrated”
+to retrieve tweet objects, with the exception of removed or
+protected tweets, by querying Twitter API directly or using
+tools like Hydrator9 or twarc.10
+Facebook and Instagram posts
+We collected Facebook and Instagram data by employ-
+ing CrowdTangle, a public tool owned and operated by
+Meta (CrowdTangle Team 2022) that allows retrieving posts
+7developer.twitter.com/en/docs/twitter-
+api/tweets/search/introduction
+8In the Italian language “elezioni” means elections.
+9github.com/DocNow/hydrator
+10github.com/DocNow/twarc
+Twitter
+19,087,594 tweets
+618,089 unique accounts
+Facebook
+1,142,812 posts
+445,461 unique accounts
+Instagram
+68,078 posts
+5,274 unique accounts
+Meta
+29,211 ads
+3,750 unique sponsors
+YouTube
+22,754 unique videos (Twitter)
+17,401 unique videos (FB)
+TikTok
+1,903 unique videos (Twitter)
+1,744 unique videos (FB)
+Table 1: Statistics of the dataset.
+elezioni
+partito democratico
+berlusconi
+renzi
+movimento 5 stelle
+salvini
+calenda
+di maio
+politiche2022
+meloni
+elezioni2022
+conte
+Table 2: A sample of Italian language keywords related to
+the 2022 election that were used to retrieve social media
+posts in our dataset.
+shared by public pages and groups with a certain amount of
+followers or that were manually added by other researchers
+on the platform.11 We queried the /posts/search end-
+point12 using the same list of keywords employed for col-
+lecting Twitter data. For each post, the API returns several
+attributes related to the post and the account (page or group)
+that shared it; the full list of attributes is available in the
+official documentation13. We retained only posts in the Ital-
+ian language by filtering on the languageCode param-
+eter: the final dataset contains 1,142,812 Facebook posts,
+shared by 445,461 unique pages and groups and generating
+over 233 M interactions (shares, comments, reactions), and
+68,078 Instagram posts, shared by 5,274 unique pages and
+generating over 97 M interactions (likes and comments). We
+provide access to the URLs and IDs14 of these posts, which
+can be used to access and retrieve those that are not removed
+or deleted, in the repository associated with this paper.
+TikTok and YouTube videos
+We augmented our dataset of social media posts by extract-
+ing metadata for TikTok and YouTube videos shared in Face-
+book15 and Twitter messages present in our dataset. For what
+concerns YouTube, we identified all external links to the
+11More
+details
+are
+available
+in
+the
+official
+documenta-
+tion:
+help.crowdtangle.com/en/articles/1140930-what-data-is-
+crowdtangle-tracking
+12github.com/CrowdTangle/API/wiki/Search
+13github.com/CrowdTangle/API/wiki
+14For each post we provide both platform and Crowdtangle ID
+that can be given as input to the GET Post ID endpoint accessi-
+ble here: github.com/CrowdTangle/API/wiki/Posts#get-postid
+15There were no links shared on Instagram.
+
+Election day
+600000
+eets
+400000
+TWe
+200000
+80000
+osts
+od
+60000
+B
+40000
+1500
+posts
+1000
+500
+1000
+ads
+750
+Meta
+500
+250
+130
+Jul 25
+Aug 19
+Sep 13
+un
+0ct 08platform and employed the official YouTube API16 to ex-
+tract video information such as the author, channel id, video
+title, description, Top 10 popular comments, etc. The result-
+ing collection yields metadata for 22,754 unique YouTube
+videos shared on Twitter and 17,401 unique YouTube videos
+shared on Facebook. For what concerns TikTok, given the
+lack of an official API, we employed pyktok Python li-
+brary17 to collect metadata about TikTok videos such as the
+title, description, length as well as information about the au-
+thor of the video. The resulting collection yields metadata
+for 1,903 unique TikTok videos shared on Twitter and 1,744
+unique TikTok videos shared on Facebook.
+Facebook and Instagram ads
+We leveraged Meta Ad Library API18 to collect all ads about
+“social issues, elections or politics” that were active on Meta
+platforms19 in the period July 1st, 2022 - October 20th,
+2022. We provide an example of a sponsored ad in Fig-
+ure 1. We queried the API with the same set of keywords
+mentioned beforehand; the API allows to search ads using
+one keyword at a time, and we queried the endpoint multi-
+ple times eventually discarding duplicated ads. The resulting
+collection contains 29,211 unique ads paid by 3,750 unique
+sponsors. For each ad, the API provides several different at-
+tributes: date of creation, period when the ad is active, name
+of the sponsor, message, platform on which the ad is active,
+lower and upper bound for the amount spent and the number
+of impressions generated, etc. In the repository associated
+with this dataset we provide access to the ID of ads, which
+can be then used to retrieve ads through Meta Ad Library
+interactive search console or API. In particular, to abide by
+Meta’s terms of use, an identification procedure is required
+to access the API endpoint, whereas the interactive search
+console only requires a Meta account to access it.
+Social media handles of political representatives
+We compiled a list of Facebook, Instagram and Twitter
+handles of elected members in the Senate and Chamber
+of deputies based on the official list released by the Ital-
+ian Ministry of Interior20. Specifically, for each represen-
+tative, we manually checked whether their official account
+was present on the three platforms. Insofar, we were able to
+match around 500 Twitter accounts, and approximately 100-
+150 Facebook and Instagram accounts. The full list is avail-
+able in the repository associated with this paper. We refer the
+interested reader to a similar useful resource presented by
+(Haman and ˇSkoln´ık 2021), who provide an online running
+database of politicians’ activity on social media (currently
+only Twitter is supported) spanning multiple countries.
+16developers.google.com/youtube/v3
+17github.com/dfreelon/pyktok
+18www.facebook.com/ads/library/api
+19These are: Facebook, Instagram, Messenger, and the Audience
+Network. Notice that only a dozen ads were placed on platforms
+other than Facebook and Instagram.
+20github.com/ondata/elezioni-politiche-2022
+Figure 3: Distributions of the number of tweets/posts and
+retweets/interactions for each Twitter, Facebook and Insta-
+gram account. Dashed lines indicate median values: 2 tweets
+and 3 retweets for Twitter accounts; 2 posts and 8 interac-
+tions for Facebook accounts; 1 post and 174 interactions for
+Instagram accounts.
+Data Characterization
+In this section, we provide a few basic descriptive statistics
+of the data presented in this work and leave more detailed
+analyses for future research.
+In Figure 2, we show the daily number of social media
+posts and ads collected in our dataset, for each platform. We
+can observe a significant increasing trend (Mann-Kendall
+P < 0.001) toward election day in all cases, with a sharp
+drop in the weeks afterward. We also notice that Twitter ac-
+tivity in our dataset is much more represented than other
+platforms, followed by Facebook and Instagram.
+In Figure 3, we show the distribution of account-wise met-
+rics for Twitter, Facebook and Instagram. Specifically, we
+show the Cumulative Distribution Function (CDF) for the
+number of tweets/posts created and retweets/interactions re-
+ceived by accounts on each platform. All distributions show
+an exponential-like behavior, with most of the accounts be-
+ing very rarely active and receiving little engagement, and
+only a minority of them exhibiting a large number of posts
+created and engagement received. Median values are shown
+by dashed lines and are available in the caption of the figure.
+In Figure 4, we show distributions of metrics for Meta ads.
+Specifically, we show the CDF of the mean amount spent
+and the mean number of impressions generated at both the
+
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+-
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+-
+0.8
+0.8
+tion
+-
+20.6
+0.6
+or
+0
+0.2
+0.2
+-
+-
+0.0
+0.0
+101
+100
+102
+103
+101
+103
+105
+Number of posts
+Number of interactions
+per Instagram account
+per Instagram accountFigure 4: Distributions of the mean amount spent and the
+mean number of impressions generated at the ad and spon-
+sor level for Meta ads. Dashed lines indicate median values:
+148.5 EUR spent for ads and by sponsors; 7498.5 impres-
+sions generated by ads and 12498.5 impressions by spon-
+sors.
+ad and sponsor level; the mean value is computed by taking
+the average between the lower and upper bound estimates of
+the amount/impressions provided by Meta Ad Library API
+for each ad. Median values are shown by dashed lines and
+are available in the caption of the figure. We refer the reader
+to (Pierri 2022) for a more detailed analysis of political ad-
+vertising on Meta platforms during the 2022 Italian general
+election.
+In Figure 5, we show the top 10 accounts on Twitter, Face-
+book and Instagram ranked by the number of tweets/posts
+created and the number of retweets/interactions received.
+We can observe notable politicians from the entire politi-
+cal spectrum as well as journalists and news outlets, but also
+supporting pages and groups. We also notice that some of the
+most active accounts do not appear among the most engaged
+ones. Finally, we can see that Italian PM Giorgia Meloni is
+the most engaged account on the three platforms.
+Potential Applications
+There are several potential applications for our dataset,
+which can consider a single platform or multiple ones at the
+same time.
+Interested researchers could further the current under-
+standing of polarization processes taking place during elec-
+tion seasons by analyzing content shared on multiple so-
+cial platforms at once. They could study whether “echo-
+chamber” effects take place on different platforms, high-
+lighting similarities and differences in their formation pro-
+cess.
+Other researchers might leverage the data in order to study
+how political candidates interacted with potential voters on
+social media platforms, thus analyzing in detail the politi-
+cal communication strategies put in place by different can-
+Figure 5: Top 10 accounts ranked by the number of
+tweets/posts created and retweets/interactions received on
+Twitter, Facebook and Instagram. Due to space limitations,
+some account names are truncated.
+didates. They could also investigate the presence of correla-
+tional effects between online signals and electoral outcomes,
+or detect the presence of toxic and hateful speech originating
+in communities of political supporters.
+Some researchers could investigate the presence of
+mis/disinformation and astroturfing campaigns taking place
+in the run-up to the election, studying patterns of similari-
+ties and differences among different platforms. They could
+also analyze how fringe and harmful content spreads across
+communities present on different platforms, and whether in-
+fluential accounts play a role in amplifying certain malicious
+narratives.
+Discussion
+We released ITA-ELECTION-2022, a large-scale dataset
+of social media posts in the Italian language discussing the
+2022 Italian General election, which took place on 25th
+September 2022, spanning multiple online platforms and
+covering a period of four months. In addition to gathering
+posts shared on Twitter, Facebook and Instagram, we col-
+lected ads sponsored on Meta platforms, we extracted meta-
+data for YouTube and TikTok videos shared on different
+platforms during the collection period, and we compiled a
+
+1.0
+1.0
+0.8
+0.8
+0.6
+0.6
+0.2
+0.2
+-
+0.0
+0.0
+103
+104
+104
+105
+106
+Mean amount (EUR) per ad
+Mean impressions per ad
+1.0
+1.0
+0.8
+0.8
+uoI
+uoI
+0.6
+0.6
+porti
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+103
+104
+105
+106
+Mean amount (EUR) per sponsor
+Mean impressions per sponsorTwitter
+Twitter
+infoitinterno
+GiorgiaMeloni
+danieledv79
+CarloCalenda
+Infinitolsacco
+GiuseppeContelT
+Divorex8
+ultimora_pol
+CenturrinoLuigi
+Mov5Stelle
+OM_VA_SH
+Fratellidltalia
+Giancar70336148
+matteosalvinimi
+Hattoriando
+ilruttosovrano
+naladrof53
+jacopo_iacoboni
+GianmarioAngius
+matteorenzi
+0.0
+0.5
+1.5
+1.0
+0.00
+0.25
+0.50
+0.75
+1.00
+Number of tweets
+1e4
+Number of retweets
+ 1e5
+Facebook
+Facebook
+CONTE E CUORE I..
+Giuseppe Conte
+Amore per Cont...
+Giorgia Meloni
+Noi con Salvini.
+II Fatto Quotid.
+NOI E MATTEO RE..
+Matteo Salvini
+Lega - Salvini
+..·
+Andrea Scanzi
+Marco Travaglio...
+la Repubblica
+Raccolta firme
+Fanpage.it
+Gianluigi Parag...
+Ultime Notizie
+.
+II Fatto Quotid...
+W IL M5S
+Conte President...
+Tu e I informaz...
+0
+2
+4
+6
+0
+4
+6
+8
+Number of posts
+1e3
+Number of interactions le6
+Instagram
+Instagram
+Affaritaliani.i...
+Giorgia Meloni
+Antonella Faggi...
+Matteo Salvini
+Roberta Ferrero...
+Sveglia Italia...
+AQTR
+Erica Rivolta
+MoVimento 5 Ste...
+Fanpage.it
+CRONACHE DI SPO...
+Carlo Calenda
+Italia Viva
+CALCIATORIBRUTT...
+AQTR
+Cronache di bas...
+Il Giornale
+Fratelli d'ltal...
+Tg2 Rai
+la Repubblica
+0
+2
+4
+6
+8
+0
+3
+1
+Number of posts
+1e2
+Number of interactions le6list of social media handles associated to political represen-
+tatives that can be used to gather further data. We described
+in detail the collection procedures carried out to build the
+dataset, and provided a few basic statistics about the col-
+lected data. We also suggested promising directions for fu-
+ture research.
+Our work is not without limitations. First, our keyword-
+based search might entail results that are not completely
+accurate, e.g., one of the terms employed for the query is
+“conte”, which might refer both to former PM Giuseppe
+Conte and football manager Antonio Conte. From another
+perspective, election-related terms might have been em-
+ployed for marketing campaigns and promoting content that
+is not pertinent to the election. However, while we are un-
+able to address these issues, which would require non-trivial
+efforts, researchers can further refine our data collection
+to meet their needs. Moreover, we performed a backward
+search to retrieve Twitter, Facebook and Instagram posts
+shared from July to September 2022, and we missed those
+that were deleted or removed during the same period. Simi-
+larly, by providing access only to IDs and URLs of collected
+posts, posts that have been removed or made private by users
+cannot be retrieved, thus limiting reproducibility analyses.
+Furthermore, we did not filter out the activity of automated
+and inauthentic accounts that might have polluted organic
+conversations around the election. Another limitation con-
+cerns Meta which, as highlighted in (Le Pochat et al. 2022),
+might not accurately label all political ads as such and our
+collection might be missing some data. Finally, the user base
+of different platforms analyzed in this work might not be
+fully representative of the actual Italian population, and this
+should be taken into consideration by future research.
+Despite these limitations, we believe that our dataset pro-
+vides fertile ground for a number of intriguing and interest-
+ing research applications, and we hope that this resource can
+advance our understanding of the interplay between online
+social media and democratic processes.
+Ethical considerations
+We performed our data collection and public release in com-
+plete agreement with the platforms’ terms of service. We ac-
+knowledge that TikTok metadata was scraped from the plat-
+form, thus potentially violating the platform’s terms of ser-
+vice, but this was due to the lack of an official API(Freelon
+2018). We do not directly share the content of social media
+posts, but rather provide access to IDs and URLs that can
+be used to retrieve the original data, with the exception of
+posts that have been deleted by platforms, and removed or
+made private by their author. We did not cause any harm nor
+expose information about individual users in the process of
+collecting and releasing the data, with the only exception of
+political representatives and a handful of popular accounts
+shown in the descriptive statistics. We understand that dis-
+closing their social media accounts might open up to poten-
+tial abuse by malicious actors, but at the same time, it en-
+ables researchers, journalists and other stakeholders to put
+important public actors, such as the members of the Italian
+Parliament and Senate, to scrutiny in order to better under-
+stand the influence of social media platforms on the demo-
+cratic process.
+Acknowledgments
+We are thankful to M.Sc. students Valeria Pant´e and Ilaria
+Saini for helping match social media accounts to politi-
+cal representatives. Work supported in part by PRIN grant
+HOPE (FP6, Italian Ministry of Education).
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+
diff --git a/IdA0T4oBgHgl3EQfCP85/content/tmp_files/2301.01986v1.pdf.txt b/IdA0T4oBgHgl3EQfCP85/content/tmp_files/2301.01986v1.pdf.txt
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+ 
+A mempolar transistor made from tellurium 
+Yifei Yang†, Lujie Xu†, Mingkun Xu†, Huan Liu, Dameng Liu, Wenrui Duan, Jing Pei, 
+Huanglong Li* 
+ 
+Y. Yang, M. Xu, J. Pei, H. Li 
+Department of Precision Instrument, Center for Brain Inspired Computing Research, 
+Tsinghua University, Beijing, 100084, China. 
+E-mail: li_huanglong@mail.tsinghua.edu.cn   
+ 
+L. Xu, W. Duan 
+School of Instrument Science and Opto Electronics Engineering, Beijing Information 
+Science and Technology University, Beijing, 100192, China. 
+ 
+L. Xu 
+Application Technology Department, Dongfang Jingyuan Electron Limited, Beijing, 
+100176, China. 
+ 
+H. Liu, D. Liu 
+State Key Laboratory of Tribology, Tsinghua University, Beijing, 100084, China. 
+            
+H. Li 
+Chinese Institute for Brain Research; Beijing, 102206, China. 
+ 
+Keywords: transistors, memristors, tellurium, reconfigurable polarity, mempolar, 
+ternary content-addressable memories, regularization methods 
+ 
+Abstract  
+The classic three-terminal electronic transistors and the emerging two-terminal ion-
+based memristors are complementary to each other in various nonconventional 
+information processing systems in a heterogeneous integration approach, such as 
+DRAM/storage-class memory hierarchy, hybrid CMOS/memristive neuromorphic 
+crossbar arrays, and so on. Recent attempts to introduce transitive functions into 
+memristors have given rise to gate-tunable memristive functions, hetero-plasticity and 
+mixed-plasticity functions. However, it remains elusive under what application 
+scenarios and in what ways transistors can benefit from the incorporation of ion-based 
+memristive effects. Here, we introduce a new type of transistor named ‘mempolar 
+transistor’ to the transistor family that has included the well-known unipolar and 
+ambipolar transistors. As its name suggests, mempolar transistor has polarity with 
+memory, reminiscent of memristor having resistance with memory. Specifically, its 
+polarity can be converted reversibly, in a nonvolatile fashion, between n-type and p-
+type depending on the history of the applied electrical stimulus. This is achieved by 
+the use of the emerging semiconducting tellurium as the electrochemically active 
+source/drain contact material, in combination with monolayer two-dimensional MoS2 
+channel, which results in a gated lateral Te/MoS2/Te memristor, or from a different 
+perspective, a transistor whose channel can be converted reversibly between n-type 
+MoS2 and p-type Te. With this unique mempolar function, our transistor holds the 
+promise for reconfigurable logic circuits and secure circuits, addressing a fundamental 
+limitation in previous implementations, that is, polarity reconfiguration was volatile 
+
+ 
+and required additional gate terminals. In addition to this manifest advantage, we 
+propose and demonstrate experimentally, a ternary content-addressable memory made 
+of only two mempolar transistors, which used to require a dozen normal transistors, 
+and by simulations, a device-inspired and hardware matched regularization method 
+‘FlipWeight’ for training artificial neural networks, which can achieve comparable 
+performance to that achieved by the prevalent ‘Dropout’ and ‘DropConnect’ methods. 
+This work represents a major advance in diversifying the functionality of transistors.  
+ 
+1. Introduction 
+Polarity is one of the most fundamental aspects of a transistor’s identity, according to 
+which transistors are most famously categorized into either n-type or p-type. The 
+pairing of n-type and p-type transistors has resulted in complementary metal-oxide-
+semiconductor (CMOS) technology, the leading technology for digital integrated 
+circuits over the past four decades. The polarity of a transistor is a reflection of the 
+comprehensive effect of its entire materials system, including the channel 
+semiconductor, impurities, source/drain contact, and so on. Typically, polarity identity 
+is determined at the fabrication stage and cannot be altered afterwards. 
+ 
+Traditionally, the performance improvements of the integrated circuits have depended 
+almost solely on the miniaturization of transistors (Moore’s Law). As miniaturization 
+is approaching its physical limits, transistor scaling is delivering performance 
+improvements at a slower pace. However, software is evolving extremely rapidly with 
+emerging applications1,2, such as artificial intelligence. Given this, the hardware that 
+cannot adapt to software will suffer from a short lifecycle and high nonrecurring 
+engineering cost3. In addition, with the ever increasing density of transistors in 
+microprocessors, power dissipation has become a huge factor challenging the 
+continuous success of the CMOS technology. Overall, both flexibility and energy 
+efficiency have become main criteria for computing fabrics. In this context, 
+incorporation into devices of functionalities that do not necessarily scale according to 
+Moore’s Law but provide additional value in different ways has become increasingly 
+pursued by semiconductor industry and academia4. 
+ 
+Different from transistor in many key aspects, memristor, experimentally discovered 
+less than two decades ago5, is one such emerging device that holds great promise for 
+low-power and adaptive electronics. Memristor is best-known as a two-terminal 
+resistor with long-term memory. In contrast to the transient electronic switching 
+(volatile) during the operation of transistor, enduring atomic structure change in the 
+switching medium can be elicited when suitable electrical stimulus is applied to 
+memristor, giving rise to nonvolatile reconfigurability of the resistance state. This 
+unique property of memristor has made it a key complementary device to transistor in 
+forming DRAM/storage-class memory hierarchy in conventional von Neumann 
+architecture and in enabling the emerging computing paradigms, such as in-memory 
+computing and neuromorphic computing6-8, where transistors and memristors are 
+integrated heterogeneously. 
+ 
+As one step further, attempts have been made to introduce transitive functions into 
+memristors. In such devices, additional control gates are positioned aside and 
+electrically insulated from the memristive channels9. These have given rise to gate-
+tunable memristive functions10-12, enabling the emulation of hetero-plasticity13-15. 
+
+ 
+Apparently, memristors and memristor-centered circuits benefit from the additional 
+transistive functions to further enhance their flexibility and augment their 
+functionalities16,17. 
+ 
+At this point, an interesting question arises as to under what application scenarios and 
+in what ways transistors can benefit from the incorporation of ion-based memristive 
+effects. Here, we propose and demonstrate that the fusion of transistive and 
+memristive functions can result in a novel type of transistor which we name 
+‘mempolar transistor’. As its name suggests, mempolar transistor has polarity with 
+memory, reminiscent of memristor. Specifically, its polarity can be converted 
+reversibly, in a nonvolatile fashion, between n-type and p-type depending on the 
+history of the applied electrical stimulus. Mempolar transistors bring major 
+improvements over the existing reconfigurable transistors18-21 in the following key 
+aspects: first, mempolar transistors are more energy-efficient because polarity 
+conversion takes place in a nonvolatile fashion, whereas in previous devices control-
+gate voltages have to be applied persistently; second, mempolar transistors have 
+smaller footprints and less fabrication complexity because they retain the classic 
+three-terminal structures without the addition of more control gates.  
+ 
+To enable mempolarity, a key design idea is to first make a two-terminal memristor 
+whose two resistance states are of n-type and p-type semiconductivity, respectively, 
+then introduce a third terminal to provide gate control over the memristive channel in 
+either polarity, as in a normal transistor. To this end, we choose monolayer two-
+dimensional (2D) MoS2 as the pristine n-type memristive medium with lateral p-type 
+tellurium (Te) electrodes, making a two-terminal lateral Te/MoS2/Te device. The 
+memristive function of this device can be foreseen (demonstrations provided) based 
+on the previous demonstrations of the electrochemically activity of Te and other 
+memristors made from it22-25. With a back gate, a mempolar transistor that can be 
+reconfigured between an n-type MoS2 transistor and a p-type Te transistor is created. 
+In addition to the aforementioned apparent advantages over the existing 
+reconfigurable transistors in logic and secure circuits, our mempolar transistor also 
+demonstrate additional value in other applications, such as ternary content-
+addressable memory (TCAM) cell made of two mempolar transistors, which used to 
+require a dozen normal transistors. Inspired by the device properties, a hardware-
+matched regularization method for mitigating the over-fitting problems in artificial 
+neural networks is also developed, which can achieve comparable performance to that 
+achieved by the prevalent ‘Dropout’ and ‘DropConnect’ methods. Our proposed 
+mempolar transistor is a valuable addition to the transistor family, enabling 
+nonvolatile fine-grain reconfigurability and supporting general-purpose hardware 
+design. 
+2. Results and discussion 
+2.1. Mempolar function and its mechanism 
+The schematic structure and the optical image of the mempolar transistor are shown in 
+figure 1a and 1b, respectively. To fabricate the device, a monolayer MoS2 flake (the 
+monolayer characteristics verified by Raman spectroscopy are shown in 
+supplementary figure S1) is mechanically exfoliated onto a heavily p-type doped Si 
+substrate with 300-nm-thick thermally oxidized SiO2. The two Te electrodes are then 
+deposited by magnetron sputtering as the source and drain terminals, followed by the 
+deposition of platinum protective layers (see Methods).  
+
+ 
+The n-type transfer characteristic of the as-fabricated device is depicted in figure 1c. 
+Under a constant drain-source voltage (Vds=Vd-Vs) of +1 V, the drain current (Id) is 
+increased by about 200 times as Vg is swept from -10 V to +10 V. Certain degree of 
+clockwise hysteresis can also be observed as Vg is swept back and forth, indicating 
+that trapping/detrapping of electrons in/from the gate oxide (SiO2) takes place during 
+the course of Vg sweeping. To elicit polarity conversion, a sufficiently large Vds pulse 
+of +30 V in amplitude and 30 s in duration is applied to the device (under Vg=0 V). 
+As aforementioned, the Te electrode under negative bias (source electrode) with 
+respect to the other can be electrochemically reduced. The induced Te2- anions may 
+migrate towards the counter electrode where they will be re-oxidized to elemental Te. 
+As this electrochemical process proceeds, this accumulated elemental Te may 
+eventually bridge the source and drain electrodes by forming local Te filament or 
+wider Te sheet. Because Te is a narrow-bandgap semiconductor with native p-type 
+conductivity, we have previously exploited this electrochemical mechanism to enable 
+filamentary resistance switching in vertical memristors where wider-bandgap (more 
+insulating) dielectrics are sandwiched between the Te electrodes23. Here, the 
+formation of Te sheet of the width as close to that of the channel as possible is 
+preferred in order to best maintain the channel geometry. Figure 1d shows the transfer 
+curve of the device after large Vds is applied. It is seen that Id is now decreased with 
+increasing Vg, verifying that transistor polarity is conversed to p-type. With the 
+conversion of polarity, the hysteresis in the transfer curve also changes direction from 
+clockwise to anti-clockwise. The p-type transfer characteristics can still be reproduced 
+after one month without degradation, confirming the nonvolatile nature of the polarity 
+conversion. The p polarity can be reversibly switched back to n polarity by applying a 
+Vds pulse of -30 V under Vg=+5 V. Gate biasing here reduces the number of free hole 
+carriers in the p-type Te channel and thus mitigates channel field screening for 
+enabling electrochemical reaction and ion drift. The reversible polarity change is 
+robustly reproduced under 100 times of polarity switching operations, as shown in 
+supplementary figure S2. Supplementary figure S3 also shows the evolutions of Id 
+during the periods of polarity-switching Vds pulses. Gradual increase (decrease) in Id 
+with time is observed in n-to-p (p-to-n) switching, which is consistent with the higher 
+conductivity of p-type Te than that of n-type MoS2.  
+In contrast to other commonly used source/drain electrode materials, Te is known to 
+be electrochemically active. In addition, Te anions are also shown to be mobile in 
+various solids. These two materials properties are key in enabling Te filament-based 
+resistance switching, as previously reported22,23. To explore the mechanisms behind 
+the unique mempolar phenomenon, we fabricate two control devices with Ti and Pt 
+source/drain electrodes that are comparatively inert. Similar measurements on their 
+transfer characteristics before and after the applications of large (+40 V, 30 s) Vds 
+pulses are performed. As shown in supplementary figure 4, polarity conversion occurs 
+in neither device.  
+ 
+
+ 
+ 
+Fig. 1 Schematic and electrical performance of the mempolar transistor. a. 
+Schematic and b. optical image of the mempolar transistor (scale bar: 2 μm). c. n-type 
+transfer curve of an as-fabricated device. d. p-type transfer curves of the device after 
+polarity switching. 
+To further identify if the atomic constitution of the channel of the mempolar transistor 
+is changed after polarity conversion, Auger electron spectroscopy (AES) analyses of 
+elements in the channel area for the as-fabricated n-type device, the p-type device 
+after polarity conversion and the restored n-type device are conducted, as shown in 
+figure 2a-c. For the as-fabricated device, it is seen that only Mo and S elements exist 
+in the channel and no observable Te element is found. However, a remarkable amount 
+of Te can be observed in the channel area after polarity conversion. Its distribution 
+looks uniform along the channel width and length except, unsurprisingly, an obvious 
+enrichment near the Te electrodes. The formation of uniform Te sheet in the channel 
+area verifies the hypothesis that polarity conversion is due to electrochemically 
+induced Te inclusion in MoS2. When the device is switched back to n-type, the 
+concentration of Te in the channel area dramatically decreases, indicating that Te 
+atoms are electrochemically extracted from the channel.  
+Figure 2d, e show the atomic force microscopy (AFM) images of the source and drain 
+electrodes before and after the device has undergone polarity conversion. Notches at 
+the electrodes can be clearly seen, which can be understood as related to 
+electrochemical reduction during n-to-p conversion and incomplete Te replenishment 
+during p-to-n conversion.  In line with the deformation of electrodes, AFM images of 
+a local area in the channel for device before and after polarity conversion reveal that 
+the as-exfoliated flat MoS2 surface (figure 2f) turns into a pretty rough  surface as the 
+device has been converted to a p-type transistor (figure 2g). This results from Te 
+
+a
+b
+Monolaver MoS,
+.V.
+MoS2
+Pt
+Pt
+Te
+Sio,
+Pt/Te
+P++ Si
+c
+p
+10-5
+10~5
+n type (pristine)
+Clockwise
+10~6
+10~6
+3
+10-7
+10-7
+Anti-clockwise
+-ptype
+p type (30 days after switch)
+108
+10~8
+-10
+-5
+0
+5
+10
+-10
+-5
+0
+5
+10
+Vg (V)
+Vg (V) 
+inclusion. Sizes and heights of the protrusions in sight can be as great as 60 μm and 6 
+nm, respectively. These protrusions disappear after the device has been converted 
+back to an n-type transistor (figure 2h).  
+To identify the chemical structures of the channel area for the as-fabricated device 
+and the converted p-type device, Raman spectroscopic studies are carried out. As seen 
+in figure 2i, the as-fabricated MoS2 channel shows two characteristic peaks at ~ 380 
+cm-1 and ~ 405 cm-1, in consistence with those of the in-plane Mo-S vibrational mode 
+(E12g) and the out-of-plane S-S vibrational mode (A1g) in monolayer MoS2, 
+respectively26. No Te-related peak is observed. However, characteristic peaks 
+corresponding to basal plane vibration (A1) and bond-stretching vibration (E2) of Te 
+chains at 123 cm−1 and 140 cm−1, respectively27, emerge after the device has been 
+converted to a p-type transistor (figure 2j). These results further support the 
+conjectured polarity conversion mechanism (schematic diagram in supplementary 
+figure S5). 
+  
+Fig. 2 Materials characterizations of the mempolar transistors. AES elemental 
+mapping images of the channel area of a mempolar transistor a. before polarity switch, 
+b. after n-to-p polarity switching and c. after being switched back from p-type to n-
+type. Scale bars for a-c: 500 nm. AFM images of the mempolar transistor d. before 
+and e. after polarity switching (scale bar: 1.5 μm). Closed-up AFM image of the 
+channel area of the mempolar transistor f. before polarity switching, g. after n-to-p 
+polarity switching and h. after being switched back from p-type to n-type (scale bar: 
+120 nm). Raman spectra collected from three different areas labeled in the SEM 
+image (scale bar: 500 nm) of the channel area of the mempolar transistor i. before 
+polarity switching and j. after n-to-p polarity switching. 
+ 
+ 
+
+a
+4.7nm
+Position1
+Mos
+Position2
+Intensity(a.u.)
+Position3
+Te
+Mo
+S
+b
+100
+200
+300
+400
+0.8nm
+--
+Ramanshift(cm-1)
+8.8nm
+C
+Positionl
+Position
+Position3
+1.5nm
+p
+e
+90nm
+95nm
+6.5mm
+Te
+Position1
+Position2
+(a.u.)
+Position3
+Intensity
+MoS,
+100
+200
+300
+400
+3nm
+5 nm
+1.2nm
+Ramanshift (cm-1) 
+2.2. A TCAM cell made of two mempolar transistors 
+Transistors with reconfigurable polarities find applications in reconfigurable logic 
+circuits20, neuromorphic circuits20,28, secure circuits21, and so on. Here, we present a 
+new application of mempolar transistors, that is, making TCAM cells. TCAM is a 
+specialized type of computer memory used in certain very-high-speed searching 
+applications. It is considered as an opposite of the more widely known random access 
+memory (RAM). In a RAM, the user supplies a memory address and the RAM returns 
+the data word stored at that address. By contrast, a TCAM is designed such that the 
+user supplies a data word and the TCAM searches its entire memory based on pattern 
+matching to see if that data word is stored anywhere in it, just like associative memory 
+in the brain. The term “ternary” refers to the ability of the memory to store and query 
+data using three different inputs: 0, 1 and X. The “X” input, which is often referred to 
+as a “don’t care” state, enables TCAM to perform broader searches, as opposed to 
+binary CAM, which performs exact-match searches using only 0s and 1s. 
+TCAM is much faster than RAM in search-intensive applications. However, there are 
+cost disadvantages to TCAM. Unlike a RAM that has simple storage cells, 
+conventional TCAMs normally have more complex circuits with large physical sizes 
+and increased power dissipation. Specifically, a single TCAM cell based on standard 
+CMOS transistor technology requires 16 transistors. Though TCAMs based on the 
+emerging nonvolatile memory device (or simply, memristor) technology have simpler 
+2T-2R cells, each employing two transistors and two memristors29,30, the memristors 
+must be integrated via the back-end-of-line process and thus the electrical parasitics is 
+worsened. In this regard, our mempolar transistor as the product of the fusion of 
+transistive and memristive functions may further simplify the structure and fabrication 
+of the TCAM cell. 
+The schematic diagram of our proposed two mempolar transistor-based TCAM cell 
+and its workings are shown in figure 3a. In this schematic, ‘ML’ refers to the match 
+line which will get charged up to the supply voltage Vdd (1 V) before the search 
+operation. WL1 and WL2 are the two write lines through which polarity switch 
+voltages (±30 V) are applied to the respective mempolar transistors during the 
+memory encoding stage. They also serve as the paths for ML discharging when a 
+match is detected during the search stage, as will soon be introduced. For the storage 
+of the 1 and 0 states, complementary polarity configurations are written (encoded) 
+into the two mempolar transistors. For the storage of the X state, both mempolar 
+transistors are written to the same polarity configuration, either p-type or n-type. SL 
+and SL
+��� are two search lines through which the searching signal and its inverse are 
+applied to the gate terminals of the respective mempolar transistors. The searching 
+signal is presented as either +10 V or -10 V voltage bias, representing data 1 or 0, 
+respectively. With the above encoding and search schemes, if a match between the 
+searching signal and the stored memory state is detected, both mempolar transistors 
+are in their ON states, which discharges the ML to the ground. However, if a 
+mismatch is detected, both mempolar transistors are in their OFF states and thus the 
+ML stays high. In the case that the TCAM cell is in the X state, no matter what the 
+searching signal is, one mempolar transistor must be in the ON state while the other is 
+turned off, thereby discharging the ML.  
+A proof-of-concept TCAM cell made with two mempolar transistors are shown in 
+figure 3b. These two devices are fabricated from MoS2 layers exfoliated on two 
+different silicon dies which are then connected with copper wires by elargol. We 
+
+ 
+experimentally demonstrate the storage of data 0/1 in this TCAM cell by converting 
+the polarity of one mempolar transitor (mempolarT1)/(mempolarT2) to p-type while 
+keeping the other (mempolarT2)/(mempolarT1) unchanged. Alternating 0 and 1 
+search data are then fed into the cell through a pair of SL and SL
+���. For each memory 
+state, the evolution of the resistance between the ML and the ground during the search 
+process is shown. Large resistance indicates that both mempolar transistors are in the 
+OFF states and therefore a mismatch is detected; on the other hand, small resistance 
+indicates that one of the two devices is turned on, corresponding to a match. The 
+resistance contrast retains as high as 102 in one thousand searches (figure 3c, d). 
+Unlike the recently reported ferroelectric TCAM cell31 in which both writing and 
+searching signal transductions share the same pathway (i.e., SL/SL
+���), our cell uses two 
+signal transduction pathways for these two operations, i.e., the SL/SL
+��� for searching 
+and the WL1/WL2 for writing. The decoupling of the search and write operations 
+prevents the stored state from being disturbed by the searching signals and therefore 
+may enable long retention time and enhance reliability.  
+ 
+Fig. 3 Two-mempolar transistors-based TCAM cell. a. Circuit diagram of the two-
+mempolar transistors-based TCAM cell during the search stage and its operating 
+mode. b. A proof-of-concept two-mempolar transistors-based TCAM cell (scale bar: 2 
+μm). Evolutions of the measured resistance between the ML and the ground with the 
+
+a
+PP△!
+b
+ML
+WL
+7S2
+(grounded)
+Mempolar
+ Mempolar
+Mempolanni
+T1
+T2
+ML(Vaa)
+MempolarT2
+WL1
+WL2
+ML
+SL
+TS
+Polarity
+Matched searching
+signal
+MempolarT1
+MempolarT2
+Stored
+Mempolar
+Mempolar
+7S
+SL
+value
+T1
+T2d
+-IUV
+1
+n
++10V
+-10V
+X
+n
+n
++10V
+-10V
+-10V
++10V
+C
+d
+Resistance between
+10M
+Resistance between
+10M
+Mismatch
+Mismatch
+Stored value 'o'
+1M
+1M
+Stored value'1'
+VML= 1V
+VML= 1V
+Match
+Match
+100k
+100k
+0
+200
+400
+600
+800
+1000
+0
+200
+400
+600
+800
+1000
+Read cycle
+Read cycle 
+searching signals alternating between matched and mismatched signals when c. ‘0’ 
+and d. ‘1’ are stored in the TCAM cell.  
+    
+2.3. Mempolar transistor-inspired method for regularizing neural networks 
+As mentioned before, the mempolar transistor with n (p) polarity can show certain 
+degree of clockwise (counter-clockwise) hysteresis in its transfer curve (figure 1c and 
+1d), which can be attributed to trapping/detrapping of carriers in/from the gate oxide 
+during the course of Vg sweeping. As seen from the transfer curves in figure 4a 
+(figure 4b), the channel conductance (under Vg=0 V) of the mempolar transistor with 
+n (p) polarity keeps decreasing (increasing) with successive forward and backward 
+voltage sweepings between 0 V to +10 V. Pulse measurements are also carried out, 
+revealing similar trends of conductance changes in devices with n and p polarities, 
+respectively (supplementary figure S6). These phenomena have been widely exploited 
+by neuromorphic engineers for emulating the long-term plasticity of synapses in the 
+training phase of neural networks16. 
+As shown in figure 4c, 10 cycles of forward and backward Vg sweeps between 0 V to 
++10 V applied to an as-fabricated n-type device (1→2) lead to dramatic decrease in its 
+baseline Id (Id under Vg=0) from about 0.7 μA to 100 nA. After the induction of this 
+long-term synaptic depression (LTD), a positive polarity switching Vds is applied 
+(2→3). The resulting p-type device has large baseline Id about 10 μA. Recall that 
+polarity conversion is elicited by applying a voltage across the source and drain 
+terminals. Therefore, this operation, in principle, should not influence the charge 
+trapping state of the gate oxide which only depends on the history of gate inputs. To 
+verify that the charge trapping state of the gate oxide is not influenced by polarity 
+conversion, we apply a negative polarity switching Vds to convert the device back to 
+n-type (3→4). It is seen that the transfer curve of the present n-type device overlap 
+that of the long-term depressed device before polarity conversion. Likewise, long-
+term synaptic potentiation (LTP) is induced in the p-type device converted from the 
+as-fabricated n-type device, followed by p-to-n conversion and then n-to-p conversion, 
+as shown in figure 4d. Results from these measurements consistently indicate that 
+polarity conversion does not influence the charge trapping state of the gate oxide but 
+can give rise to reverse effect to what long-term plasticity induces before polarity 
+conversion. Specifically, the strength of the reverse effect is monotonically dependent 
+on the pre-accumulated long-term plasticity. 
+Drawing inspiration from this device behavior, we propose a new algorithm 
+‘FlipWeight’ and demonstrate its application by simulations in the context of 
+regularizing neural networks. Traditionally, training a deep neural network (DNN) 
+that can generalize well to new data is a challenging task. This is because a typical 
+DNN has so many parameters (over-parameterized) and limited available data, 
+exhibiting a significant tendency toward overfitting on the training dataset. 
+Approaches to reduce error in generalizing to out-of-sample data points are referred to 
+as regularization methods32. Generally speaking, the rationale behind regularization is 
+constraining the complexity of the DNN model by either reducing the number of 
+synaptic connections or reducing values of synaptic weights. Dropout33,34 and its 
+variant DropConnect35 are two of the most widely used regularization methods. As 
+their names suggest, these two methods randomly drop a number of neurons or 
+connections for each batch during training. With this, not all but only a fraction of 
+weights are updated. After the training on a batch of samples, the omitted neurons 
+(and connections attached to it) or omitted connections in the last batch of training are 
+
+ 
+recovered. A new set of neurons or connections are randomly selected and omitted in 
+the next training batch. Unlike these two methods, our method adapts the idea of 
+reducing values of synaptic weights to improve generalization performance.    
+ 
+ 
+Fig. 4 Gradual channel conductance changes in mempolar transistors and 
+flipping between the high and low conductance states via polarity switching. a. 
+Gradual decrease in the baseline Id (Id under Vg=0) of an n-type mempolar transistor 
+under successive positive Vg sweep. b. Gradual increase in the baseline Id  of a p-type 
+mempolar transistor under successive positive Vg sweep. Flipping back and forth 
+between the high and low conductance states c. in an n-type mempolar transistor 
+before which the channel conductance has been gradually tuned to be relatively low 
+and d. in a p-type mempolar transistor before which the channel conductance has been 
+gradually tuned to be relatively high.  
+Large weights in a DNN are a sign of a complex network that has a tendency to 
+overfit the training data36. Therefore, we consider randomly selecting large weights 
+that are over a threshold and scaling them down according to a certain rule before 
+each training batch starts. A weight scaling rule that naturally matches the properties 
+of our mempolar transistor is inverse scaling with respect to the initial weight value. 
+After this pre-treatment, the standard backpropagation (BP) method is used to 
+calculate the gradients for all the weights in the DNN. Except those pre-treated 
+weights, all the other weights are directly updated using the calculated gradients. The 
+pre-treated weights are first recovered to their original values (also supported by 
+device functions) and then updated using the corresponding gradients calculated from 
+the pre-treated DNN. In the next training batch, a new set of large weights are 
+randomly selected and scaled down, followed by the same procedure of BP 
+calculations and weight update.  
+
+a
+b
+3x106
+4x10°
+Sweep
+Sweep
+1
+1
+3x10-6
+2
+2x10~6
+-3
+2x10-6
+4
+5
+5
+1x106
+1x10~6
+0
+0
+0
+2
+4
+6
+8
+10
+0
+2
+4
+6
+8
+10
+V (V)
+Vg (V)
+c
+n-type
+1-2LTDtraining
+d
+2
+p-type1-2LTPtraining
+3
+1x10~5
+2-3 n-p switching
+1x10
+2--3 p-n switching
+3-4 p-n switching
+3—-4 n-p switching
+10 times
+sweep
+1x106
+1x106
+La
+10timessweep
+3
+1x10-7
+1x10
+2
+1x10*8
+-2
+0
+2
+4
+6
+8
+10
+12
+-2
+0
+2
+4
+6
+8
+10
+12
+Vg(V)
+Vg(V) 
+We benchmark this FlipWeight method against the Dropout and DropConnect 
+methods on a five-layer convolutional neural network (CNN), as shown in figure 5a. 
+We point out that the FlipWeight method is only used in the fully-connected (FC) 
+layers, like Dropout and DropConnect. As presented in figure 5b and 5c, CNN trained 
+with either Dropout or DropConnect or FlipWeight technique can achieve higher 
+validation accuracies and lower loss values compared to the baseline that is 
+implemented without any regularization technique. The gaps between the training 
+curves and the validation curves are also reduced dramatically with the use of these 
+regularization 
+techniques, 
+verifying 
+their 
+effectiveness 
+in 
+improving 
+the 
+generalization performance. Notably, although the final validation accuracy and loss 
+value of CNN implemented with FlipWeight method are similar to those of the CNNs 
+implemented with Dropout and DropConnect, it is evident that our FlipWeight 
+method results in faster convergence than do the other two methods. This can be 
+understood from the fact that all the weights are updated in each training iteration in 
+CNN regularized by our FlipWeight method, while connections omitted in the 
+Dropout or DropConnect approach are simply not involved in weight updating. 
+We also compare the mean, standard deviation and sum of absolute value of weights 
+between CNN models regularized by different methods, as shown in figure 5d-f. It is 
+seen that our FlipWeight method leads to overall excitatory connections (positive 
+mean) while the connections in baseline CNN without regularization and CNNs 
+regularized by Dropout and DropConnect are overall inhibitory (negative mean). Our 
+FlipWeight method also leads to the largest standard deviation of weights and sum of 
+absolute values of weights. The weight distributions are also visualized. The truncated 
+Gaussian distribution for weight initialization is shown in figure 5g. After training, the 
+baseline model without regularization shows a wide distribution of weights among 
+which many have large values (figure 5g). For Dropout and DropConnect, the weight 
+distributions are tighter and peak near zero (figure 5h, i). In stark contrast, the weight 
+distribution in CNN regularized by FlipWeight method is multimodal and even tighter, 
+whose several peaks are close to zero (figure 5j). These results demonstrate the 
+uniqueness of our FlipWeight method that can achieve state-of-the-art performance.   
+
+ 
+ 
+ 
+
+a
+Layerl: Convolution
+Layer2: Convolution
+Layer3: FC
+Layer4: FC Layer5: FC
+Kernel: 4x4
+Kernel:4x4
+12 channels
+Output
+64 units
+12 channels
+(8x8x12)
+Input
+256 units
+(16x16x12)
+(32x32x3)
+b
+c
+100
+Train:Baseline
+Train:Dropout
+3
+Train:Dropconnect
+Train:Flipw
+Accuracy
+Train:Baseline..
+OSS
+Val:Baseline
+60
+Val:Dropout
+TrainDropout
+Val:Dropconnect
+Train:Dropconnect
+Val:Flipw
+40
+Val:Baseline
+1
+Val:Dropout
+20
+Val:Dropconnect
+Val:Flipw
+0
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+100
+Enoch
+Epoch1e-2
+le-1
+1e4
+4
+6
+ of absolute values
+6
+2
+Standard deviation
+5
+5
+0
+4
+4
+3
+Sum
+2
+OV
+g
+lel
+h
+1e2
+Weight Histogram: Initial , Baseline
+Weight Histogram: Dropout
+0.8
+6
+5
+0.6
+4
+0.4
+3
+2
+0.2
+0
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.41e2
+le3
+Weight Histogram: Dropconnect
+Weight Histogram: Flip W
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.4 
+Fig. 5 Performance and characteristics of the CNNs regularized by different 
+methods. a. Schematic illustration of the adopted network structure for image 
+recognition. Comparison of the b. convergence curves and c. loss curves obtained 
+from unregularized model and models regularized by different methods. Comparison 
+of the d. mean, e. stand deviation and f. sums of absolute values of synaptic weights 
+among different models after training, where “Initial” denotes the untrained model, 
+“Base” denotes unregularized baseline model, “Dout”, “DCon” and “FlipW” denote 
+models regularized by Dropout, DropConnect and FlipW methods, respectively. 
+Comparison of weight distribution among g. the untrained model and the 
+unregularized baseline model, and models regularized by h. Dropout, i. DropConnect, 
+and j. FlipWeight methods, respectively. 图 j FlipWeight 
+ 
+3. Conclusion 
+In summary, we introduce the emerging ion-based memristive functions into the 
+purely electronic transistors and demonstrate a new type of transistor named 
+‘mempolar transistor’ whose polarity can be run-time switched between n-type and p-
+type in a non-volatile manner. This novel transistor function is achieved by the use of 
+the emerging semiconducting Te as the electrochemically active source/drain contact 
+material, in combination with monolayer MoS2 channel, which results in a gated 
+lateral Te/MoS2/Te memristor, or from a different perspective, a transistor whose 
+channel can be converted reversibly between n-type MoS2 and p-type Te. Mempolar 
+transistors address a key drawback of the previously showcased transistors with 
+reconfigurable polarities, that is, polarity reconfiguration was volatile and realized via 
+electrostatic control from additional gate terminals. When used in reconfigurable logic 
+circuits or secure circuits, mempolar transistors will potentially mitigate the problems 
+of excessive energy consumption, extensive hardware overhead and massive 
+interconnections. In addition to these manifest advantage, we design and demonstrate 
+experimentally a TCAM made of only two mempolar transistors, which used to 
+require a dozen normal transistors. We also develop and demonstrate by simulations a 
+device-inspired regularization method for training ANNs, which achieves state-of-the-
+art performance. This work broadens the functionality of transistors and provides the 
+implication that rich technological opportunities are available for the fusion between 
+electronics and ionics. 
+ 
+4. Methods 
+Device fabrication: The MoS2 flakes were mechanically exfoliated from bulk crystals 
+(purchased from Six Carbon Technology, Inc.) onto the SiO2 (300 nm)/Si substrate. 
+The source and drain electrodes made from 50 nm Te and 20 nm Pt protective layers 
+were deposited by magnetron sputtering after the standard electron-beam lithography 
+patterning.  
+Electrical measurements: Cyclic quasi-DC voltage sweep measurements were 
+performed by the Keysight B1500A semiconductor analysis system. The Keysight 
+B1530A waveform generator/fast measurement unit is used to perform the pulse 
+measurements.  
+
+ 
+Materials characterizations: The AES analyses were performed by a scanning auger 
+microprobe (PHI710, ULVAC). The morphology characterizations of the devices 
+were performed by an AFM (DIMENSION ICON, BRUKER) in ScanAsyst mode. 
+The Raman spectra were obtained on a single-gating micro-Raman spectrometer 
+(Horiba-JY T64000) excited with 532 nm laser. 
+Neural network model parameterization, training and tests: We evaluated the 
+performance of various regularization methods in CIFAR-10 pattern classification 
+tasks37. The adopted network structure was [Input-12C4-12C4-768FC-256FC-64FC-
+10] (C: convolution, FC: fully-connected layer). We modelled the flipping of a 
+connection weight from its current large value to a small value according to the 
+physical process of polarity switching-induced channel conductance change in the 
+mempolar transistor and by simplifying this process as obtaining the reciprocal of the 
+initial large value (inverse scaling of weight). All CNN models were trained using the 
+adaptive moment estimation (Adam) optimizer38 for 160 epochs with batch size of 
+100 and an initial learning rate of 0.0005. The retaining proportions for models 
+regularized by Dropout or DropConnect were set to 0.5. For the FlipWeight method, 
+connections in the FC layers with weights larger than a threshold value are inversely 
+scaled to their reciprocals with the effect of a scaling factor before each training epoch 
+starts. 0.06 was found to be a suitable threshold value. The hyper-parameters were 
+kept the same for all tested models in this work. The simulations were performed by 
+Tensorflow1.15.0 on 4 RTX 2080Ti GPUs.   
+Supporting Information  
+Supporting Information is available from the author. 
+ 
+Acknowledgments  
+Y. Y., L. X. and M. X. contributed equally to this work. H. L. conceived the idea. Y.Y. 
+and L.X. performed the device fabrication and measurements under the supervision of 
+H.L. and W.D.. M.X. conducted the neural network simulations under the supervision 
+of J.P.. H. Liu and D.L. assisted the device fabrication. Y.Y., M.X. and H.L. wrote 
+this manuscript. This research was supported by National Natural Science Foundation 
+(grant nos. 61974082, 61704096, 61836004), National Key R&D Program of China 
+(2021ZD0200300, 2018YFE0200200), Youth Elite Scientist Sponsorship (YESS) 
+Program of China Association for Science and Technology (CAST) (no. 
+2019QNRC001), Tsinghua-IDG/McGovern Brain-X program, Beijing science and 
+technology program (grant nos. Z181100001518006 and Z191100007519009), 
+Suzhou-Tsinghua innovation leading program 2016SZ0102, CETC Haikang Group-
+Brain Inspired Computing Joint Research Center.  
+ 
+Competing interests 
+The authors declare no competing interests. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+References 
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+
diff --git a/IdA0T4oBgHgl3EQfCP85/content/tmp_files/load_file.txt b/IdA0T4oBgHgl3EQfCP85/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf,len=959
+page_content='A mempolar transistor made from tellurium  Yifei Yang†, Lujie Xu†, Mingkun Xu†, Huan Liu, Dameng Liu, Wenrui Duan, Jing Pei,  Huanglong Li*  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Pei, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Li  Department of Precision Instrument, Center for Brain Inspired Computing Research,  Tsinghua University, Beijing, 100084, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  E-mail: li_huanglong@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='cn  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Duan  School of Instrument Science and Opto Electronics Engineering, Beijing Information  Science and Technology University, Beijing, 100192, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Xu  Application Technology Department, Dongfang Jingyuan Electron Limited, Beijing,  100176, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Liu  State Key Laboratory of Tribology, Tsinghua University, Beijing, 100084, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Li  Chinese Institute for Brain Research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Beijing, 102206, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Keywords: transistors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' memristors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' tellurium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' reconfigurable polarity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' mempolar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  ternary content-addressable memories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' regularization methods  Abstract  The classic three-terminal electronic transistors and the emerging two-terminal ion-  based memristors are complementary to each other in various nonconventional  information processing systems in a heterogeneous integration approach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' such as  DRAM/storage-class memory hierarchy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' hybrid CMOS/memristive neuromorphic  crossbar arrays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Recent attempts to introduce transitive functions into  memristors have given rise to gate-tunable memristive functions, hetero-plasticity and  mixed-plasticity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' However, it remains elusive under what application  scenarios and in what ways transistors can benefit from the incorporation of ion-based  memristive effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Here, we introduce a new type of transistor named ‘mempolar  transistor’ to the transistor family that has included the well-known unipolar and  ambipolar transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' As its name suggests, mempolar transistor has polarity with  memory, reminiscent of memristor having resistance with memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Specifically, its  polarity can be converted reversibly, in a nonvolatile fashion, between n-type and p-  type depending on the history of the applied electrical stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' This is achieved by  the use of the emerging semiconducting tellurium as the electrochemically active  source/drain contact material, in combination with monolayer two-dimensional MoS2  channel, which results in a gated lateral Te/MoS2/Te memristor, or from a different  perspective, a transistor whose channel can be converted reversibly between n-type  MoS2 and p-type Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' With this unique mempolar function, our transistor holds the  promise for reconfigurable logic circuits and secure circuits, addressing a fundamental  limitation in previous implementations, that is, polarity reconfiguration was volatile  and required additional gate terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In addition to this manifest advantage, we  propose and demonstrate experimentally, a ternary content-addressable memory made  of only two mempolar transistors, which used to require a dozen normal transistors,  and by simulations, a device-inspired and hardware matched regularization method  ‘FlipWeight’ for training artificial neural networks, which can achieve comparable  performance to that achieved by the prevalent ‘Dropout’ and ‘DropConnect’ methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  This work represents a major advance in diversifying the functionality of transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Introduction  Polarity is one of the most fundamental aspects of a transistor’s identity, according to  which transistors are most famously categorized into either n-type or p-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The  pairing of n-type and p-type transistors has resulted in complementary metal-oxide-  semiconductor (CMOS) technology, the leading technology for digital integrated  circuits over the past four decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The polarity of a transistor is a reflection of the  comprehensive effect of its entire materials system, including the channel  semiconductor, impurities, source/drain contact, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Typically, polarity identity  is determined at the fabrication stage and cannot be altered afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Traditionally, the performance improvements of the integrated circuits have depended  almost solely on the miniaturization of transistors (Moore’s Law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' As miniaturization  is approaching its physical limits, transistor scaling is delivering performance  improvements at a slower pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' However, software is evolving extremely rapidly with  emerging applications1,2, such as artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Given this, the hardware that  cannot adapt to software will suffer from a short lifecycle and high nonrecurring  engineering cost3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In addition, with the ever increasing density of transistors in  microprocessors, power dissipation has become a huge factor challenging the  continuous success of the CMOS technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Overall, both flexibility and energy  efficiency have become main criteria for computing fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In this context,  incorporation into devices of functionalities that do not necessarily scale according to  Moore’s Law but provide additional value in different ways has become increasingly  pursued by semiconductor industry and academia4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Different from transistor in many key aspects, memristor, experimentally discovered  less than two decades ago5, is one such emerging device that holds great promise for  low-power and adaptive electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Memristor is best-known as a two-terminal  resistor with long-term memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In contrast to the transient electronic switching  (volatile) during the operation of transistor, enduring atomic structure change in the  switching medium can be elicited when suitable electrical stimulus is applied to  memristor, giving rise to nonvolatile reconfigurability of the resistance state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' This  unique property of memristor has made it a key complementary device to transistor in  forming DRAM/storage-class memory hierarchy in conventional von Neumann  architecture and in enabling the emerging computing paradigms, such as in-memory  computing and neuromorphic computing6-8, where transistors and memristors are  integrated heterogeneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  As one step further, attempts have been made to introduce transitive functions into  memristors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In such devices, additional control gates are positioned aside and  electrically insulated from the memristive channels9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' These have given rise to gate-  tunable memristive functions10-12, enabling the emulation of hetero-plasticity13-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Apparently, memristors and memristor-centered circuits benefit from the additional  transistive functions to further enhance their flexibility and augment their  functionalities16,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  At this point, an interesting question arises as to under what application scenarios and  in what ways transistors can benefit from the incorporation of ion-based memristive  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Here, we propose and demonstrate that the fusion of transistive and  memristive functions can result in a novel type of transistor which we name  ‘mempolar transistor’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' As its name suggests, mempolar transistor has polarity with  memory, reminiscent of memristor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Specifically, its polarity can be converted  reversibly, in a nonvolatile fashion, between n-type and p-type depending on the  history of the applied electrical stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Mempolar transistors bring major  improvements over the existing reconfigurable transistors18-21 in the following key  aspects: first, mempolar transistors are more energy-efficient because polarity  conversion takes place in a nonvolatile fashion, whereas in previous devices control-  gate voltages have to be applied persistently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' second, mempolar transistors have  smaller footprints and less fabrication complexity because they retain the classic  three-terminal structures without the addition of more control gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  To enable mempolarity, a key design idea is to first make a two-terminal memristor  whose two resistance states are of n-type and p-type semiconductivity, respectively,  then introduce a third terminal to provide gate control over the memristive channel in  either polarity, as in a normal transistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' To this end, we choose monolayer two-  dimensional (2D) MoS2 as the pristine n-type memristive medium with lateral p-type  tellurium (Te) electrodes, making a two-terminal lateral Te/MoS2/Te device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The  memristive function of this device can be foreseen (demonstrations provided) based  on the previous demonstrations of the electrochemically activity of Te and other  memristors made from it22-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' With a back gate, a mempolar transistor that can be  reconfigured between an n-type MoS2 transistor and a p-type Te transistor is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  In addition to the aforementioned apparent advantages over the existing  reconfigurable transistors in logic and secure circuits, our mempolar transistor also  demonstrate additional value in other applications, such as ternary content-  addressable memory (TCAM) cell made of two mempolar transistors, which used to  require a dozen normal transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Inspired by the device properties, a hardware-  matched regularization method for mitigating the over-fitting problems in artificial  neural networks is also developed, which can achieve comparable performance to that  achieved by the prevalent ‘Dropout’ and ‘DropConnect’ methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Our proposed  mempolar transistor is a valuable addition to the transistor family, enabling  nonvolatile fine-grain reconfigurability and supporting general-purpose hardware  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Results and discussion  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Mempolar function and its mechanism  The schematic structure and the optical image of the mempolar transistor are shown in  figure 1a and 1b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' To fabricate the device, a monolayer MoS2 flake (the  monolayer characteristics verified by Raman spectroscopy are shown in  supplementary figure S1) is mechanically exfoliated onto a heavily p-type doped Si  substrate with 300-nm-thick thermally oxidized SiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The two Te electrodes are then  deposited by magnetron sputtering as the source and drain terminals, followed by the  deposition of platinum protective layers (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  The n-type transfer characteristic of the as-fabricated device is depicted in figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Under a constant drain-source voltage (Vds=Vd-Vs) of +1 V, the drain current (Id) is  increased by about 200 times as Vg is swept from -10 V to +10 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Certain degree of  clockwise hysteresis can also be observed as Vg is swept back and forth, indicating  that trapping/detrapping of electrons in/from the gate oxide (SiO2) takes place during  the course of Vg sweeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' To elicit polarity conversion, a sufficiently large Vds pulse  of +30 V in amplitude and 30 s in duration is applied to the device (under Vg=0 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  As aforementioned, the Te electrode under negative bias (source electrode) with  respect to the other can be electrochemically reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The induced Te2- anions may  migrate towards the counter electrode where they will be re-oxidized to elemental Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  As this electrochemical process proceeds, this accumulated elemental Te may  eventually bridge the source and drain electrodes by forming local Te filament or  wider Te sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Because Te is a narrow-bandgap semiconductor with native p-type  conductivity, we have previously exploited this electrochemical mechanism to enable  filamentary resistance switching in vertical memristors where wider-bandgap (more  insulating) dielectrics are sandwiched between the Te electrodes23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Here, the  formation of Te sheet of the width as close to that of the channel as possible is  preferred in order to best maintain the channel geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Figure 1d shows the transfer  curve of the device after large Vds is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' It is seen that Id is now decreased with  increasing Vg, verifying that transistor polarity is conversed to p-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' With the  conversion of polarity, the hysteresis in the transfer curve also changes direction from  clockwise to anti-clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The p-type transfer characteristics can still be reproduced  after one month without degradation, confirming the nonvolatile nature of the polarity  conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The p polarity can be reversibly switched back to n polarity by applying a  Vds pulse of -30 V under Vg=+5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Gate biasing here reduces the number of free hole  carriers in the p-type Te channel and thus mitigates channel field screening for  enabling electrochemical reaction and ion drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The reversible polarity change is  robustly reproduced under 100 times of polarity switching operations, as shown in  supplementary figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Supplementary figure S3 also shows the evolutions of Id  during the periods of polarity-switching Vds pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Gradual increase (decrease) in Id  with time is observed in n-to-p (p-to-n) switching, which is consistent with the higher  conductivity of p-type Te than that of n-type MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  In contrast to other commonly used source/drain electrode materials, Te is known to  be electrochemically active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In addition, Te anions are also shown to be mobile in  various solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' These two materials properties are key in enabling Te filament-based  resistance switching, as previously reported22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' To explore the mechanisms behind  the unique mempolar phenomenon, we fabricate two control devices with Ti and Pt  source/drain electrodes that are comparatively inert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Similar measurements on their  transfer characteristics before and after the applications of large (+40 V, 30 s) Vds  pulses are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' As shown in supplementary figure 4, polarity conversion occurs  in neither device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' 1 Schematic and electrical performance of the mempolar transistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Schematic and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' optical image of the mempolar transistor (scale bar: 2 μm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' n-type  transfer curve of an as-fabricated device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' p-type transfer curves of the device after  polarity switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  To further identify if the atomic constitution of the channel of the mempolar transistor  is changed after polarity conversion, Auger electron spectroscopy (AES) analyses of  elements in the channel area for the as-fabricated n-type device, the p-type device  after polarity conversion and the restored n-type device are conducted, as shown in  figure 2a-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' For the as-fabricated device, it is seen that only Mo and S elements exist  in the channel and no observable Te element is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' However, a remarkable amount  of Te can be observed in the channel area after polarity conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Its distribution  looks uniform along the channel width and length except, unsurprisingly, an obvious  enrichment near the Te electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The formation of uniform Te sheet in the channel  area verifies the hypothesis that polarity conversion is due to electrochemically  induced Te inclusion in MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' When the device is switched back to n-type, the  concentration of Te in the channel area dramatically decreases, indicating that Te  atoms are electrochemically extracted from the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Figure 2d, e show the atomic force microscopy (AFM) images of the source and drain  electrodes before and after the device has undergone polarity conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Notches at  the electrodes can be clearly seen, which can be understood as related to  electrochemical reduction during n-to-p conversion and incomplete Te replenishment  during p-to-n conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In line with the deformation of electrodes, AFM images of  a local area in the channel for device before and after polarity conversion reveal that  the as-exfoliated flat MoS2 surface (figure 2f) turns into a pretty rough  surface as the  device has been converted to a p-type transistor (figure 2g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' This results from Te  a  b  Monolaver MoS,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  MoS2  Pt  Pt  Te  Sio,  Pt/Te  P++ Si  c  p  10-5  10~5  n type (pristine)  Clockwise  10~6  10~6  3  10-7  10-7  Anti-clockwise  ptype  p type (30 days after switch)  108  10~8  10  5  0  5  10  10  5  0  5  10  Vg (V)  Vg (V)  inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Sizes and heights of the protrusions in sight can be as great as 60 μm and 6  nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' These protrusions disappear after the device has been converted  back to an n-type transistor (figure 2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  To identify the chemical structures of the channel area for the as-fabricated device  and the converted p-type device, Raman spectroscopic studies are carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' As seen  in figure 2i, the as-fabricated MoS2 channel shows two characteristic peaks at ~ 380  cm-1 and ~ 405 cm-1, in consistence with those of the in-plane Mo-S vibrational mode  (E12g) and the out-of-plane S-S vibrational mode (A1g) in monolayer MoS2,  respectively26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' No Te-related peak is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' However, characteristic peaks  corresponding to basal plane vibration (A1) and bond-stretching vibration (E2) of Te  chains at 123 cm−1 and 140 cm−1, respectively27, emerge after the device has been  converted to a p-type transistor (figure 2j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' These results further support the  conjectured polarity conversion mechanism (schematic diagram in supplementary  figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' 2 Materials characterizations of the mempolar transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' AES elemental  mapping images of the channel area of a mempolar transistor a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' before polarity switch,  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' after n-to-p polarity switching and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' after being switched back from p-type to n-  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Scale bars for a-c: 500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' AFM images of the mempolar transistor d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' before  and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' after polarity switching (scale bar: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='5 μm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Closed-up AFM image of the  channel area of the mempolar transistor f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' before polarity switching, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' after n-to-p  polarity switching and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' after being switched back from p-type to n-type (scale bar:  120 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Raman spectra collected from three different areas labeled in the SEM  image (scale bar: 500 nm) of the channel area of the mempolar transistor i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' before  polarity switching and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' after n-to-p polarity switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  a  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='7nm  Position1  Mos  Position2  Intensity(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=')  Position3  Te  Mo  S  b  100  200  300  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='8nm  --  Ramanshift(cm-1)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='8nm  C  Positionl  Position  Position3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='5nm  p  e  90nm  95nm  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='5mm  Te  Position1  Position2  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=')  Position3  Intensity  MoS,  100  200  300  400  3nm  5 nm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='2nm  Ramanshift (cm-1)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' A TCAM cell made of two mempolar transistors  Transistors with reconfigurable polarities find applications in reconfigurable logic  circuits20, neuromorphic circuits20,28, secure circuits21, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Here, we present a  new application of mempolar transistors, that is, making TCAM cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' TCAM is a  specialized type of computer memory used in certain very-high-speed searching  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' It is considered as an opposite of the more widely known random access  memory (RAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In a RAM, the user supplies a memory address and the RAM returns  the data word stored at that address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' By contrast, a TCAM is designed such that the  user supplies a data word and the TCAM searches its entire memory based on pattern  matching to see if that data word is stored anywhere in it, just like associative memory  in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The term “ternary” refers to the ability of the memory to store and query  data using three different inputs: 0, 1 and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The “X” input, which is often referred to  as a “don’t care” state, enables TCAM to perform broader searches, as opposed to  binary CAM, which performs exact-match searches using only 0s and 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  TCAM is much faster than RAM in search-intensive applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' However, there are  cost disadvantages to TCAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Unlike a RAM that has simple storage cells,  conventional TCAMs normally have more complex circuits with large physical sizes  and increased power dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Specifically, a single TCAM cell based on standard  CMOS transistor technology requires 16 transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Though TCAMs based on the  emerging nonvolatile memory device (or simply, memristor) technology have simpler  2T-2R cells, each employing two transistors and two memristors29,30, the memristors  must be integrated via the back-end-of-line process and thus the electrical parasitics is  worsened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In this regard, our mempolar transistor as the product of the fusion of  transistive and memristive functions may further simplify the structure and fabrication  of the TCAM cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  The schematic diagram of our proposed two mempolar transistor-based TCAM cell  and its workings are shown in figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In this schematic, ‘ML’ refers to the match  line which will get charged up to the supply voltage Vdd (1 V) before the search  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' WL1 and WL2 are the two write lines through which polarity switch  voltages (±30 V) are applied to the respective mempolar transistors during the  memory encoding stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' They also serve as the paths for ML discharging when a  match is detected during the search stage, as will soon be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' For the storage  of the 1 and 0 states, complementary polarity configurations are written (encoded)  into the two mempolar transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' For the storage of the X state, both mempolar  transistors are written to the same polarity configuration, either p-type or n-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' SL  and SL  ��� are two search lines through which the searching signal and its inverse are  applied to the gate terminals of the respective mempolar transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The searching  signal is presented as either +10 V or -10 V voltage bias, representing data 1 or 0,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' With the above encoding and search schemes, if a match between the  searching signal and the stored memory state is detected, both mempolar transistors  are in their ON states, which discharges the ML to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' However, if a  mismatch is detected, both mempolar transistors are in their OFF states and thus the  ML stays high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In the case that the TCAM cell is in the X state, no matter what the  searching signal is, one mempolar transistor must be in the ON state while the other is  turned off, thereby discharging the ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  A proof-of-concept TCAM cell made with two mempolar transistors are shown in  figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' These two devices are fabricated from MoS2 layers exfoliated on two  different silicon dies which are then connected with copper wires by elargol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' We  experimentally demonstrate the storage of data 0/1 in this TCAM cell by converting  the polarity of one mempolar transitor (mempolarT1)/(mempolarT2) to p-type while  keeping the other (mempolarT2)/(mempolarT1) unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Alternating 0 and 1  search data are then fed into the cell through a pair of SL and SL  ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' For each memory  state, the evolution of the resistance between the ML and the ground during the search  process is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Large resistance indicates that both mempolar transistors are in the  OFF states and therefore a mismatch is detected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' on the other hand, small resistance  indicates that one of the two devices is turned on, corresponding to a match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The  resistance contrast retains as high as 102 in one thousand searches (figure 3c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Unlike the recently reported ferroelectric TCAM cell31 in which both writing and  searching signal transductions share the same pathway (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=', SL/SL  ���), our cell uses two  signal transduction pathways for these two operations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=', the SL/SL  ��� for searching  and the WL1/WL2 for writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The decoupling of the search and write operations  prevents the stored state from being disturbed by the searching signals and therefore  may enable long retention time and enhance reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' 3 Two-mempolar transistors-based TCAM cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Circuit diagram of the two-  mempolar transistors-based TCAM cell during the search stage and its operating  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' A proof-of-concept two-mempolar transistors-based TCAM cell (scale bar: 2  μm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Evolutions of the measured resistance between the ML and the ground with the  a  PP△!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
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+page_content='Read cycle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='Read cycle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='searching signals alternating between matched and mismatched signals when c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' ‘0’  and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' ‘1’ are stored in the TCAM cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Mempolar transistor-inspired method for regularizing neural networks  As mentioned before, the mempolar transistor with n (p) polarity can show certain  degree of clockwise (counter-clockwise) hysteresis in its transfer curve (figure 1c and  1d), which can be attributed to trapping/detrapping of carriers in/from the gate oxide  during the course of Vg sweeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' As seen from the transfer curves in figure 4a  (figure 4b), the channel conductance (under Vg=0 V) of the mempolar transistor with  n (p) polarity keeps decreasing (increasing) with successive forward and backward  voltage sweepings between 0 V to +10 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Pulse measurements are also carried out,  revealing similar trends of conductance changes in devices with n and p polarities,  respectively (supplementary figure S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' These phenomena have been widely exploited  by neuromorphic engineers for emulating the long-term plasticity of synapses in the  training phase of neural networks16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  As shown in figure 4c, 10 cycles of forward and backward Vg sweeps between 0 V to  +10 V applied to an as-fabricated n-type device (1→2) lead to dramatic decrease in its  baseline Id (Id under Vg=0) from about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='7 μA to 100 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' After the induction of this  long-term synaptic depression (LTD), a positive polarity switching Vds is applied  (2→3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The resulting p-type device has large baseline Id about 10 μA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Recall that  polarity conversion is elicited by applying a voltage across the source and drain  terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Therefore, this operation, in principle, should not influence the charge  trapping state of the gate oxide which only depends on the history of gate inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' To  verify that the charge trapping state of the gate oxide is not influenced by polarity  conversion, we apply a negative polarity switching Vds to convert the device back to  n-type (3→4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' It is seen that the transfer curve of the present n-type device overlap  that of the long-term depressed device before polarity conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Likewise, long-  term synaptic potentiation (LTP) is induced in the p-type device converted from the  as-fabricated n-type device, followed by p-to-n conversion and then n-to-p conversion,  as shown in figure 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Results from these measurements consistently indicate that  polarity conversion does not influence the charge trapping state of the gate oxide but  can give rise to reverse effect to what long-term plasticity induces before polarity  conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Specifically, the strength of the reverse effect is monotonically dependent  on the pre-accumulated long-term plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Drawing inspiration from this device behavior, we propose a new algorithm  ‘FlipWeight’ and demonstrate its application by simulations in the context of  regularizing neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Traditionally, training a deep neural network (DNN)  that can generalize well to new data is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' This is because a typical  DNN has so many parameters (over-parameterized) and limited available data,  exhibiting a significant tendency toward overfitting on the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Approaches to reduce error in generalizing to out-of-sample data points are referred to  as regularization methods32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Generally speaking, the rationale behind regularization is  constraining the complexity of the DNN model by either reducing the number of  synaptic connections or reducing values of synaptic weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Dropout33,34 and its  variant DropConnect35 are two of the most widely used regularization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' As  their names suggest, these two methods randomly drop a number of neurons or  connections for each batch during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' With this, not all but only a fraction of  weights are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' After the training on a batch of samples, the omitted neurons  (and connections attached to it) or omitted connections in the last batch of training are  recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' A new set of neurons or connections are randomly selected and omitted in  the next training batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Unlike these two methods, our method adapts the idea of  reducing values of synaptic weights to improve generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' 4 Gradual channel conductance changes in mempolar transistors and  flipping between the high and low conductance states via polarity switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Gradual decrease in the baseline Id (Id under Vg=0) of an n-type mempolar transistor  under successive positive Vg sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Gradual increase in the baseline Id  of a p-type  mempolar transistor under successive positive Vg sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Flipping back and forth  between the high and low conductance states c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' in an n-type mempolar transistor  before which the channel conductance has been gradually tuned to be relatively low  and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' in a p-type mempolar transistor before which the channel conductance has been  gradually tuned to be relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Large weights in a DNN are a sign of a complex network that has a tendency to  overfit the training data36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Therefore, we consider randomly selecting large weights  that are over a threshold and scaling them down according to a certain rule before  each training batch starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' A weight scaling rule that naturally matches the properties  of our mempolar transistor is inverse scaling with respect to the initial weight value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  After this pre-treatment, the standard backpropagation (BP) method is used to  calculate the gradients for all the weights in the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Except those pre-treated  weights, all the other weights are directly updated using the calculated gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The  pre-treated weights are first recovered to their original values (also supported by  device functions) and then updated using the corresponding gradients calculated from  the pre-treated DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In the next training batch, a new set of large weights are  randomly selected and scaled down, followed by the same procedure of BP  calculations and weight update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
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+page_content='Vg(V)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='Vg(V)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='We benchmark this FlipWeight method against the Dropout and DropConnect  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='methods on a five-layer convolutional neural network (CNN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' as shown in figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  We point out that the FlipWeight method is only used in the fully-connected (FC)  layers, like Dropout and DropConnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' As presented in figure 5b and 5c, CNN trained  with either Dropout or DropConnect or FlipWeight technique can achieve higher  validation accuracies and lower loss values compared to the baseline that is  implemented without any regularization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The gaps between the training  curves and the validation curves are also reduced dramatically with the use of these  regularization  techniques,  verifying  their  effectiveness  in  improving  the  generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Notably, although the final validation accuracy and loss  value of CNN implemented with FlipWeight method are similar to those of the CNNs  implemented with Dropout and DropConnect, it is evident that our FlipWeight  method results in faster convergence than do the other two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' This can be  understood from the fact that all the weights are updated in each training iteration in  CNN regularized by our FlipWeight method, while connections omitted in the  Dropout or DropConnect approach are simply not involved in weight updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  We also compare the mean, standard deviation and sum of absolute value of weights  between CNN models regularized by different methods, as shown in figure 5d-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' It is  seen that our FlipWeight method leads to overall excitatory connections (positive  mean) while the connections in baseline CNN without regularization and CNNs  regularized by Dropout and DropConnect are overall inhibitory (negative mean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Our  FlipWeight method also leads to the largest standard deviation of weights and sum of  absolute values of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The weight distributions are also visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The truncated  Gaussian distribution for weight initialization is shown in figure 5g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' After training, the  baseline model without regularization shows a wide distribution of weights among  which many have large values (figure 5g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' For Dropout and DropConnect, the weight  distributions are tighter and peak near zero (figure 5h, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In stark contrast, the weight  distribution in CNN regularized by FlipWeight method is multimodal and even tighter,  whose several peaks are close to zero (figure 5j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' These results demonstrate the  uniqueness of our FlipWeight method that can achieve state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  a  Layerl: Convolution  Layer2: Convolution  Layer3: FC  Layer4: FC Layer5: FC  Kernel: 4x4  Kernel:4x4  12 channels  Output  64 units  12 channels  (8x8x12)  Input  256 units  (16x16x12)  (32x32x3)  b  c  100  Train:Baseline  Train:Dropout  3  Train:Dropconnect  Train:Flipw  Accuracy  Train:Baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='.  OSS  Val:Baseline  60  Val:Dropout  TrainDropout  Val:Dropconnect  Train:Dropconnect  Val:Flipw  40  Val:Baseline  1  Val:Dropout  20  Val:Dropconnect  Val:Flipw  0  0  20  40  60  80  100  0  20  40  60  80  100  Enoch  Epoch1e-2  le-1  1e4  4  6  of absolute values  6  2  Standard deviation  5  5  0  4  4  3  Sum  2  OV  g  lel  h  1e2  Weight Histogram: Initial , Baseline  Weight Histogram: Dropout  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='8  6  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='6  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
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+page_content='4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' 5 Performance and characteristics of the CNNs regularized by different  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Schematic illustration of the adopted network structure for image  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Comparison of the b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' convergence curves and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' loss curves obtained  from unregularized model and models regularized by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Comparison  of the d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' mean, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' stand deviation and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' sums of absolute values of synaptic weights  among different models after training, where “Initial” denotes the untrained model,  “Base” denotes unregularized baseline model, “Dout”, “DCon” and “FlipW” denote  models regularized by Dropout, DropConnect and FlipW methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Comparison of weight distribution among g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' the untrained model and the  unregularized baseline model, and models regularized by h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Dropout, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' DropConnect,  and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' FlipWeight methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' 图 j FlipWeight  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Conclusion  In summary, we introduce the emerging ion-based memristive functions into the  purely electronic transistors and demonstrate a new type of transistor named  ‘mempolar transistor’ whose polarity can be run-time switched between n-type and p-  type in a non-volatile manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' This novel transistor function is achieved by the use of  the emerging semiconducting Te as the electrochemically active source/drain contact  material, in combination with monolayer MoS2 channel, which results in a gated  lateral Te/MoS2/Te memristor, or from a different perspective, a transistor whose  channel can be converted reversibly between n-type MoS2 and p-type Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Mempolar  transistors address a key drawback of the previously showcased transistors with  reconfigurable polarities, that is, polarity reconfiguration was volatile and realized via  electrostatic control from additional gate terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' When used in reconfigurable logic  circuits or secure circuits, mempolar transistors will potentially mitigate the problems  of excessive energy consumption, extensive hardware overhead and massive  interconnections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' In addition to these manifest advantage, we design and demonstrate  experimentally a TCAM made of only two mempolar transistors, which used to  require a dozen normal transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' We also develop and demonstrate by simulations a  device-inspired regularization method for training ANNs, which achieves state-of-the-  art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' This work broadens the functionality of transistors and provides the  implication that rich technological opportunities are available for the fusion between  electronics and ionics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Methods  Device fabrication: The MoS2 flakes were mechanically exfoliated from bulk crystals  (purchased from Six Carbon Technology, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=') onto the SiO2 (300 nm)/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  The source and drain electrodes made from 50 nm Te and 20 nm Pt protective layers  were deposited by magnetron sputtering after the standard electron-beam lithography  patterning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Electrical measurements: Cyclic quasi-DC voltage sweep measurements were  performed by the Keysight B1500A semiconductor analysis system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The Keysight  B1530A waveform generator/fast measurement unit is used to perform the pulse  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Materials characterizations: The AES analyses were performed by a scanning auger  microprobe (PHI710, ULVAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The morphology characterizations of the devices  were performed by an AFM (DIMENSION ICON, BRUKER) in ScanAsyst mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  The Raman spectra were obtained on a single-gating micro-Raman spectrometer  (Horiba-JY T64000) excited with 532 nm laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Neural network model parameterization, training and tests: We evaluated the  performance of various regularization methods in CIFAR-10 pattern classification  tasks37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The adopted network structure was [Input-12C4-12C4-768FC-256FC-64FC-  10] (C: convolution, FC: fully-connected layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' We modelled the flipping of a  connection weight from its current large value to a small value according to the  physical process of polarity switching-induced channel conductance change in the  mempolar transistor and by simplifying this process as obtaining the reciprocal of the  initial large value (inverse scaling of weight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' All CNN models were trained using the  adaptive moment estimation (Adam) optimizer38 for 160 epochs with batch size of  100 and an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The retaining proportions for models  regularized by Dropout or DropConnect were set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' For the FlipWeight method,  connections in the FC layers with weights larger than a threshold value are inversely  scaled to their reciprocals with the effect of a scaling factor before each training epoch  starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='06 was found to be a suitable threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The hyper-parameters were  kept the same for all tested models in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' The simulations were performed by  Tensorflow1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='0 on 4 RTX 2080Ti GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Supporting Information  Supporting Information is available from the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Acknowledgments  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' conceived the idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' performed the device fabrication and measurements under the supervision of  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='. M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' conducted the neural network simulations under the supervision  of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='. H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Liu and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' assisted the device fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' wrote  this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' This research was supported by National Natural Science Foundation  (grant nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' 61974082, 61704096, 61836004), National Key R&D Program of China  (2021ZD0200300, 2018YFE0200200), Youth Elite Scientist Sponsorship (YESS)  Program of China Association for Science and Technology (CAST) (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  2019QNRC001), Tsinghua-IDG/McGovern Brain-X program, Beijing science and  technology program (grant nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content=' Z181100001518006 and Z191100007519009),  Suzhou-Tsinghua innovation leading program 2016SZ0102, CETC Haikang Group-  Brain Inspired Computing Joint Research Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
+page_content='  Competing interests  The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdA0T4oBgHgl3EQfCP85/content/2301.01986v1.pdf'}
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@@ -0,0 +1,1065 @@
+1
+Cost-Effective Two-Stage Network Slicing for
+Edge-Cloud Orchestrated Vehicular Networks
+Wen Wu‡, Kaige Qu⋆, Peng Yang∗, Ning Zhang†, Xuemin (Sherman) Shen⋆, and Weihua Zhuang⋆
+Frontier Research Center, Peng Cheng Laboratory, Shenzhen, China‡
+Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Canada⋆
+School of Electronic Information and Communications, Huazhong University of Science and Technology, China∗
+Department of Electrical and Computer Engineering, University of Windsor, Windsor, Canada†
+Email: wuw02@pcl.ac.cn‡, {k2qu, sshen, wzhuang}@uwaterloo.ca⋆,
+yangpeng@hust.edu.cn∗, and ning.zhang@uwindsor.ca†
+Abstract—In this paper, we study a network slicing problem
+for edge-cloud orchestrated vehicular networks, in which the edge
+and cloud servers are orchestrated to process computation tasks
+for reducing network slicing cost while satisfying the quality
+of service requirements. We propose a two-stage network slicing
+framework, which consists of 1) network planning stage in a large
+timescale to perform slice deployment, edge resource provision-
+ing, and cloud resource provisioning, and 2) network operation
+stage in a small timescale to perform resource allocation and
+task dispatching. Particularly, we formulate the network slicing
+problem as a two-timescale stochastic optimization problem to
+minimize the network slicing cost. Since the problem is NP-hard
+due to coupled network planning and network operation stages,
+we develop a Two timescAle netWork Slicing (TAWS) algorithm
+by collaboratively integrating reinforcement learning (RL) and
+optimization methods, which can jointly make network planning
+and operation decisions. Specifically, by leveraging the timescale
+separation property of decisions, we decouple the problem into
+a large-timescale network planning subproblem and a small-
+timescale network operation subproblem. The former is solved
+by an RL method, and the latter is solved by an optimization
+method. Simulation results based on real-world vehicle traffic
+traces show that the TAWS can effectively reduce the network
+slicing cost as compared to the benchmark scheme.
+I. INTRODUCTION
+To make autonomous driving from a mere vision to reality,
+future vehicular networks are required to support various
+Internet of vehicles (IoV) services, such as object detection,
+in-vehicle infotainment, and safety message dissemination [1].
+Those IoV services have diversified quality of service (QoS)
+requirements in terms of delay, throughput, reliability, etc.
+Emerging network slicing is deemed as a de-facto solution to
+support diversified IoV services in vehicular networks. Its ba-
+sic idea is to construct multiple isolated logical sub-networks
+(i.e., slices) for different services on top of the physical
+network, thereby facilitating flexible, agile, and cost-effective
+service provisioning. Starting from the fifth-generation (5G)
+era, standardization efforts from the 3rd generation partnership
+project (3GPP) body, e.g., Releases 15-17 [2]–[4], and proof-
+of-concept systems, e.g., Orion [5], have fuelled the maturity
+of network slicing. In the coming 6G era, advanced network
+slicing techniques are expected to play an increasingly impor-
+tant role [6]–[8].
+In the literature, significant research efforts have been
+devoted to network slicing. Ye et al. investigated a radio
+spectrum resource slicing problem, in which radio spectrum
+is sliced between macro base stations (MBSs) and small
+BSs (SBSs) [9]. To achieve efficient resource allocation, a
+deep learning-based algorithm was proposed to jointly allo-
+cate radio spectrum and transmit power in a slicing-based
+network [10]. The previous work in
+[11] considered the
+resource provisioning problem and proposed a constrained
+learning algorithm to solve it. However, this work differs from
+the existing works in several important aspects. Firstly, the
+existing works focus on utilizing resources on the network
+edge, low-cost cloud resources are yet to be considered. As
+a remedy, a certain amount of computation tasks processed
+at the congested BSs can be dispatched to the remote cloud,
+i.e., task dispatching, such that system cost can be reduced.
+Secondly, network slicing includes two stages: 1) network
+planning stage to provision network resources for slices in
+the large timescale, and 2) network operation stage to allocate
+the reserved resources to end users in the small timescale [3],
+[12]. The existing works mainly decouple network slicing
+into two independent stages, while the interaction between
+them is seldom considered. Hence, designing a cost-effective
+network slicing scheme should take cloud resources and such
+interaction relationship into consideration.
+Optimizing network slicing performance in dynamic vehic-
+ular networks faces the following challenges. Firstly, network
+planning and operation decisions are nested. Large-timescale
+network planning decisions (e.g., resource reservation), will
+condition small-timescale network operation decisions (e.g.,
+resource allocation). Meanwhile, the performance achieved
+in the network operation stage will also affect the decision-
+making in the network planning stage, which is difficult to
+be solved by conventional optimization methods. Secondly,
+since vehicle traffic density varies temporal-spatially, net-
+work planning decisions need to be made to optimize long-
+term performance in the slice lifecycle while accommodating
+such network dynamics. Deep reinforcement learning (RL)
+is considered as a plausible solution for long-term stochastic
+optimization.
+In this paper, we first propose a cost-effective two-stage
+arXiv:2301.03358v1  [cs.NI]  31 Dec 2022
+
+2
+network slicing framework for edge-cloud orchestrated vehic-
+ular networks, by considering nested network planning and
+operation stages and effectively leveraging cloud resources.
+We then apply a network slicing cost model that accounts
+for slice deployment, resource provision, slice configuration
+adjustment, and QoS satisfaction. Based on the model, we
+formulate the network slicing problem as a two-timescale
+stochastic optimization problem to minimize the network
+slicing cost. Second, to solve the problem, we develop a
+learning-based algorithm, named Two timescAle netWork
+Slicing (TAWS). The TAWS exploits the timescale separation
+structure of decision variables and decouples the problem into
+two subproblems in different timescales. Regarding the large-
+timescale network planning subproblem, an RL algorithm is
+designed to minimize network slicing cost via optimizing slice
+deployment, edge resource provisioning, and cloud resource
+provisioning. Regarding the small-timescale network operation
+subproblem, an optimization algorithm is designed to mini-
+mize average service delay via optimizing resource allocation
+and task dispatching. In addition, the achieved service delay in
+the network operation stage is incorporated into the reward of
+the RL-based network planning algorithm, thereby capturing
+the interaction between two stages and enabling closed-loop
+network control. Simulation results on real-world vehicle
+traces demonstrate that the proposed algorithm outperforms
+the benchmark scheme in terms of reducing network slicing
+cost.
+The remainder of this paper is organized as follows. The
+system model and problem formulation are presented in Sec-
+tions II and III, respectively. Section IV describes the proposed
+TAWS algorithm. Simulation results are given in Section V,
+along with the conclusion in Section VI.
+II. SYSTEM MODEL
+A. Network Model
+As shown in Fig. 1, the network slicing framework consists
+of several components.
+Physical network: A two-tier cellular network is deployed
+for serving on-road vehicles. The set of BSs is denoted by
+M, including the set of MBSs denoted by Mm and the set
+of SBSs denoted by Ms, i.e., M = Mm ∪Ms. Each BS has
+a circular coverage and is equipped with an edge server. In
+the considered scenario, vehicles driving on the road generate
+computation tasks over time, which are offloaded to roadside
+BSs. Those tasks can be either processed at edge servers or
+dispatched to the remote cloud server via backbone networks.
+Once completed, computation results are sent back to vehicles.
+Network slice: Multiple network slices are constructed
+on top of the physical vehicular network. We consider K
+delay-sensitive services with differentiated delay requirements,
+denoted by set K. Let θk, ∀k ∈ K denote the tolerable delay
+of service k. For example, the tolerable delay of objective
+detection service is 100 ms [13], whereas the tolerable delay
+of in-vehicle infotainment can be up to several hundreds of
+milliseconds.
+Network controller: A hierarchical network control archi-
+tecture is adopted, including an upper-layer software defined
+Slice 1
+Slice N
+Network 
+Slices
+...
+Physical
+Network 
+SBS
+MBS
+Backbone
+Transmission
+Cloud 
+Server
+Edge Server
+Switch
+Vehicle
+Computation 
+Task
+Control
+ Link
+SDN 
+Controller
+Fig. 1.
+Network slicing for edge-cloud orchestrated vehicular networks.
+networking (SDN) controller that connects to all BSs, and
+lower-layer local network controllers located at BSs. Those
+controllers are in charge of network information collection and
+making network slicing decisions.
+B. Two-Stage Network Slicing Framework
+We present a two-stage network slicing framework for the
+considered network. Firstly, a network planning stage operates
+in the large timescale (referred to as planning windows) to
+reserve resources at specific network nodes for the constructed
+slices. The duration of each planning window is denoted by
+Tp. At each planning window, the SDN controller collects the
+average vehicle traffic density information in the considered
+area, based on which planning decisions are made. Secondly,
+the network operation stage operates in the small timescale
+(referred to as operation slots) to dynamically allocate the
+reserved resources to vehicles according to real-time vehicles’
+service requests and network conditions. The duration of each
+operation slot is denoted by To. A planning window includes
+multiple operation slots, i.e., Tp/To ∈ Z+. At each operation
+slot, the local network controller at each BS collects real-time
+service requests and channel conditions of its associated vehi-
+cles, based on which operation decisions are made. Decision
+structures in two stages are detailed respectively as follows.
+1) Network Planning Decision Structure: The planning
+window is indexed by w ∈ W = {1, 2, ..., W}, and plan-
+ning decisions in planning window w include the following
+components.
+Slice deployment decision, denoted by ow ∈ RMs×1. Each
+element is a binary variable, i.e.,
+ow
+m ∈ {0, 1}, m ∈ Ms.
+(1)
+If SBS m is activated for slice deployment, we have ow
+m = 1;
+otherwise, ow
+m = 0. When service demands are low, deploying
+slices at a selective subset of BSs can reduce network slicing
+cost as compared to deploying slices at all BSs while guaran-
+teeing slices’ service level agreements (SLAs). This is because
+running network slicing requires resource virtualization, which
+incurs network operating costs. For service continuity consid-
+eration, we assume that MBSs that cover the entire area are
+always activated. Note that only when a BS is activated for
+slice deployment, edge resources at the BS can be provisioned.
+Edge resource provisioning decision, including radio spec-
+trum and computing resource provisioning at all BSs for
+all slices, denoted by Bw ∈ RK×M and Cw ∈ RK×M,
+
+3
+respectively. The corresponding elements
+{bw
+k,m, cw
+k,m} ∈ Z+, ∀k ∈ K, m ∈ M,
+(2)
+represent the number of subcarriers and edge virtual machine
+(VM) instances provisioned for slice k at BS m, where Z+
+denotes the set of positive integers.1 The bandwidth of a
+subcarrier is denoted by β, and the computing capability of
+an edge VM is denoted by Fe. Due to the limitation of edge
+resources, the following capacity constraints are imposed:
+ow
+m
+�
+k∈K
+bw
+k,m ≤ Bm, ow
+m
+�
+k∈K
+cw
+k,m ≤ Cm, ∀m ∈ M,
+(3)
+where Bm and Cm represent the total numbers of subcarriers
+and VM instances at BS m, respectively.
+Cloud resource provisioning decision, denoted by hw ∈
+RK×1. Each element
+hw
+k ∈ Z+, ∀k ∈ K
+(4)
+denotes the number of cloud VM instances reserved for slice k.
+The computing capability of a cloud VM is denoted by Fc.
+2) Network Operation Decision Structure: Let t ∈ T =
+{1, 2, ..., T} denote the index of operation slots within a
+planning window. At operation slot t, the following decisions
+are determined for each slice k.
+Radio spectrum allocation decision, denoted by yt
+k
+∈
+RN t×1. The reserved radio spectrum at each BS is allocated to
+active vehicles within BS’s coverage for task offloading. Due
+to vehicle mobility, the number of vehicles varies across time.
+Let N t denote the set of active vehicles in operation slot t,
+and N t = |N t|. For simplicity, each vehicle associates to the
+nearest BS. Let N t
+m denote the set of active vehicles associated
+to BS m at operation slot t, and yt
+k,n ∈ R+ represents the
+fraction of radio spectrum allocated to vehicle n. The total
+amount of the allocated bandwidth should not exceed the
+reserved number of subcarriers at the corresponding BS, i.e.,
+�
+n∈N tm
+yt
+k,n ≤ bw
+k,m, ∀m ∈ Mw.
+(5)
+Here, Mw denotes the set of the activated BSs in window w.
+Task dispatching decision, denoted by xt
+k
+∈
+ZM w×1.
+The BS receives computation tasks uploaded from its as-
+sociated vehicles. The task arrivals of vehicles follow an
+arbitrary stochastic process. Let at
+k,n denote the number of
+the generated tasks of vehicle n in operation slot t, and
+the aggregated computation workload at BS m is given by
+At
+k,m = �
+n∈N tm at
+k,n. Processing all tasks at BSs with limited
+computing resources may incur prohibitive high queuing delay,
+and hence a portion of computation tasks can be dispatched to
+the remote cloud via backbone networks. Let xt
+k,m represent
+the number of dispatched tasks from BS m in slice k, i.e.,
+xt
+k,m ∈ {0, 1, 2, ..., At
+k,m}, ∀m ∈ Mw.
+(6)
+The operation decisions impact service delay at each oper-
+ation slot, which is analyzed in the following subsection.
+1Memory resource is also allocated to the VM instance to enable task
+processing, which is matched to its allocated computing resource.
+C. Service Delay Model
+The service delay includes task offloading delay and task
+processing delay at either the edge or the cloud. For service
+k, the following delay analysis is adopted.
+Task offloading delay: The transmission rate of one sub-
+carrier from vehicle n to its associated BS is given by
+Rt
+n = β log2
+�
+1 +
+Pvgt
+n
+βNo+βI
+�
+, where Pv, gt
+n, No, and I repre-
+sent vehicle’s transmission power, instantaneous channel gain,
+noise spectrum density, and interference spectrum density,
+respectively. With the allocated radio spectrum yt
+k,nbw
+k,m, the
+task offloading delay of vehicle n is given by dt
+k,n,o =
+ξk
+yt
+k,nbw
+k,mRtn , ∀n ∈ N t
+m, where ξk (in bits) denotes the task
+data size of service k.
+Edge processing delay: Given the task dispatching de-
+cision, At
+k,m − xt
+k,m tasks are processed at BS m. Let
+Qt
+k,m (in bits) denote the amount of the backlogged tasks
+at BS m. Taking task computation delay and queuing delay
+into account, edge processing delay at BS m is given by
+dt
+k,m,e = (Qt
+k,m+(At
+k,m−xt
+k,m+1)ξk/2)ηk
+cw
+k,mFe
+, ∀m ∈ Mw, where ηk
+(in cycles/bit) denotes task computation intensity of service k,
+and cw
+k,mFe is the computing capability of BS m with cw
+k,m
+provisioned edge VMs. The task backlog at BS m is updated
+by Qt+1
+k,m =
+�
+Qt
+k,m + (At
+k,m − xt
+k,m)ξk − cw
+k,mFeTo/ηk
+�+
+,
+where [x]+ = max {x, 0}.
+Cloud processing delay: For BS m, xt
+k,m tasks are dis-
+patched via backbone networks and then processed at the
+cloud, whose delay is given by dt
+k,m,c = dt
+r + ξkηk
+hw
+k Fc , where
+dt
+r denotes the round trip time in the backbone network. The
+second term represents the task processing delay in the cloud.
+Note that the queuing delay at the cloud is negligible as multi-
+core cloud servers can parallelly process different tasks.
+As such, the average delay for each computation task is
+given by
+Dt
+k(xt
+k, yt
+k) =
+�
+m∈Mw
+�
+n∈N tm
+dt
+k,n,o
+�
+m∈Mw N tm
++
+�
+m∈Mw
+dt
+k,m,e
+�
+At
+k,m − xt
+k,m
+�
++ dt
+k,m,cxt
+k,m
+�
+m∈Mw At
+k,m
+.
+(7)
+In the above equation, the first term represents the average task
+offloading delay for each task, and the second term represents
+the average task processing delay taking workload distribution
+between the edge and cloud servers into account. By averaging
+all operation slots, the average service delay is given by ¯Dw
+k =
+1
+T
+�T
+t=1 Dt
+k(xt
+k, yt
+k).
+D. Network Slicing Cost Model
+The following network slicing cost model is adopted for
+slicing performance evaluation, including several components.
+Slice deployment cost: The cost is because running network
+slices at BSs incurs the overhead of resource virtualization,
+which is given by Φw
+d = qd
+�
+m∈Ms ow
+m. Here, qd denotes the
+unit cost of deploying network slices at a BS.
+Resource provisioning cost: The cost component character-
+izes resource provisioning cost of radio spectrum resources,
+
+4
+edge computing resources, and cloud computing resources.
+For simplicity, we assume the unit costs of a subcarrier, an
+edge VM instance, and a cloud VM instance are the same,
+denoted by qr > 0. The resource provisioning cost is given
+by Φw
+p = qr
+�
+k∈K
+�
+hw
+k + �
+m∈M
+�
+ow
+mbw
+k,m + ow
+mcw
+k,m
+��
+.
+Slice adjustment cost: The cost component characterizes the
+difference between two subsequent planning decisions, i.e., the
+cost for adjusting the amount of the reserved spectrum and
+computing resources. For computing resources, VM instances
+can be resized via advanced virtualization techniques in prac-
+tical systems, e.g., Kubernetes [14]. Here, qs represents the
+unit price of adjusting a unit of reserved network resources.
+Hence, the slice adjustment cost is given by
+Φw
+s =qs1
+�
+ow−1
+k,m = 1 ∧ ow
+k,m = 1
+�
+·
+�
+k∈K
+��
+hw
+k − hw−1
+k
+�+
++
+�
+m∈M
+��
+bw
+k,m − bw−1
+k,m
+�+
++
+�
+cw
+k,m − cw−1
+k,m
+�+��
+,
+(8)
+where
+1 {·}
+is
+an
+indicator
+function
+and
+1
+�
+ow−1
+k,m = 1 ∧ ow
+k,m = 1
+�
+indicates that slice k is deployed
+in the previous and current planning windows.
+SLA revenue: The cost component characterizes the benefit
+caused by QoS satisfaction, i.e., the achieved service delay of
+each slice. The piece-wise SLA revenue function is denoted
+by
+Ωk (D) =
+�
+�
+�
+�
+�
+�
+�
+qb,
+if D < θ
+′
+k,
+qb
+�
+D−θ
+′
+k
+θk−θ′
+k
+�
+,
+if θ
+′
+k ≤ D ≤ θk,
+−qp,
+if D > θk.
+(9)
+Here, qb > 0 is the highest unit revenue once a slice’s SLA
+is satisfied, and qp > 0 is the unit penalty once the slice’s
+SLA is violated. Obviously, qp > qb for discouraging slice’s
+SLA violation. In addition, θ
+′
+k < θk represents the threshold
+achieving the highest revenue. For simplicity, we set θ
+′
+k =
+θk/2 in the simulation. The overall SLA revenue of all slices
+is given by Φw
+q = �
+k∈K Ωk
+� ¯Dw
+k
+�
+.
+Taking all cost components into account, the overall network
+slicing cost in the entire slice lifecycle (i.e., all planning win-
+dows) is given by Φ (ow, Bw, Cw, hw, {xt
+k, yt
+k}t∈T ,k∈K) =
+�
+w∈W
+�
+Φw
+d + Φw
+p + Φw
+s − Φw
+q
+�
+, which is adopted to evalu-
+ate network slicing performance.
+III. PROBLEM FORMULATION
+The network slicing problem aims to minimize the network
+slicing cost via determining network planning decisions at
+each planning window and network operation decisions at each
+operation slot for each slice, which is formulated as:
+P0 :
+min
+{ow,Bw,Cw,hw}w∈W
+{xt
+k,yt
+k}t∈T ,k∈K,w∈W
+�
+w∈W
+Φ (ow, Bw, Cw, hw)
+s.t. (1), (2), (3), (4), (5), and (6). (10a)
+In Problem P0, the network planning and operation decision
+making are coupled in two timescales, which should be jointly
+optimized. To address the challenge, we first decouple the
+problem into a large-timescale network planning subproblem
+and multiple small-timescale network operation subproblems.
+Subproblem 1: Network planning subproblem is to mini-
+mize the network slicing cost across all the planning windows,
+which is formulated as:
+P1 :
+min
+{ow,Bw,
+Cw,hw}w∈W
+�
+w∈W
+Φ (ow, Bw, Cw, hw)
+s.t. (1), (2), (3), and (4).
+(11a)
+Addressing the above subproblem requires network traffic
+information of all planning windows, which is difficult to
+be known a priori. To solve it, we leverage an RL method
+to design a network planning algorithm, which makes online
+decisions under spatial-temporally varying vehicle traffic.
+Subproblem 2: Network operation subproblem is to sched-
+ule network resources of each slice to active vehicles with
+random task arrivals with the objective of minimizing average
+service delay, which is formulated as:
+P2 : min
+xt
+k,yt
+k
+Dt
+k(xt
+k, yt
+k)
+s.t. (5) and (6).
+(12a)
+In the above subproblem, radio spectrum resource allocation
+and task dispatching decisions jointly impact the service
+delay performance. To solve the problem, we analyze the
+subproblem property and design an optimization algorithm to
+make real-time network operation decisions.
+IV. LEARNING-BASED NETWORK SLICING ALGORITHM
+In this section, we solve two subproblems in Sections IV-A
+and IV-B, respectively. Finally, we present the TWAS algo-
+rithm for jointly optimizing planning and operation decisions
+in Section IV-C.
+A. Network Operation Optimization
+We can observe that the radio spectrum allocation de-
+cision only impacts offloading delay component, and the
+task dispatching decision only impacts the computation delay
+component. Moreover, both decisions are independent in each
+BS. Hence, the radio spectrum allocation and task dispatching
+decisions can be optimized individually at each BS.
+1) Radio Spectrum Allocation Optimization: From (7), the
+radio spectrum allocation optimization problem is equivalent
+to minimizing the task offloading delay at each BS, i.e.,
+Pr
+m : min
+yt
+k
+�
+n∈N tm
+ξk
+yt
+k,nbw
+k,mRtn
+s.t. (5).
+(13a)
+The objective function can be proved to be convex since its
+second-order derivative is positive. In addition, the constraint
+is convex. Hence, problem Pr
+m is a convex optimization
+problem. Using the Karush-Kuhn-Tucker conditions [15], the
+optimal radio spectrum resource allocation decision is
+(yt
+k,n)⋆ =
+�
+1/Rtn
+�
+i∈N tm
+�
+1/Rt
+i
+, ∀n ∈ N t
+m.
+(14)
+
+5
+2) Task Dispatching Optimization: Similarly, from (7), task
+dispatching optimization is to minimize the task processing
+delay, which is formulated as:
+Pw
+m : min
+xt
+k,m
+dt
+k,m,e
+�
+At
+k,m − xt
+k,m
+�
++ dt
+k,m,cxt
+k,m
+s.t. (6).
+(15a)
+The above objective function can be rewritten as
+Ψ(xt
+k,m) = dt
+k,m,e
+�
+At
+k,m − xt
+k,m
+�
++ dt
+k,m,cxt
+k,m
+= ν1ξk
+2
+(xt
+k,m)2 +
+�
+νt
+2 − ν1ν3 − ξkAk,mν1
+2
+�
+xt
+k,m
++ ν1νt
+3At
+k,m.
+(16)
+Here, ν1 =
+ηk
+cw
+k,mFe
+> 0, νt
+2 = dt
+r +
+ηkξk
+hw
+k Fc , and ν3 =
+Qk,m + Ak,m+1
+2
+ξk. Since the second-order derivative of the
+objective function ∂2Ψ(xt
+k,m)/∂2xt
+k,m = νt
+1ξk > 0, the
+problem is a convex optimization problem [15]. The optimal
+task dispatching decision is given by
+(xt
+k,m)⋆ = 2νt
+2 + ξkν1Ak,m − 2ν1νt
+3
+2ν1ξk
+, ∀m ∈ Mw.
+(17)
+B. Network Planing Optimization
+The network planning problem is a stochastic optimization
+problem to minimize the network slicing cost, which can be
+transformed into a Markov decision process (MDP) [11]. The
+components of the MDP are defined as follows.
+1) Action, which is consistent with planning decisions,
+including slice deployment, radio spectrum and computing
+resource provisioning at BSs, and cloud computing resource
+provisioning, i.e., Aw
+=
+{ow, Bw, Cw, hw}. The action
+dimension is Ms + 2KM + K.
+2) State, which includes average vehicle traffic density in
+the current planning window and the planning decisions in the
+previous window due to the switching cost. The entire area is
+divided into J disjoint regions, and the average vehicle traffic
+density of all regions is denoted by Λw ∈ RJ×1. As such, the
+state is given by Sw = {Λw, ow−1, Bw−1, Cw−1, hw−1}. The
+state dimension is 2KM + M + K + J.
+3) Reward, which is defined as the inverse of the net-
+work slicing cost in the current planning window, i.e.,
+Rw (Sw, Aw) = −Φ (ow, Bw, Cw, hw) . Note that minimiz-
+ing the network slicing cost is equivalent to maximizing the
+cumulative reward.
+Upon state Sw, the learning agent takes action Aw, and the
+corresponding reward Rw (Sw, Aw) is obtained, along with
+the state evolves into new state Sw+1. With the above setting,
+our goal is to obtain an optimal planning policy π⋆ ∈ Π
+which makes decisions based on the observed state, thereby
+maximizing the expected long-term cumulative reward. As
+such, problem P2 can be formulated as the following MDP:
+P′
+2 : max
+π∈Π E
+�
+lim
+W →∞
+W
+�
+w=1
+(ϕ)wRw (Sw, Aw) |π
+�
+,
+(18a)
+where ϕ > 0 is the discount factor. Since vehicle traffic density
+is continuous, the action-state space can be prohibitively large.
+To address this issue, an RL algorithm can be adopted.
+Algorithm 1: TAWS algorithm.
+1 for training episode =1, 2, ... do
+2
+for planning window w = 1, 2, ..., W do
+3
+Generate planning decisions via the actor network;
+4
+for each slice in parallel do
+5
+for operation slot t = 1, 2, ..., T do
+6
+for each BS in parallel do
+7
+Make radio spectrum allocation and task
+dispatching decisions by (14) and (17);
+8
+Calculate the instantaneous service delay;
+9
+Measure the average service delay within the
+planning window;
+10
+Collect vehicle traffic density of all regions, and
+observe reward Rw and new state Sw+1;
+11
+Store transition {Sw, Aw, Rw, Sw+1} in the
+experience replay buffer;
+12
+Sample a random minibatch of transitions from the
+experience replay buffer;
+13
+Update the weights of neural networks;
+C. Proposed TAWS Algorithm
+We present the TAWS algorithm to jointly solve the entire
+network slicing problem P0, collaboratively integrating RL
+and optimization methods. The core idea of TAWS is to adopt
+an RL method for network planning decision making and an
+optimization method for network operation decision making.
+The service delay performance is measured at the end of each
+planning window and then incorporated into the reward in
+the RL framework, such that the interaction between network
+planning and operation stages can be captured. The TAWS
+algorithm is shown in Algorithm 1.
+The RL method is based on the deep deterministic policy
+gradient (DDPG) algorithm [16], [17], which consists of
+four neural networks, i.e., actor evaluation network, critic
+evaluation network, actor target network, and critic target
+network. In the initialization phase, all neural networks and
+the environment are initialized. The procedure of the TAWS
+is two-step: 1) Network slicing decisions are generated and
+executed. The actor network outputs the planning decisions
+Aw, which is clipped to feasible decision space. The network
+operation decisions are generated via the optimization method,
+and the service delay performance is measured at the end
+of each planning window. The reward Rw can be obtained
+and the new state can be observed Sw+1. The transition tuple
+{Sw, Aw, Rw, Sw+1} is stored in the experience replay buffer
+for updating neural networks; and 2) Neural networks are
+updated. A mini-batch of transitions are randomly sampled
+from the experience replay buffer to update the weights of
+neural networks. Specifically, the critic network is updated
+by minimizing the loss function, and the actor network is
+updated via the policy gradient method. Then, actor and critic
+target networks are updated by slowly copying the weights of
+evaluation networks.
+V. SIMULATION RESULTS
+We evaluate the performance of the proposed algorithm on
+real-world vehicle traffic traces in urban vehicular networks.
+We consider a 1,000×1,000 m2 simulation area, which is
+
+6
+Table I
+SIMULATION PARAMETERS.
+Parameter
+Value
+Parameter
+Value
+No
+−174 dBm
+I
+−164 dBm
+Pv
+27 dBm
+β
+20 MHz
+dr
+0.15 sec
+J
+16
+To
+1 sec
+Tp
+10 min
+Fc
+100 GHz
+Fe
+10 GHz
+Bm
+10
+Cm
+10
+ξ1, ξ2
+{0.6, 2} Mbit
+η1, η2
+{1000, 200} cycles/bit
+θ1, θ2
+{100, 200} ms
+θ
+′
+1, θ
+′
+2
+{50, 100} ms
+0
+100
+200
+300
+400
+500
+Training Episodes
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+Overall System Cost
+Five-Point Moving Average
+(a) Convergence
+1.4
+1.6
+1.8
+2
+Task Arrival Rate (Packet/sec)
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+Overall System Cost
+Proposed
+Short Term Optimization
+(b) Network slicing cost
+Fig. 2.
+Performance of the proposed TWAS algorithm.
+covered by two SBSs and an MBS. Each SBS has a coverage
+radius of 300 m, and the MBS located in the centre covers
+the entire simulation area. The vehicle traffic density of the
+simulation area is measured by a unit of a small region
+of 250×250 m2, i.e., J = 16. This dataset is collected by
+Didi Chuxing GAIA Initiative2 and contains vehicle traces in
+the second ring road in Xi’an collected from taxis that are
+equipped with GPS devices. The periods of a planning window
+and an operation slot are set to 10 minutes and 1 second,
+respectively. The period of the slice lifecycle is set to 4
+hours, including 24 planning windows. The task arrivals of
+two services both follow Poisson processes with different task
+arrival rates. We construct two slices for supporting two types
+of delay-sensitive services. One is an object detect service
+whose service delay requirement is 100 ms, while the other
+is an in-vehicle infotainment service whose service delay
+requirement is 200 ms. Regarding the TWAS algorithm, the
+neuron units in hidden layers of both actor and critic networks
+are set to 128 and 64. Important simulation parameters are
+summarized in Table I.
+As shown in Fig. 2(a), we present the overall network slicing
+cost with respect to training episodes. All simulation points are
+processed by a five-point moving average in order to highlight
+the convergence trend of the proposed algorithm. It can be
+seen that the proposed algorithm converges after 500 training
+episodes.
+As shown in Fig. 2(b), we compare the performance of
+the proposed algorithm and a short term optimization bench-
+mark. The basic idea of the benchmark is to minimize the
+network slicing cost at each individual planning window. Since
+planning decisions are discrete, a simple exhaustive searching
+method is adopted to obtain the optimal one-shot planning
+decisions. Firstly, it can be seen that the proposed algorithm
+can greatly reduce the network slicing cost as compared to the
+benchmark. Specifically, when the task arrival rate is 2 packets
+per second, the proposed algorithm can reduce the network
+slicing cost by 23%. The reason is that the proposed algorithm
+takes the switching cost between two consequent planning
+windows into account, while the benchmark scheme does
+2Didi Chuxing Dataset: https://gaia.didichuxing.com.
+not. Secondly, the overall network slicing cost increases with
+the increase of the task arrival rate, because more radio and
+computing resources are consumed in heavy traffic scenarios.
+VI. CONCLUSION
+In this paper, we have investigated a network slicing prob-
+lem in edge-cloud orchestrated vehicular networks. A two-
+stage network slicing algorithm, named TWAS, has been
+proposed to jointly make network planning and operation
+decisions in an online fashion. The TAWS can adapt to
+network dynamics in different timescales, including spatial-
+temporally varying vehicle traffic density and random task
+arrivals. Simulation results demonstrat that the TAWS can re-
+duce the network slicing cost as compared to the conventional
+scheme. For the future work, we aim to determine the optimal
+planning window size for minimizing the network slicing cost
+under vehicular network dynamics.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf,len=515
+page_content='1  Cost-Effective Two-Stage Network Slicing for  Edge-Cloud Orchestrated Vehicular Networks  Wen Wu‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Kaige Qu⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Peng Yang∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Ning Zhang†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Xuemin (Sherman) Shen⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' and Weihua Zhuang⋆  Frontier Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Peng Cheng Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' China‡  Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' University of Waterloo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Waterloo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Canada⋆  School of Electronic Information and Communications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Huazhong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' China∗  Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' University of Windsor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Windsor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Canada†  Email: wuw02@pcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='cn‡, {k2qu, sshen, wzhuang}@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='ca⋆,  yangpeng@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='cn∗, and ning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='zhang@uwindsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='ca†  Abstract—In this paper, we study a network slicing problem  for edge-cloud orchestrated vehicular networks, in which the edge  and cloud servers are orchestrated to process computation tasks  for reducing network slicing cost while satisfying the quality  of service requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' We propose a two-stage network slicing  framework, which consists of 1) network planning stage in a large  timescale to perform slice deployment, edge resource provision-  ing, and cloud resource provisioning, and 2) network operation  stage in a small timescale to perform resource allocation and  task dispatching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Particularly, we formulate the network slicing  problem as a two-timescale stochastic optimization problem to  minimize the network slicing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Since the problem is NP-hard  due to coupled network planning and network operation stages,  we develop a Two timescAle netWork Slicing (TAWS) algorithm  by collaboratively integrating reinforcement learning (RL) and  optimization methods, which can jointly make network planning  and operation decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Specifically, by leveraging the timescale  separation property of decisions, we decouple the problem into  a large-timescale network planning subproblem and a small-  timescale network operation subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The former is solved  by an RL method, and the latter is solved by an optimization  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Simulation results based on real-world vehicle traffic  traces show that the TAWS can effectively reduce the network  slicing cost as compared to the benchmark scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' INTRODUCTION  To make autonomous driving from a mere vision to reality,  future vehicular networks are required to support various  Internet of vehicles (IoV) services, such as object detection,  in-vehicle infotainment, and safety message dissemination [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Those IoV services have diversified quality of service (QoS)  requirements in terms of delay, throughput, reliability, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Emerging network slicing is deemed as a de-facto solution to  support diversified IoV services in vehicular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Its ba-  sic idea is to construct multiple isolated logical sub-networks  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', slices) for different services on top of the physical  network, thereby facilitating flexible, agile, and cost-effective  service provisioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Starting from the fifth-generation (5G)  era, standardization efforts from the 3rd generation partnership  project (3GPP) body, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', Releases 15-17 [2]–[4], and proof-  of-concept systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', Orion [5], have fuelled the maturity  of network slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' In the coming 6G era, advanced network  slicing techniques are expected to play an increasingly impor-  tant role [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  In the literature, significant research efforts have been  devoted to network slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' investigated a radio  spectrum resource slicing problem, in which radio spectrum  is sliced between macro base stations (MBSs) and small  BSs (SBSs) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' To achieve efficient resource allocation, a  deep learning-based algorithm was proposed to jointly allo-  cate radio spectrum and transmit power in a slicing-based  network [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The previous work in  [11] considered the  resource provisioning problem and proposed a constrained  learning algorithm to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' However, this work differs from  the existing works in several important aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Firstly, the  existing works focus on utilizing resources on the network  edge, low-cost cloud resources are yet to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' As  a remedy, a certain amount of computation tasks processed  at the congested BSs can be dispatched to the remote cloud,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', task dispatching, such that system cost can be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Secondly, network slicing includes two stages: 1) network  planning stage to provision network resources for slices in  the large timescale, and 2) network operation stage to allocate  the reserved resources to end users in the small timescale [3],  [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The existing works mainly decouple network slicing  into two independent stages, while the interaction between  them is seldom considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Hence, designing a cost-effective  network slicing scheme should take cloud resources and such  interaction relationship into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Optimizing network slicing performance in dynamic vehic-  ular networks faces the following challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Firstly, network  planning and operation decisions are nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Large-timescale  network planning decisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', resource reservation), will  condition small-timescale network operation decisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=',  resource allocation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Meanwhile, the performance achieved  in the network operation stage will also affect the decision-  making in the network planning stage, which is difficult to  be solved by conventional optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Secondly,  since vehicle traffic density varies temporal-spatially, net-  work planning decisions need to be made to optimize long-  term performance in the slice lifecycle while accommodating  such network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Deep reinforcement learning (RL)  is considered as a plausible solution for long-term stochastic  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  In this paper, we first propose a cost-effective two-stage  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='03358v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='NI]  31 Dec 2022  2  network slicing framework for edge-cloud orchestrated vehic-  ular networks, by considering nested network planning and  operation stages and effectively leveraging cloud resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  We then apply a network slicing cost model that accounts  for slice deployment, resource provision, slice configuration  adjustment, and QoS satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Based on the model, we  formulate the network slicing problem as a two-timescale  stochastic optimization problem to minimize the network  slicing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Second, to solve the problem, we develop a  learning-based algorithm, named Two timescAle netWork  Slicing (TAWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The TAWS exploits the timescale separation  structure of decision variables and decouples the problem into  two subproblems in different timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Regarding the large-  timescale network planning subproblem, an RL algorithm is  designed to minimize network slicing cost via optimizing slice  deployment, edge resource provisioning, and cloud resource  provisioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Regarding the small-timescale network operation  subproblem, an optimization algorithm is designed to mini-  mize average service delay via optimizing resource allocation  and task dispatching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' In addition, the achieved service delay in  the network operation stage is incorporated into the reward of  the RL-based network planning algorithm, thereby capturing  the interaction between two stages and enabling closed-loop  network control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Simulation results on real-world vehicle  traces demonstrate that the proposed algorithm outperforms  the benchmark scheme in terms of reducing network slicing  cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The  system model and problem formulation are presented in Sec-  tions II and III, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Section IV describes the proposed  TAWS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Simulation results are given in Section V,  along with the conclusion in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' SYSTEM MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Network Model  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' 1, the network slicing framework consists  of several components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Physical network: A two-tier cellular network is deployed  for serving on-road vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The set of BSs is denoted by  M, including the set of MBSs denoted by Mm and the set  of SBSs denoted by Ms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', M = Mm ∪Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Each BS has  a circular coverage and is equipped with an edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' In  the considered scenario, vehicles driving on the road generate  computation tasks over time, which are offloaded to roadside  BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Those tasks can be either processed at edge servers or  dispatched to the remote cloud server via backbone networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Once completed, computation results are sent back to vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Network slice: Multiple network slices are constructed  on top of the physical vehicular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' We consider K  delay-sensitive services with differentiated delay requirements,  denoted by set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Let θk, ∀k ∈ K denote the tolerable delay  of service k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' For example, the tolerable delay of objective  detection service is 100 ms [13], whereas the tolerable delay  of in-vehicle infotainment can be up to several hundreds of  milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Network controller: A hierarchical network control archi-  tecture is adopted, including an upper-layer software defined  Slice 1  Slice N  Network  Slices  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Physical  Network  SBS  MBS  Backbone  Transmission  Cloud  Server  Edge Server  Switch  Vehicle  Computation  Task  Control  Link  SDN  Controller  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Network slicing for edge-cloud orchestrated vehicular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  networking (SDN) controller that connects to all BSs, and  lower-layer local network controllers located at BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Those  controllers are in charge of network information collection and  making network slicing decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Two-Stage Network Slicing Framework  We present a two-stage network slicing framework for the  considered network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Firstly, a network planning stage operates  in the large timescale (referred to as planning windows) to  reserve resources at specific network nodes for the constructed  slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The duration of each planning window is denoted by  Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' At each planning window, the SDN controller collects the  average vehicle traffic density information in the considered  area, based on which planning decisions are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Secondly,  the network operation stage operates in the small timescale  (referred to as operation slots) to dynamically allocate the  reserved resources to vehicles according to real-time vehicles’  service requests and network conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The duration of each  operation slot is denoted by To.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' A planning window includes  multiple operation slots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', Tp/To ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' At each operation  slot, the local network controller at each BS collects real-time  service requests and channel conditions of its associated vehi-  cles, based on which operation decisions are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Decision  structures in two stages are detailed respectively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  1) Network Planning Decision Structure: The planning  window is indexed by w ∈ W = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', W}, and plan-  ning decisions in planning window w include the following  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Slice deployment decision, denoted by ow ∈ RMs×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Each  element is a binary variable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=',  ow  m ∈ {0, 1}, m ∈ Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (1)  If SBS m is activated for slice deployment, we have ow  m = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  otherwise, ow  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' When service demands are low, deploying  slices at a selective subset of BSs can reduce network slicing  cost as compared to deploying slices at all BSs while guaran-  teeing slices’ service level agreements (SLAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' This is because  running network slicing requires resource virtualization, which  incurs network operating costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' For service continuity consid-  eration, we assume that MBSs that cover the entire area are  always activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Note that only when a BS is activated for  slice deployment, edge resources at the BS can be provisioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Edge resource provisioning decision, including radio spec-  trum and computing resource provisioning at all BSs for  all slices, denoted by Bw ∈ RK×M and Cw ∈ RK×M,  3  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The corresponding elements  {bw  k,m, cw  k,m} ∈ Z+, ∀k ∈ K, m ∈ M,  (2)  represent the number of subcarriers and edge virtual machine  (VM) instances provisioned for slice k at BS m, where Z+  denotes the set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='1 The bandwidth of a  subcarrier is denoted by β, and the computing capability of  an edge VM is denoted by Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Due to the limitation of edge  resources, the following capacity constraints are imposed:  ow  m  �  k∈K  bw  k,m ≤ Bm, ow  m  �  k∈K  cw  k,m ≤ Cm, ∀m ∈ M,  (3)  where Bm and Cm represent the total numbers of subcarriers  and VM instances at BS m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Cloud resource provisioning decision, denoted by hw ∈  RK×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Each element  hw  k ∈ Z+, ∀k ∈ K  (4)  denotes the number of cloud VM instances reserved for slice k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  The computing capability of a cloud VM is denoted by Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  2) Network Operation Decision Structure: Let t ∈ T =  {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', T} denote the index of operation slots within a  planning window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' At operation slot t, the following decisions  are determined for each slice k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Radio spectrum allocation decision, denoted by yt  k  ∈  RN t×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The reserved radio spectrum at each BS is allocated to  active vehicles within BS’s coverage for task offloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Due  to vehicle mobility, the number of vehicles varies across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Let N t denote the set of active vehicles in operation slot t,  and N t = |N t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' For simplicity, each vehicle associates to the  nearest BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Let N t  m denote the set of active vehicles associated  to BS m at operation slot t, and yt  k,n ∈ R+ represents the  fraction of radio spectrum allocated to vehicle n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The total  amount of the allocated bandwidth should not exceed the  reserved number of subcarriers at the corresponding BS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=',  �  n∈N tm  yt  k,n ≤ bw  k,m, ∀m ∈ Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (5)  Here, Mw denotes the set of the activated BSs in window w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Task dispatching decision, denoted by xt  k  ∈  ZM w×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  The BS receives computation tasks uploaded from its as-  sociated vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The task arrivals of vehicles follow an  arbitrary stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Let at  k,n denote the number of  the generated tasks of vehicle n in operation slot t, and  the aggregated computation workload at BS m is given by  At  k,m = �  n∈N tm at  k,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Processing all tasks at BSs with limited  computing resources may incur prohibitive high queuing delay,  and hence a portion of computation tasks can be dispatched to  the remote cloud via backbone networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Let xt  k,m represent  the number of dispatched tasks from BS m in slice k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=',  xt  k,m ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', At  k,m}, ∀m ∈ Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (6)  The operation decisions impact service delay at each oper-  ation slot, which is analyzed in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  1Memory resource is also allocated to the VM instance to enable task  processing, which is matched to its allocated computing resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Service Delay Model  The service delay includes task offloading delay and task  processing delay at either the edge or the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' For service  k, the following delay analysis is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Task offloading delay: The transmission rate of one sub-  carrier from vehicle n to its associated BS is given by  Rt  n = β log2  �  1 +  Pvgt  n  βNo+βI  �  , where Pv, gt  n, No, and I repre-  sent vehicle’s transmission power, instantaneous channel gain,  noise spectrum density, and interference spectrum density,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' With the allocated radio spectrum yt  k,nbw  k,m, the  task offloading delay of vehicle n is given by dt  k,n,o =  ξk  yt  k,nbw  k,mRtn , ∀n ∈ N t  m, where ξk (in bits) denotes the task  data size of service k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Edge processing delay: Given the task dispatching de-  cision, At  k,m − xt  k,m tasks are processed at BS m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Let  Qt  k,m (in bits) denote the amount of the backlogged tasks  at BS m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Taking task computation delay and queuing delay  into account, edge processing delay at BS m is given by  dt  k,m,e = (Qt  k,m+(At  k,m−xt  k,m+1)ξk/2)ηk  cw  k,mFe  , ∀m ∈ Mw, where ηk  (in cycles/bit) denotes task computation intensity of service k,  and cw  k,mFe is the computing capability of BS m with cw  k,m  provisioned edge VMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The task backlog at BS m is updated  by Qt+1  k,m =  �  Qt  k,m + (At  k,m − xt  k,m)ξk − cw  k,mFeTo/ηk  �+  ,  where [x]+ = max {x, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Cloud processing delay: For BS m, xt  k,m tasks are dis-  patched via backbone networks and then processed at the  cloud, whose delay is given by dt  k,m,c = dt  r + ξkηk  hw  k Fc , where  dt  r denotes the round trip time in the backbone network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The  second term represents the task processing delay in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Note that the queuing delay at the cloud is negligible as multi-  core cloud servers can parallelly process different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  As such, the average delay for each computation task is  given by  Dt  k(xt  k, yt  k) =  �  m∈Mw  �  n∈N tm  dt  k,n,o  �  m∈Mw N tm  +  �  m∈Mw  dt  k,m,e  �  At  k,m − xt  k,m  �  + dt  k,m,cxt  k,m  �  m∈Mw At  k,m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (7)  In the above equation, the first term represents the average task  offloading delay for each task, and the second term represents  the average task processing delay taking workload distribution  between the edge and cloud servers into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' By averaging  all operation slots, the average service delay is given by ¯Dw  k =  1  T  �T  t=1 Dt  k(xt  k, yt  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Network Slicing Cost Model  The following network slicing cost model is adopted for  slicing performance evaluation, including several components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Slice deployment cost: The cost is because running network  slices at BSs incurs the overhead of resource virtualization,  which is given by Φw  d = qd  �  m∈Ms ow  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Here, qd denotes the  unit cost of deploying network slices at a BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Resource provisioning cost: The cost component character-  izes resource provisioning cost of radio spectrum resources,  4  edge computing resources, and cloud computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  For simplicity, we assume the unit costs of a subcarrier, an  edge VM instance, and a cloud VM instance are the same,  denoted by qr > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The resource provisioning cost is given  by Φw  p = qr  �  k∈K  �  hw  k + �  m∈M  �  ow  mbw  k,m + ow  mcw  k,m  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Slice adjustment cost: The cost component characterizes the  difference between two subsequent planning decisions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', the  cost for adjusting the amount of the reserved spectrum and  computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' For computing resources, VM instances  can be resized via advanced virtualization techniques in prac-  tical systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', Kubernetes [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Here, qs represents the  unit price of adjusting a unit of reserved network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Hence, the slice adjustment cost is given by  Φw  s =qs1  �  ow−1  k,m = 1 ∧ ow  k,m = 1  � �  k∈K  ��  hw  k − hw−1  k  �+  +  �  m∈M  ��  bw  k,m − bw−1  k,m  �+  +  �  cw  k,m − cw−1  k,m  �+��  ,  (8)  where  1 {·}  is  an  indicator  function  and  1  �  ow−1  k,m = 1 ∧ ow  k,m = 1  �  indicates that slice k is deployed  in the previous and current planning windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  SLA revenue: The cost component characterizes the benefit  caused by QoS satisfaction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', the achieved service delay of  each slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The piece-wise SLA revenue function is denoted  by  Ωk (D) =  �  �  �  �  �  �  �  qb,  if D < θ  ′  k,  qb  �  D−θ  ′  k  θk−θ′  k  �  ,  if θ  ′  k ≤ D ≤ θk,  −qp,  if D > θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (9)  Here, qb > 0 is the highest unit revenue once a slice’s SLA  is satisfied, and qp > 0 is the unit penalty once the slice’s  SLA is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Obviously, qp > qb for discouraging slice’s  SLA violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' In addition, θ  ′  k < θk represents the threshold  achieving the highest revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' For simplicity, we set θ  ′  k =  θk/2 in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The overall SLA revenue of all slices  is given by Φw  q = �  k∈K Ωk  � ¯Dw  k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Taking all cost components into account, the overall network  slicing cost in the entire slice lifecycle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', all planning win-  dows) is given by Φ (ow, Bw, Cw, hw, {xt  k, yt  k}t∈T ,k∈K) =  �  w∈W  �  Φw  d + Φw  p + Φw  s − Φw  q  �  , which is adopted to evalu-  ate network slicing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' PROBLEM FORMULATION  The network slicing problem aims to minimize the network  slicing cost via determining network planning decisions at  each planning window and network operation decisions at each  operation slot for each slice, which is formulated as:  P0 :  min  {ow,Bw,Cw,hw}w∈W  {xt  k,yt  k}t∈T ,k∈K,w∈W  �  w∈W  Φ (ow, Bw, Cw, hw)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' (1), (2), (3), (4), (5), and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' (10a)  In Problem P0, the network planning and operation decision  making are coupled in two timescales, which should be jointly  optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' To address the challenge, we first decouple the  problem into a large-timescale network planning subproblem  and multiple small-timescale network operation subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Subproblem 1: Network planning subproblem is to mini-  mize the network slicing cost across all the planning windows,  which is formulated as:  P1 :  min  {ow,Bw,  Cw,hw}w∈W  �  w∈W  Φ (ow, Bw, Cw, hw)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' (1), (2), (3), and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (11a)  Addressing the above subproblem requires network traffic  information of all planning windows, which is difficult to  be known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' To solve it, we leverage an RL method  to design a network planning algorithm, which makes online  decisions under spatial-temporally varying vehicle traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Subproblem 2: Network operation subproblem is to sched-  ule network resources of each slice to active vehicles with  random task arrivals with the objective of minimizing average  service delay, which is formulated as:  P2 : min  xt  k,yt  k  Dt  k(xt  k, yt  k)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (12a)  In the above subproblem, radio spectrum resource allocation  and task dispatching decisions jointly impact the service  delay performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' To solve the problem, we analyze the  subproblem property and design an optimization algorithm to  make real-time network operation decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' LEARNING-BASED NETWORK SLICING ALGORITHM  In this section, we solve two subproblems in Sections IV-A  and IV-B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Finally, we present the TWAS algo-  rithm for jointly optimizing planning and operation decisions  in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Network Operation Optimization  We can observe that the radio spectrum allocation de-  cision only impacts offloading delay component, and the  task dispatching decision only impacts the computation delay  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Moreover, both decisions are independent in each  BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Hence, the radio spectrum allocation and task dispatching  decisions can be optimized individually at each BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  1) Radio Spectrum Allocation Optimization: From (7), the  radio spectrum allocation optimization problem is equivalent  to minimizing the task offloading delay at each BS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=',  Pr  m : min  yt  k  �  n∈N tm  ξk  yt  k,nbw  k,mRtn  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (13a)  The objective function can be proved to be convex since its  second-order derivative is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' In addition, the constraint  is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Hence, problem Pr  m is a convex optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Using the Karush-Kuhn-Tucker conditions [15], the  optimal radio spectrum resource allocation decision is  (yt  k,n)⋆ =  �  1/Rtn  �  i∈N tm  �  1/Rt  i  , ∀n ∈ N t  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (14)  5  2) Task Dispatching Optimization: Similarly, from (7), task  dispatching optimization is to minimize the task processing  delay, which is formulated as:  Pw  m : min  xt  k,m  dt  k,m,e  �  At  k,m − xt  k,m  �  + dt  k,m,cxt  k,m  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (15a)  The above objective function can be rewritten as  Ψ(xt  k,m) = dt  k,m,e  �  At  k,m − xt  k,m  �  + dt  k,m,cxt  k,m  = ν1ξk  2  (xt  k,m)2 +  �  νt  2 − ν1ν3 − ξkAk,mν1  2  �  xt  k,m  + ν1νt  3At  k,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (16)  Here, ν1 =  ηk  cw  k,mFe  > 0, νt  2 = dt  r +  ηkξk  hw  k Fc , and ν3 =  Qk,m + Ak,m+1  2  ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Since the second-order derivative of the  objective function ∂2Ψ(xt  k,m)/∂2xt  k,m = νt  1ξk > 0, the  problem is a convex optimization problem [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The optimal  task dispatching decision is given by  (xt  k,m)⋆ = 2νt  2 + ξkν1Ak,m − 2ν1νt  3  2ν1ξk  , ∀m ∈ Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  (17)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Network Planing Optimization  The network planning problem is a stochastic optimization  problem to minimize the network slicing cost, which can be  transformed into a Markov decision process (MDP) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The  components of the MDP are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  1) Action, which is consistent with planning decisions,  including slice deployment, radio spectrum and computing  resource provisioning at BSs, and cloud computing resource  provisioning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', Aw  =  {ow, Bw, Cw, hw}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The action  dimension is Ms + 2KM + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  2) State, which includes average vehicle traffic density in  the current planning window and the planning decisions in the  previous window due to the switching cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The entire area is  divided into J disjoint regions, and the average vehicle traffic  density of all regions is denoted by Λw ∈ RJ×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' As such, the  state is given by Sw = {Λw, ow−1, Bw−1, Cw−1, hw−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The  state dimension is 2KM + M + K + J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  3) Reward, which is defined as the inverse of the net-  work slicing cost in the current planning window, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=',  Rw (Sw, Aw) = −Φ (ow, Bw, Cw, hw) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Note that minimiz-  ing the network slicing cost is equivalent to maximizing the  cumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Upon state Sw, the learning agent takes action Aw, and the  corresponding reward Rw (Sw, Aw) is obtained, along with  the state evolves into new state Sw+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' With the above setting,  our goal is to obtain an optimal planning policy π⋆ ∈ Π  which makes decisions based on the observed state, thereby  maximizing the expected long-term cumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' As  such, problem P2 can be formulated as the following MDP:  P′  2 : max  π∈Π E  �  lim  W →∞  W  �  w=1  (ϕ)wRw (Sw, Aw) |π  �  ,  (18a)  where ϕ > 0 is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Since vehicle traffic density  is continuous, the action-state space can be prohibitively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  To address this issue, an RL algorithm can be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Algorithm 1: TAWS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  1 for training episode =1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' do  2  for planning window w = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', W do  3  Generate planning decisions via the actor network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  4  for each slice in parallel do  5  for operation slot t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', T do  6  for each BS in parallel do  7  Make radio spectrum allocation and task  dispatching decisions by (14) and (17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  8  Calculate the instantaneous service delay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  9  Measure the average service delay within the  planning window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  10  Collect vehicle traffic density of all regions, and  observe reward Rw and new state Sw+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  11  Store transition {Sw, Aw, Rw, Sw+1} in the  experience replay buffer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  12  Sample a random minibatch of transitions from the  experience replay buffer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  13  Update the weights of neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Proposed TAWS Algorithm  We present the TAWS algorithm to jointly solve the entire  network slicing problem P0, collaboratively integrating RL  and optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The core idea of TAWS is to adopt  an RL method for network planning decision making and an  optimization method for network operation decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  The service delay performance is measured at the end of each  planning window and then incorporated into the reward in  the RL framework, such that the interaction between network  planning and operation stages can be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The TAWS  algorithm is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  The RL method is based on the deep deterministic policy  gradient (DDPG) algorithm [16], [17], which consists of  four neural networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', actor evaluation network, critic  evaluation network, actor target network, and critic target  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' In the initialization phase, all neural networks and  the environment are initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The procedure of the TAWS  is two-step: 1) Network slicing decisions are generated and  executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The actor network outputs the planning decisions  Aw, which is clipped to feasible decision space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The network  operation decisions are generated via the optimization method,  and the service delay performance is measured at the end  of each planning window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The reward Rw can be obtained  and the new state can be observed Sw+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The transition tuple  {Sw, Aw, Rw, Sw+1} is stored in the experience replay buffer  for updating neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' and 2) Neural networks are  updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' A mini-batch of transitions are randomly sampled  from the experience replay buffer to update the weights of  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Specifically, the critic network is updated  by minimizing the loss function, and the actor network is  updated via the policy gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Then, actor and critic  target networks are updated by slowly copying the weights of  evaluation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' SIMULATION RESULTS  We evaluate the performance of the proposed algorithm on  real-world vehicle traffic traces in urban vehicular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  We consider a 1,000×1,000 m2 simulation area, which is  6  Table I  SIMULATION PARAMETERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Parameter  Value  Parameter  Value  No  −174 dBm  I  −164 dBm  Pv  27 dBm  β  20 MHz  dr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='15 sec  J  16  To  1 sec  Tp  10 min  Fc  100 GHz  Fe  10 GHz  Bm  10  Cm  10  ξ1, ξ2  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='6, 2} Mbit  η1, η2  {1000, 200} cycles/bit  θ1, θ2  {100, 200} ms  θ  ′  1, θ  ′  2  {50, 100} ms  0  100  200  300  400  500  Training Episodes  1000  1500  2000  2500  3000  3500  4000  4500  Overall System Cost  Five-Point Moving Average  (a) Convergence  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='8  2  Task Arrival Rate (Packet/sec)  0  200  400  600  800  1000  1200  1400  1600  Overall System Cost  Proposed  Short Term Optimization  (b) Network slicing cost  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  Performance of the proposed TWAS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  covered by two SBSs and an MBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Each SBS has a coverage  radius of 300 m, and the MBS located in the centre covers  the entire simulation area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The vehicle traffic density of the  simulation area is measured by a unit of a small region  of 250×250 m2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=', J = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' This dataset is collected by  Didi Chuxing GAIA Initiative2 and contains vehicle traces in  the second ring road in Xi’an collected from taxis that are  equipped with GPS devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The periods of a planning window  and an operation slot are set to 10 minutes and 1 second,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The period of the slice lifecycle is set to 4  hours, including 24 planning windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The task arrivals of  two services both follow Poisson processes with different task  arrival rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' We construct two slices for supporting two types  of delay-sensitive services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' One is an object detect service  whose service delay requirement is 100 ms, while the other  is an in-vehicle infotainment service whose service delay  requirement is 200 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Regarding the TWAS algorithm, the  neuron units in hidden layers of both actor and critic networks  are set to 128 and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Important simulation parameters are  summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' 2(a), we present the overall network slicing  cost with respect to training episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' All simulation points are  processed by a five-point moving average in order to highlight  the convergence trend of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' It can be  seen that the proposed algorithm converges after 500 training  episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' 2(b), we compare the performance of  the proposed algorithm and a short term optimization bench-  mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The basic idea of the benchmark is to minimize the  network slicing cost at each individual planning window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Since  planning decisions are discrete, a simple exhaustive searching  method is adopted to obtain the optimal one-shot planning  decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Firstly, it can be seen that the proposed algorithm  can greatly reduce the network slicing cost as compared to the  benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Specifically, when the task arrival rate is 2 packets  per second, the proposed algorithm can reduce the network  slicing cost by 23%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The reason is that the proposed algorithm  takes the switching cost between two consequent planning  windows into account, while the benchmark scheme does  2Didi Chuxing Dataset: https://gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='didichuxing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Secondly, the overall network slicing cost increases with  the increase of the task arrival rate, because more radio and  computing resources are consumed in heavy traffic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' CONCLUSION  In this paper, we have investigated a network slicing prob-  lem in edge-cloud orchestrated vehicular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' A two-  stage network slicing algorithm, named TWAS, has been  proposed to jointly make network planning and operation  decisions in an online fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' The TAWS can adapt to  network dynamics in different timescales, including spatial-  temporally varying vehicle traffic density and random task  arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' Simulation results demonstrat that the TAWS can re-  duce the network slicing cost as compared to the conventional  scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
+page_content=' For the future work, we aim to determine the optimal  planning window size for minimizing the network slicing cost  under vehicular network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE1T4oBgHgl3EQfrwV8/content/2301.03358v1.pdf'}
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+KEK-TH-2493
+Sub-GeV dark matter search at ILC beam dumps
+Kento Asai(a,b), Sho Iwamoto(c), Maxim Perelstein(d),
+Yasuhito Sakaki(e,f), and Daiki Ueda(g,h)
+(a) Institute for Cosmic Ray Research (ICRR), The University of Tokyo, Kashiwa,
+Chiba 277–8582, Japan
+(b) Department of Physics, Faculty of Engineering Science, Yokohama National
+University, Yokohama 240–8501, Japan
+(c) Institute for Theoretical Physics, ELTE E¨otv¨os Lor´and University,
+Budapest H-1117, Hungary
+(d) Department of Physics, LEPP, Cornell University, Ithaca, NY 14853, USA
+(e) Radiation Science Center, High Energy Accelerator Research Organization (KEK),
+Ibaraki 305–0801, Japan
+(f) The Graduate University for Advanced Studies (SOKENDAI),
+Hayama 240–0193, Japan
+(g) Theory Center, High Energy Accelerator Research Organization (KEK),
+1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
+(h) Center for High Energy Physics, Peking University, Beijing 100871, China
+Abstract
+Light dark matter particles may be produced in electron and positron beam dumps
+of the International Linear Collider (ILC). We propose an experimental setup to
+search for such events, the Beam-Dump eXperiment at the ILC (ILC-BDX). The
+setup consists of a muon shield placed behind the beam dump, followed by a multi-
+layer tracker and an electromagnetic calorimeter. The calorimeter can detect electron
+recoils due to elastic scattering of dark matter particles produced in the dump, while
+the tracker is sensitive to decays of excited dark-sector states into the dark matter
+particle. We study the production, decay and scattering of sub-GeV dark matter
+particles in this setup in several models with a dark photon mediator. Taking into
+account beam-related backgrounds due to neutrinos produced in the beam dump as
+well as the cosmic-ray background, we evaluate the sensitivity reach of the ILC-BDX
+experiment. We find that the ILC-BDX will be able to probe interesting regions of
+the model parameter space and, in many cases, reach well below the relic target.
+arXiv:2301.03816v1  [hep-ph]  10 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Beam dump experiment
+4
+3
+Expected backgrounds
+7
+3.1
+Beam-induced background . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.2
+Cosmic-ray background . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+4
+Examples of detectable DM models
+11
+4.1
+Pseudo-Dirac DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+4.1.1
+Small mass-splitting
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+4.1.2
+Large mass-splitting
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+4.2
+Scalar elastic DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+4.3
+Scalar inelastic DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+4.4
+Majorana DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+4.5
+Existing constraints and projected sensitivities at other experiments . . . .
+18
+5
+Summary
+19
+A Dark photon production cross sections
+21
+A.1 Pair-annihilation production . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+A.2 Bremsstrahlung production
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+B DM-electron recoil cross sections
+22
+C Neutrino-induced background
+23
+C.1 Irreducible background . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+24
+C.2 Reducible background
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+1
+Introduction
+The International Linear Collider (ILC) has been proposed as the next energy-frontier
+facility in particle physics. The physics program of the ILC is well established [1–3]: it in-
+cludes measurements of the Higgs boson couplings with unprecedented precision, searches
+1
+
+for new physics in rare Higgs decays and other channels, and precise top-mass determina-
+tion, among other topics. Most studies of the physics potential of the ILC to date focus on
+experiments with the detector situated at the main interaction point, where the electron
+and positron beams collide head-on, yielding the maximum collision energy.
+Recently, it was suggested that, in parallel with the experiments at the main interaction
+point, the ILC can pursue a complementary physics program using its beam dumps [4–10].
+Particle collisions within the beam dumps occur in fixed-target kinematics, with typical
+effective energies of O(10) GeV or below. At the same time, since every beam-electron
+and beam-positron interacts with the dump, the beam-dump experiments can accumulate
+enormous integrated luminosity. This makes the beam dump the ideal place to search for
+relatively light new particles with very small couplings to the Standard Model (SM).
+The new physics targeted by the beam-dump experimental program is very well mo-
+tivated theoretically [11, 12]. Previously studied examples include visibly-decaying dark
+photons, axion-like particles, vector bosons associated with gauged lepton flavor symme-
+tries, and heavy neutral leptons. In all these cases, it was demonstrated that beam-dump
+experiments at the ILC can probe model parameters well beyond the reach of the currently
+available experiments and are competitive with proposed future dedicated facilities.
+In this paper, we propose a new search for light dark matter using the ILC beam dumps.
+It is well known that a stable particle with mass in the MeV–GeV range, interacting with
+the SM via a dark photon mediator, is an attractive dark matter (DM) candidate. With
+reasonable model parameters, its thermal relic density matches the observed DM abundance
+and it is consistent with all astrophysical and experimental constraints. The DM particles
+can be pair-produced at the ILC beam dump via virtual dark-photon exchanges or in decays
+of dark photons. An electromagnetic calorimeter placed ∼ 100 m downstream of the beam
+dump, behind a lead shield is designed to search for the DM by detecting elastic scattering
+of the DM particles on the electrons in the detector material. In addition, in many models
+such as inelastic DM [13], the DM particle is the lowest-lying state of a multiplet with small
+mass splittings. A long-lived excited state in the same multiplet as the DM, produced at
+the beam dump, may propagate through the lead shield and decay into the DM and charged
+SM particles. The charged particles can then be detected by a tracker, providing a “visible
+decay” signature of the DM. We will estimate the experimental reach in both electron-recoil
+and visible-decay signal.
+The proposed experimental setup is illustrated in Fig. 1. The concept of this experiment
+is the same as the Beam-Dump eXperiment (BDX) proposed at JLab [14], and we, therefore,
+call our proposal “ILC-BDX.” The advantage of the ILC, compared to the original BDX
+proposal, is the higher beam energy as well as availability of both electron and positron
+beams.#1 We also note that the experimental setup used here is essentially identical to
+#1An additional advantage is the availability of polarized beams at the ILC. We will not explore conse-
+2
+
+ldec
+Beam dump
+Muon shield
+lsh
+ldump
+rdet
+Detector
+z
+Lead
+Concrete
+Water
+e±
+ldet
+Decay volume (Multi-layer tracker)
+rdec
+e+
+e−
+χ
+χ
+e±
+Nucleus
+e±
+χ
+χ
+γ*
+A′ 
+A′ 
+e−
+e−
+A′ *
+χ
+χ
+A′ *
+e−
+e+
+χ2
+χ1
+Figure 1: The ILC-BDX experimental setup consists of the main beam dump, a muon
+shield, a decay volume, and a detector. A multi-layer tracker is placed in the decay volume
+to measure the charged tracks. The DM particles may be produced at electron and positron
+beam dumps via pair-annihilation of positron on an atomic electron, or via bremsstrahlung.
+The resulting DM particles can scatter off electrons in the detector and yield observable
+electron-recoil events. In models where the DM particle is part of a multiplet, an excited
+DM state may be produced. It can then decay into a lighter DM state and SM particles in
+the decay volume, producing a visible signature in the tracker.
+that proposed previously for searches for visible dark photons and axion-like particles.
+(Detailed design of the detector may be different to optimize the sensitivity for various
+searches, but this is outside the scope of this study.) The ILC-BDX can be pursued in
+parallel to the previously discussed searches, adding to the rich physics program at the ILC
+beam dumps. It is important to emphasize that this program can be pursued in parallel
+with the exploration of the Higgs and electroweak physics at the main IP, and can broaden
+the capabilities of the ILC to search for physics beyond the Standard Model (BSM) at a
+modest additional cost.
+The rest of the paper is organized as follows. In Section 2, we introduce the setup
+of our proposed experiment for sub-GeV DM search at the ILC beam dumps, and list
+formulae used to calculate the number of expected signal events. In Section 3, we discuss
+the two types of background (BG) events that occur in this setup, i.e., beam-induced and
+cosmic-ray backgrounds, and estimate the number of events expected from each BG. In
+Section 4, we introduce five DM models used as benchmarks, i.e., pseudo-Dirac DM with
+small and large mass-splitting, scalar elastic and inelastic DM, and Majorana DM. We
+then present our estimates of the ILC-BDX sensitivity reach for each model. The main
+findings of our analysis are summarized in Section 5. The Appendix contains the formulae
+for cross sections of the relevant dark-photon production processes (Appendix A) and the
+quences of polarization in this paper, but note that it can be particularly useful in characterizing the new
+physics if a non-SM signal is observed.
+3
+
+DM-electron elastic scattering (Appendix B), as well as some details of our estimates of
+neutrino-induced BG rates (Appendix C).
+2
+Beam dump experiment
+We adopt the same experimental setup as that of Ref. [9], which is illustrated in Fig. 1.
+For both electron and positron beams, the ILC main beam dumps are planned as absorbers
+consisting of water cylinders along the beam axes with the length of ldump = 11 m [15].
+The proposed setup consists of a muon shield with the length of lsh = 70 m made of lead, a
+cylindrical decay volume with ldec = 50 m and radius of rdec = 3 m with a multi-layer tracker
+installed, and a cylindrical detector with the radius of rdet = 2 m and the length of ldet =
+0.64 m, behind either of the beam dumps. The multi-layer tracker is designed to detect
+the visible-decay signal, while the cylindrical detector is assumed to be an electromagnetic
+(EM) calorimeter made of CsI(Tl) scintillating crystals (n(det)
+e−
+= 1.1 × 1024 cm−3) designed
+to detect recoil electrons caused by incoming boosted DM particles (electron-recoil signals).
+We focus on the 250 GeV ILC (ILC-250) with the beam energy of Ebeam = 125 GeV in
+the lab frame, and the number of incident electrons and positrons into the beam dump of
+Ne± = 4×1021/year [1,16–19]. As studied comprehensively in Refs. [5,6,9,20], this setup was
+found very sensitive to long-lived BSM particles decaying into visible SM particles thanks
+to its thick shield. We will see that, with the calorimeter as a recoil-electron detector, it is
+also sensitive to DM particle production.
+We study two types of signal events associated with different production mechanisms of
+DM particles. If the DM particles are produced in the water beam dump by a BSM interac-
+tion (such as dark photons), they are highly boosted and, passing through the muon shield,
+scatter off electrons in the calorimeter. Such electron recoils, detected by the calorimeter,
+are a typical signal of DM production. On the other hand, the DM particles may also be
+produced in in-flight decays of heavier DM-sector particles if, for example, the DM is the
+lowest-lying state of a multiplet. The heavier particle, which we denote by χ2, can pass
+through the muon shield and then eventually decay into a DM particle χ1 and SM particles.
+If the decay happens in the decay volume, charged tracks may be observed in the tracker.
+This visible-decay signature is used as an additional signal of DM production.
+In this work, we focus on five DM models with the dark photon mediator to provide a
+benchmark study. They are pseudo-Dirac-fermion DM with small or large mass-splitting,
+scalar elastic DM, scalar inelastic DM, and Majorana-fermion DM. The details of each
+model are discussed in Section 4. Among the models, the pseudo-Dirac DM with small
+mass-splitting, scalar DM, and Majorana DM can be observed as electron-recoil events.
+Meanwhile, the pseudo-Dirac DM with large mass-splitting can produce visible-decay events
+in addition to electron-recoil signature; specifically, due to the pseudo-Dirac nature, a
+4
+
+produced dark photon A′ decays mainly into χ2 ¯χ1 (or χ1 ¯χ2) and the heavier DM-sector
+particle χ2 may decay visibly in the decay volume.
+The number of signal events is schematically given by
+Nsignal = Ne± × li × nj × σij→A′→DM × Acc,
+(2.1)
+where we consider a particle i in the shower interacting with a particle j in the material
+of the beam dump to produce DM particles through an on-shell dark photon A′ as the
+mediator. The track length le± of a shower electron and positron is provided in Ref. [6],
+nj is the number density of j, and Acc denotes the detector acceptance discussed below.
+More specifically#2,
+N pair
+signal = Ne±
+�
+dEe+ dle+
+dEe+ · ne− · σ(e+e− → A′) · Br(A′ → χ¯χ) · Acc,
+(2.2)
+N brems
+signal = Ne±
+�
+i=e−,e+
+�
+dEi
+dli
+dEi
+· nN
+�
+dEA′
+� π
+0
+dθA′ d2σ(iN → iA′N)
+dEA′dθA′
+· Br(A′ → χ¯χ) · Acc
+(2.3)
+for pair-annihilation e+e− → A′ and bremsstrahlung e±N → e±A′N with a target nucleus
+N, respectively. The cross sections σ on the right-hand side are provided in Appendix A.
+Here θA′ is the emission angle of A′ with respect to the direction of the e± beam in the lab
+frame. In all the models we consider, it is assumed that dark photons decay exclusively
+into the DM-sector particles, i.e., Br(A′ → χ¯χ) = 1, with χ being a DM sector particle
+(χ1 and/or χ2). This is justified since decays to SM final states are suppressed by a small
+mixing parameter ϵ2.
+Let us estimate the acceptance for each type of signal. For a visible-decay event to be
+observed, the visible decay of χ2 must occur in the decay volume. Noting that A′ mainly
+decays into a χ1-χ2 pair in the models we consider, we approximate the acceptance by
+Acc(decay) =
+� rdec/(ldump+lsh)
+0
+dθχ
+� ldec
+0
+dzdPang
+dθχ
+· dPdec
+dz
+· Θ(rdec − rdec
+⊥ ).
+(2.4)
+Here, we assume that the decay A′ → χ2 ¯χ1 (or χ1 ¯χ2) is immediate and χ2 exclusively
+decays into electrons because of the small mass difference ∆ ≡ mχ2 − mχ1 < 2mµ. The lab
+frame distribution of the χ2 emission angle θχ is approximated by#3
+dPang
+dθχ
+= sin θχ · 1
+2
+�
+mA′
+EA′ − pA′ cos θχ
+�2
+,
+(2.5)
+#2In Ref. [9], this calculation scheme is referred to as the coarse-grained integration method.
+#3The angular distribution in the CM-frame is given by dPang/d cos θCM
+χ
+= 1/2 when the polarization of
+the dark photon is averaged.
+5
+
+and dPdec/dz denotes the probability of χ2 to decay at the position z (the horizontal axis
+in Fig. 1), i.e.,
+dPdec
+dz
+=
+1
+l(lab)
+χ2
+exp
+�
+−ldump + lsh + z
+l(lab)
+χ2
+�
+,
+l(lab)
+χ2
+= pχ2
+mχ2
+1
+Γχ2
+(2.6)
+with l(lab)
+χ2
+denoting the lab frame decay length of χ2. The lab-frame momentum of χ2 is
+approximated by pχ2 ≈ (EA′ + pA′ cos θχ)/2, where the mass-splitting is neglected. The
+momentum of A′ is obtained by pA′ =
+�
+E2
+A′ − m2
+A′, and Γχ2 (mχ2) is the total decay width
+(the mass) of χ2. The Heaviside function Θ in Eq. (2.4) governs the radial requirement on
+the decay position of χ2, i.e., the radial deviation
+rdec
+⊥
+≈
+�
+θ2
+e + θ2
+A′ + θ2
+χ · (ldump + lsh + z)
+(2.7)
+must be smaller than the radius rdec of the multi-layer tracker. In Eq. (2.7), θe is the angle
+of the beam-oriented e± with respect to the beam axis, θA′ is the production angle of A′
+and is equal to 0 for pair-annihilation, and θχ is the emission angle of χ2 at the decay of
+A′. We estimate θe by Monte Carlo simulations [6] and use the mean value
+θe = 16 mrad · GeV/Ee±.
+(2.8)
+Similarly, we estimate the acceptance for electron-recoil signal as
+Acc(recoil) =
+� rdet/(ldump+lsh+ldec)
+0
+dθχ
+dPang
+dθχ
+· Θ(rdet − rrec
+⊥ ) · Precoil,
+(2.9)
+where the radial deviation is approximately given by
+rrec
+⊥ =
+�
+θ2
+e + θ2
+A′ + θ2
+χ · (ldump + lsh + ldec),
+(2.10)
+and Precoil is the probability of electron recoil
+Precoil = n(det)
+e− ldet
+� E+
+e
+E−
+e
+dER
+dσrecoil
+dER
+· Θ(ER − Emin)
+(2.11)
+given by the electron number density n(det)
+e−
+of the detector, the length ldet of detector,
+kinematically allowed maximum (minimum) recoil energy E+
+e (E−
+e ) given by Eq. (B.4),
+and the effective recoil cross section approximated by
+dσrecoil
+dER
+≈ dσ(χ1e− → χ2e−)
+dER
++ dσ(χ2e− → χ1e−)
+dER
+exp
+�
+−ldump + lsh + ldec
+l(lab)
+χ2
+�
+(2.12)
+for models with χ2 and
+dσrecoil
+dER
+≈ 2 × dσ(χe− → χe−)
+dER
+(2.13)
+for the other models. The analytical formulae for the differential cross section of the DM-
+electron scattering are shown in Appendix B. Here, the electron recoil energy ER is required
+to be larger than the threshold Emin = 1 GeV to reduce BG events.
+6
+
+E d�
+dE
+[1/cm2/incident-e]
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+Figure 2: The beam-induced neutrino fluxes at the end of the muon shield (left) and at the
+EM calorimeter location (right).
+3
+Expected backgrounds
+Background (BG) events in this experiment are classified into two types: beam-induced
+and beam-unrelated. The beam-induced BG events arise from the SM particles produced
+by the injected beam at the main beam dump, while the beam-unrelated ones are mainly
+due to cosmic-ray muons.
+3.1
+Beam-induced background
+The beam injected into the main beam dumps produces SM particles. In addition to light
+particles such as pions, neutrons, and muons, because of the initial high-energy beams,
+tau-leptons and heavy mesons (D, B, and Bc) are also produced by the shower photon
+hitting nuclei [9]#4. Both light and heavy particle decays can produce neutrinos, which
+pass through the muon shield and reach the detectors to generate the beam-induced BG
+events for both the electron-recoil and visible-decay signals.
+The neutrino fluxes are calculated with Monte Carlo simulation. We use PHITS 3.25 [21]
+for production and transport of SM particles other than heavy mesons. For heavy meson
+production, the differential production cross sections obtained by PYTHIA 8.3 [22] are
+implemented into PHITS; see Ref. [9] for details. In Fig. 2, we show the neutrino fluxes per
+#4Heavy meson production through the electromagnetic-shower photons is taken into account, which is
+the dominant channel of heavy meson production in our setup.
+7
+
+electron injection at the end of the muon shield in the left panel and at the detector behind
+the decay volume in the right panel. Tau-type neutrino fluxes are negligible, compared
+to electron- and muon-types, because the beam energy is not large enough to produce a
+considerable number of source particles such as tau-leptons and Ds. Due to the stopping
+power of pions, many pions at rest are produced in the beam dump so that the neutrino flux
+increases in the low-energy regions. The electron-type neutrinos in a high-energy regime
+mainly arise from the decays of Ds.
+A source of irreducible BG events for the electron-recoil searches are neutrino-electron
+scattering,#5
+νe− → νe−,
+¯νe− → ¯νe−,
+(3.1)
+where beam-originated neutrinos scatter on the atomic electrons in the detector material.
+The recoil electrons produce electromagnetic showers in the calorimeter, which are difficult
+to distinguish from the DM-electron recoil events. From the neutrino flux in the right panel
+of Fig. 2, the number of neutrino-originated electron-recoil events is estimated to be around
+1/year by imposing Emin = 1 GeV; see Appendix C for details. Neutrino-nucleon scattering
+is another source of beam-induced BG for the electron-recoil search. We list the possible
+processes below. Note that, in contrast to the neutrino-electron scattering, these processes
+provide recoil nucleons in the calorimeter, making this component of the BG potentially
+reducible depending on the design of the calorimeter.
+• Quasi-elastic scattering — Neutrinos produced at the beam dump or in the muon
+shield may interact with material in the calorimeter through the following quasi-
+elastic scattering processes:
+CC :
+νℓn → ℓ−p,
+¯νℓp → ℓ+n,
+NC :
+νp → νp,
+¯νp → ¯νp,
+νn → νn,
+¯νn → ¯νn,
+where CC (NC) denotes charged (neutral) current interaction. Among these, the
+processes νen → e−p and ¯νep → e+n will produce an electromagnetic shower in the
+calorimeter and thus potentially be misidentified as an electron-recoil signal. The
+number of such events is conservatively estimated as 2×102/year using the simulated
+neutrino flux (Fig. 2 right); see Appendix C for details. Other processes are without
+electromagnetic showers, and thus we expect them to be distinguishable from the
+signal events.
+• Neutral pion production — In the detector, muon-type neutrinos may produce reso-
+nant neutral pions (π0) decaying into photons:
+CC :
+νµn → µ−pπ0,
+¯νµp → µ+nπ0,
+#5The processes νµe− → µνe and ντe− → τνe do not contribute because the neutrino flux above the
+threshold energy, Eν ≥ (m2
+µ − m2
+e)/2me ≃ 10.8 GeV and Eν ≥ (m2
+τ − m2
+e)/2me ≃ 3 TeV, is negligible.
+8
+
+NC :
+νµp → νµpπ0,
+¯νµp → ¯νµpπ0,
+νµn → νµnπ0,
+¯νµn → ¯νµnπ0.
+The electromagnetic showers from the photons, which is accompanied by a recoil
+nucleon, may be misidentified as electron recoils. As detailed in Appendix C, the
+number of the single-π0 production is conservatively estimated as 2 × 103/year from
+the flux (Fig. 2 right), but this BG may be reducible depending on the calorimeter
+design by, for example, using the information on the radius of the electromagnetic
+shower.
+Since these BG events are due to misidentification of electron recoils, we postpone quanti-
+tative studies to the detector-design stage. Instead, in our plots displaying the ILC-BDX
+sensitivity of the electron-recoil searches, we show curves corresponding to 10, 100, and
+1000 signal events in a 10-year ILC run. These choices illustrate a range of plausible sce-
+narios for the degree to which beam-induced background from neutrino-nucleon scattering
+can be controlled in practice.
+Lastly, we consider the potential beam-induced BG events for the visible-decay signal.
+The main source will be the following processes involving strange mesons:
+CC :
+νµn → µ−K0p,
+νµn → µ−K0Σ+,
+NC :
+νµn → νµK0Λ0,
+νµp → νµK0Σ+,
+νµn → νµK0Σ0.
+Neutral kaons produced at the end of the muon shield and the surrounding walls of the
+decay volume can decay in the decay volume to yield charged tracks mimicking the visible-
+decay signal. In Ref. [23] of the SHiP experiment, this class of neutrino-induced BG events
+is evaluated using Geant4 [24]. We infer the number of charged tracks from neutral kaon
+decays in the decay volume by comparing the number of neutrinos between the ILC beam
+dump and the SHiP experiment. In the SHiP experiment with 2 × 1020 protons on target,
+7 × 1017 neutrinos with momentum between 2 GeV and 100 GeV are expected, which
+results in ∼ 104 pairs of charged tracks from neutral kaon decays; 99.4% of them are
+rejected by using the topology of oppositely-charged two tracks thanks to the fact that
+the BG-originated tracks generally do not point to the beam-target interaction points [23],
+and around 60 events are expected to remain as the BG#6. At the ILC-BDX, the number
+of neutrinos with momentum between 2 GeV and 100 GeV is estimated as 1.4 × 1017 per
+10-year run (cf. Fig. 2 left), and the ILC-BDX expects 12 BG events with pairs of charged
+tracks per 10-year run. Consequently, in the following analysis of ILC-BDX visible-decay
+searches, we expect 12 BG events and show the expected exclusion limit sensitivity with
+95% confidence-level (C.L.), which corresponds to 7.4 signal events in a 10-year run.
+#6The SHiP experiment has a veto system to reduce the BG further. We do not discuss it because the
+reduction strongly depends on the detector setup.
+9
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+Kinetic energy [GeV]
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+CwxZp5EtjNhODPvawvzs9owx+mvUSFUliMo
+Xl0zSXFlC7CphOhgaOcW2BcC/tWymdM472
+S6oQzpeiFfRPajgPmhBuet3guHt61etcdOs
+41sku2SMHJCB9ckH+kEsyIJw8O6z4Ww6L27
+L3Xa9qtV16pkf5I3cn/8ByQixVw=</late
+xit>
+Cosmic muon flux
+<latexit sha1_bas
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+=">AB+XicbZBLS8NAFIUnPmt9RV26GS
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+RL6D9x40IRt/4Td/4b0zSIrwMDH+fcy1x
+OGDOqjed9OCura+sbm4Wt4vbO7t6+e3DY
+1tIqTFpYMqm6IdKEUFahpGurEiIeM
+dMJY5F3pkRpKsWdmcUk4GgkaEQxMqk1c
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+NUcuO/9ocSWE2EwQ1r3fC82QYKUoZiReb
+FvNYkRnqAR6aUoECc6SL5/A0dYwki
+p9wsDM/b6RIK71jIfpJEdmrH9nC/O/rGd
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+XADmqAFMJiCB/AEnp3EeXRenNfl6IqT7
+xyBH3LePgF6rpPk</latexit>
+at sea level
+<latexit sha1_base64="YWgWeCRY0yn
+RdXyaZAjmo8VAiIw=">AB83icbZBLS8NAFIVv6qvWV9Wlm8EiIuSVKV2V3Djso
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+0MhElC/yKHhfJVwX6s659XL21qleZbXUYQjOIZTcKAOTbiBFrSBQP8ATPVmw9W
+i/W63K0YOU7h/BD1tsnwUmR1Q=</latexit>
+at beam dump area
+<latexit sha1_base64="jL2NRA9kIJI
+sYSbiQdhAXAgOiYM=">AB+nicbVDLSsNAFJ34rPWV6tLNYBHERUmqUrsruHFZwT
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+Xvr0XqxXhfRJSv7KMfsN4+AZ5Sk+s=</latexit>
+Figure 3: The cosmic-muon fluxes at sea level (dashed line) and the beam dump area (black
+band). The uncertainly in the latter flux is due to the indeterminate density of subsurface
+materials in the Kitakami Mountains at the experimental site. The subsurface materials
+are assumed to be soil and granite, with an average density of 2.0 to 2.4 g/cm3.
+3.2
+Cosmic-ray background
+The beam-unrelated BG is mainly due to cosmic rays. Figure 3 shows the fluxes of cosmic
+muons at sea level and the beam dump area evaluated by EXPACS [25–27] and PHITS [21]#7.
+The kinetic energy loss of the cosmic muon from the ground surface to a depth of ∼120
+meters below the ground surface is estimated as ρ × ⟨dE/dx⟩ × 120 m ∼ 50 GeV with the
+mass density of the ground ρ ∼ 2.2 g/cm3 and the stopping power of muon ⟨dE/dx⟩ ∼
+2 MeV · cm2/g. Due to this large energy loss, the muon flux at the beam dump area is 200
+times smaller than that on the ground. Then, the number of the cosmic-muon BG events
+for 10 years is estimated as
+N BG
+cos ∼ O(10) · ϵveto.
+(3.2)
+This estimate arises from the following factors:
+O(10) ∼ 10 year
+(operation time)
+× 10−4 muon/cm2/s
+(cosmic muon flux at beam dump area)
+× 1002 cm2
+(detector area from top view)
+× 10−4/muon
+(hit rate per cosmic muon)
+× 1312 × 5 bunch/s
+(bunch number per second)
+× 100 ns/bunch.
+(time window per bunch)
+#7Muon-induced neutrons avoid the muon veto and may become a BG event. Evaluation of this effect is
+left for future work.
+10
+
+Time
+1312 bunches
+1312 bunches
+0.73 ms
+200 ms
+100 ns
+~600 ns
+time window
+Figure 4: A schematic picture of the ILC beam [17,18]. The number of bunches per pulse
+is 1312, the beam pulse length is 0.73 ms, the pulse repetition rate is 1/200 ms−1 = 5 Hz,
+and the bunch spacing is ∼600 ns. The number of the beam-unrelated BG events can be
+reduced by imposing 100 ns time window per bunch.
+The hit rate is the probability that a cosmic-ray muon on an iron block of size 10 cm ×
+10 cm×10 cm will cause an energy deposition above the threshold value of 1 GeV, which is
+evaluated in the Monte Carlo simulation. Detectors with smaller cell sizes are sufficiently
+realistic, and the above estimation of the hit rate is conservative.
+The time structure
+of the ILC bunches [17, 18] is represented in Fig. 4.
+The factor ϵveto is the reduction
+factor by the cosmic-muon veto, which is typically much smaller than 10%.
+The deep
+underground location of the detector and coincidence time window significantly reduce the
+beam-unrelated BG. Consequently, we will neglect the cosmic-muon BG in the rest of this
+study.
+4
+Examples of detectable DM models
+We evaluate the sensitivity of the ILC electron and positron beam dump experiments to DM
+particles using the formulae in Section 2. Taking into account the beam-induced BG events
+coming from the SM neutrinos, we show the prospects of the ILC-BDX for electron-recoil
+and visible-decay searches. For the electron-recoil searches, we illustrate parameter spaces
+in which more than 10, 102, and 103 signal events are expected at a 10-year ILC-BDX run.
+For the visible-decay searches, regions with more than 7.4 signal events in a 10-year run are
+shown, which corresponds to a 95% C.L. exclusion. Throughout this work, we assume that
+the beam-unrelated BG events are negligible, as indicated by the estimates in Section 3.
+As benchmark models, we consider a class of DM models in which the DM field is
+charged under a new “dark” gauge symmetry, U(1)D.
+At low energy, where U(1)D is
+spontaneously broken and the dark photon A′ acquires a mass mA′, the relevant terms of
+11
+
+the Lagrangian are given by
+L ⊃ −1
+4FµνF µν − 1
+4F ′
+µνF ′µν + 1
+2m2
+A′A′
+µA′µ − ϵ
+2F ′
+µνF µν − gDA′
+µJµ
+χ − eAµJµ
+EM ,
+(4.1)
+where Fµν (F ′
+µν) is the photon (dark photon) field strength, gD is the U(1)D gauge coupling,
+Jµ
+χ (Jµ
+EM) is the DM (electromagnetic-matter) current, and ϵ parametrizes the kinetic mixing
+between the photon and the dark photon. By the redefinition Aµ → Aµ − ϵA′
+µ, the gauge
+kinetic terms become canonical, and the interaction terms read
+Lint = −gDA′
+µJµ
+χ + ϵeA′
+µJµ
+EM − eAµJµ
+EM .
+(4.2)
+We consider five models for the nature of the DM: pseudo-Dirac-fermion DM with small
+or large mass-splitting, scalar elastic DM, scalar inelastic DM, and Majorana-fermion DM.
+The DM current Jµ
+χ for each of these models is listed below, and the recoil profiles for
+χe → χe scattering in the lab frame are summarized in Appendix B.
+The DM models used in this study have four or five free parameters: the dark photon
+mass mA′, the DM mass mχ1, the dark fine structure constant αD ≡ g2
+D/(4π), the kinetic
+mixing parameter ϵ, and the mass difference ∆ = mχ2 − mχ1 of DMs if χ2 is present. The
+DM particles remain in chemical equilibrium with the SM plasma in the early universe
+before freezing out. The DM relic density is determined by the cross section of the pair-
+annihilation process DM + DM ↔ SM + SM. For mχ ≪ mA′, the annihilation cross section
+for these models can be parametrized as σv ∝ y/m2
+χ with y ≡ ϵ2αD(mχ/mA′)4. The cross
+section for which the relic density matches the observed value defines the “relic target” in the
+(mχ, y) space. Following the common practice, we will use this two-dimensional parameter
+space to illustrate the sensitivity of the ILC beam dump searches. Note however that the
+signal event rates at the ILC-BDX do not depend exclusively on these two parameters and
+thus further assumptions are necessary to represent the reach on the (mχ, y) plane. These
+assumptions will be specified in the captions of each of our reach plots.
+4.1
+Pseudo-Dirac DM
+Pseudo-Dirac inelastic DM#8 is described by a pair of two-component Weyl fermions (η, ξ)
+that have opposite unit charge under U(1)D. Both a U(1)D-conserving Dirac mass mD and
+a U(1)D-breaking Majorana mass mM are present in the low-energy theory, since the U(1)D
+symmetry is spontaneously broken. Namely, the Lagrangian has the mass terms
+−L ⊃ mDηξ + 1
+2mM(η2 + ξ2) + H.c.
+(4.3)
+#8If the DM χ is a Dirac fermion, the DM current is given by Jµ
+χ ∝ ¯χγµχ, and it annihilates to SM
+in s-wave two-to-two processes. Constraints from the cosmic microwave background (CMB) power in-
+jection have ruled out s-wave two-to-two annihilating DM lighter than O(10) GeV as the thermal DM
+candidate [28]. This motivates pseudo-Dirac DM as the simplest viable model of sub-GeV fermionic DM.
+12
+
+Here and below, we assume mD ≫ mM > 0. The mass eigenstates are then given by
+χ1 =
+i
+√
+2(η − ξ),
+χ2 = 1
+√
+2(η + ξ)
+(4.4)
+with masses mχ1,2 = mD ∓ mM, and the DM current becomes off-diagonal:
+Jµ
+χ = i¯χ2γµχ1 + H.c.
+(4.5)
+The ILC-BDX search strategy depends on the mass difference ∆ ≡ mχ2 − mχ1 = 2mM.
+For ∆ > 2me (large mass-splitting), the heavier DM particle χ2 can decay into χ1e−e+
+with the partial decay width [29,30]
+Γ(χ2 → χ1e−e+) ≃ 4ϵ2ααD∆5
+15πm4
+A′
+,
+(4.6)
+which results in a visible-decay signal.
+Otherwise, if ∆ < 2me (small mass-splitting),
+χ2 does not decay inside the apparatus because of the small width of the main decay
+channel χ2 → χ1 + 3γ. This region of parameter space is accessible only by searches for
+the electron-recoil signal.
+Figure 5 summarizes our results for the small mass-splitting case, where we take the limit
+∆ ≃ 0 and fix αD = 0.5 and mA′ = 3mχ. The three red solid lines show the sensitivity of the
+ILC-BDX recoil-electron search with √s = 250 GeV. The lines correspond to 10, 102, and
+103 signal events with 10-year statistics.#9 The three lines illustrate the range of plausible
+scenarios for the level of reducible beam-induced backgrounds affecting the search, see
+Section 3. The shaded regions are excluded by past experiments listed in Section 4.5, and
+the solid black lines are the relic target for pseudo-Dirac DM.
+Figure 6 shows the result for larger mass-splitting, where we fix ∆ = 0.1mχ1 as well
+as αD = 0.1 and mA′ = 3mχ1.
+Consequently, the visible-decay signal is expected for
+mχ1 ≳ 1 MeV. It is noteworthy that the search for visible decays of χ2 provides a unique
+sensitivity for this model, well beyond that achievable at proposed dedicated future facilities
+such as LDMX. This strongly motivates the inclusion of a multi-layer tracker in the decay
+volume (see Fig. 1) as a key component of the ILC-BDX design.
+Below, we discuss the physics that determines the structure of the reach curves in Figs. 5
+and 6.
+4.1.1
+Small mass-splitting: ∆ < 2me
+Both pair-annihilation and bremsstrahlung are included as the source of signal events,
+where on-shell dark photons are produced and decay into a χ2-χ1 pair, and either χ2 or χ1
+is detected as electron recoil.
+#9We also checked that the visible decay signals coming from Z decays at the Giga-Z program [35] of
+the ILC are not significant for mχ ≲ 1 GeV.
+13
+
+10-3
+10-2
+10-1
+100
+10-16
+10-15
+10-14
+10-13
+10-12
+10-11
+10-10
+10-9
+10-8
+10-7
+mχ [GeV]
+y = ϵ2αD(mχ/mA')4
+BaBar
+Relic Target
+Belle II (extrapolated)
+LDMX
+ILC-250 10-year
+103
+102
+10
+ILC (recoil)
+BDX (recoil)
+(a) electron beam dump
+10-3
+10-2
+10-1
+100
+10-16
+10-15
+10-14
+10-13
+10-12
+10-11
+10-10
+10-9
+10-8
+10-7
+mχ [GeV]
+y = ϵ2αD(mχ/mA')4
+BaBar
+Relic Target
+Belle II (extrapolated)
+LDMX
+ILC-250 10-year
+103
+102
+10
+ILC (recoil)
+BDX (recoil)
+(b) positron beam dump
+Figure 5: Projected sensitivity reach of a 10-year ILC-250 run in the pseudo-Dirac DM
+model with small mass-splitting, ∆ ≪ min(me, mχ1). It is assumed that αD = 0.5 and
+mA′ = 3mχ. On each panel, the three red solid lines show the sensitivity of the ILC-BDX
+recoil-electron search, corresponding to 10, 102, and 103 signal events. The black solid line
+shows DM relic targets [31]. The shaded grey region is excluded by the past experiments;
+see Section 4.5. The dashed lines show the sensitivity of the BDX recoil-electron search
+(purple) [29], LDMX (blue) [31], and Belle II experiment (green) [12,32].
+• Pair-annihilation (recoil-electron) — The DM with mass less than ≃ 10−2 GeV cannot
+be detected because of the threshold Emin = 1 GeV. This is because decays of the
+dark photon with mass less than √2meEmin cannot contribute to signal events since
+the energy of the produced dark photon is Elab
+A′ ≃ Ee+ ≃ m2
+A′/2me. Also, the DM
+with mass larger than ≃ 10−1 GeV cannot be produced from decays of the dark
+photon because the dark photon mass cannot exceed √2meEbeam. In the positron
+beam dump for the DM mass of √2meEbeam/3, primary positron beam production
+dominates, and a peak structure arises.
+• Bremsstrahlung (recoil-electron) — For the DM mass smaller than ∼ 10−2 GeV, the
+number of the recoil-electron events is suppressed by the threshold Emin = 1 GeV.
+Since the expected decay angle of the dark photon is (π/2) · (mA′/EA′), the dark
+photon energy has to satisfy mA′ · (π/2) · (ldump + lsh + ldec)/rdet ≲ EA′ to obtain
+sufficient angular acceptance. However, the minimum energy of the dark photon is
+EA′ ∼ Emin = 1 GeV because of the threshold, and the signal events from the DM
+with mass smaller than ∼ 10−2 GeV are suppressed even if the angular acceptance
+14
+
+10-3
+10-2
+10-1
+100
+10-16
+10-15
+10-14
+10-13
+10-12
+10-11
+10-10
+10-9
+10-8
+10-7
+mχ1 [GeV]
+y = ϵ2αD(mχ1/mA')4
+Belle II (extrapolated)
+LDMX
+BDX(decay)
+Relic Target
+ILC-250 10-year
+103
+102
+10
+ILC (recoil)
+ILC (decay)
+(a) electron beam dump
+10-3
+10-2
+10-1
+100
+10-16
+10-15
+10-14
+10-13
+10-12
+10-11
+10-10
+10-9
+10-8
+10-7
+mχ1 [GeV]
+y = ϵ2αD(mχ1/mA')4
+Belle II (extrapolated)
+LDMX
+Relic Target
+ILC-250 10-year
+103
+102
+10
+ILC (recoil)
+ILC (decay)
+BDX(decay)
+(b) positron beam dump
+Figure 6: Projected sensitivity reach of a 10-year ILC-250 run in the pseudo-Dirac DM
+model with a large mass-splitting, ∆ = 0.1mχ1.
+It is assumed that αD = 0.1 and
+mA′ = 3mχ1. The notation is similar to Fig. 5; in addition, dot-dashed lines show the
+sensitivity of the ILC-BDX decay-signal search (95% C.L. exclusion).
+Also shown are
+expected sensitivities of the BDX visible-decay search (purple-dashed lines) [14, 29], the
+LDMX (blue) [33], and Belle II (green) [34].
+holds. For the DM with mass larger than 0.1 GeV, the angular acceptance becomes
+worse, and the sensitivity rapidly decreases.
+4.1.2
+Large mass-splitting: ∆ > 2me
+The new feature in this case is the availability of the visible-decay signal. Here, we discuss
+the sensitivity reach for this channel, highlighting each of the DM decay processes. The
+results of the electron recoil searches are similar to the case of the small mass-splitting.
+• Pair-annihilation (visible-decay) — For ∆ = 0.1mχ1, the heavier DM with the mass
+smaller than ≃ 10−2 GeV cannot decay, and the visible-decay signals do not arise.
+Similar to the recoil-electron search, the DM with mass larger than 10−1 GeV can-
+not be produced because the dark photon mass cannot exceed √2meEbeam. In the
+positron beam dump experiment, for mχ1 = √2meEbeam/3, the peak structure arises
+due to the primary positron beam.
+• Bremsstrahlung (visible-decay) — Similar to the pair-annihilation process, for ∆ =
+15
+
+10-3
+10-2
+10-1
+100
+10-16
+10-15
+10-14
+10-13
+10-12
+10-11
+10-10
+10-9
+10-8
+10-7
+mχ [GeV]
+y = ϵ2αD(mχ/mA')4
+Relic Target
+Belle II (extrapolated)
+SuperCDMS
+SNOLAB
+SENSEI
+LDMX
+ILC-250 10-year
+103
+102
+10
+ILC (recoil)
+(a) electron beam dump
+10-3
+10-2
+10-1
+100
+10-16
+10-15
+10-14
+10-13
+10-12
+10-11
+10-10
+10-9
+10-8
+10-7
+mχ [GeV]
+y = ϵ2αD(mχ/mA')4
+Relic Target
+Belle II (extrapolated)
+SuperCDMS
+SNOLAB
+SENSEI
+LDMX
+ILC-250 10-year
+103
+102
+10
+ILC (recoil)
+(b) positron beam dump
+Figure 7: The results for the scalar elastic DM. It is assumed that αD = 0.5 and mA′ =
+3mχ. The notation is the same as in Fig. 5; in addition, the gray lines show the expected
+sensitivities of SENSEI [32] and SuperCDMS [12,32].
+0.1mχ1, the heavier DM with the mass smaller than ≃ 10−2 GeV cannot decay and
+produce the visible-decay signals. For 10−1 GeV ≲ mχ, the number of signal events
+rapidly decreases because the angular acceptance becomes worse.
+4.2
+Scalar elastic DM
+The model with a scalar elastic DM is described by a complex scalar field χ that is SM-
+singlet with unit U(1)D-charge. It is safe from CMB bounds since the DM annihilation
+is in p-wave and thus with a suppressed rate. The DM current, originating in the kinetic
+term |Dµχ|2, is given by
+Jµ
+χ = i(χ∗∂µχ − χ∂µχ∗) .
+(4.7)
+The analysis results are shown in Fig. 7 with the same notation as in Fig. 5. For the non-
+relativistic DM-SM fermion elastic scattering cross section, there is no velocity suppression,
+and the constraints from direct detection experiments are significant.
+16
+
+10-3
+10-2
+10-1
+100
+10-16
+10-15
+10-14
+10-13
+10-12
+10-11
+10-10
+10-9
+10-8
+10-7
+mχ [GeV]
+y = ϵ2αD(mχ/mA')4
+Relic Target
+Belle II (extrapolated)
+LDMX
+ILC-250 10-year
+103
+102
+10
+ILC (recoil)
+(a) electron beam dump
+10-3
+10-2
+10-1
+100
+10-16
+10-15
+10-14
+10-13
+10-12
+10-11
+10-10
+10-9
+10-8
+10-7
+mχ [GeV]
+y = ϵ2αD(mχ/mA')4
+Relic Target
+Belle II (extrapolated)
+LDMX
+ILC-250 10-year
+103
+102
+10
+ILC (recoil)
+(b) positron beam dump
+Figure 8: The results for the scalar inelastic DM. The prescription αD = 0.5, mA′ = 3mχ
+is adopted. Notation is the same as in Fig. 5.
+4.3
+Scalar inelastic DM
+The scalar inelastic DM is characterized by the dark photon current
+Jµ
+χ = χ1∂µχ2 − χ2∂µχ1
+(4.8)
+involving two real scalar fields χ1,2 that are quasi-degenerate but have non-zero mass dif-
+ference ∆ = mχ2 − mχ1 > 0 after the spontaneous symmetry breaking of U(1)D.#10 We
+focus only on the small mass-splitting case since the large mass-splitting case of the scalar
+inelastic DM is similar to that of the pseudo-Dirac inelastic DM. The results of our analysis
+are shown in Fig. 8, with the same notation as in Fig. 5.
+4.4
+Majorana DM
+For a Majorana fermion χ, the DM current can be an axial-vector as follows:
+Jµ
+χ = 1
+2 ¯χγµγ5χ.
+(4.9)
+#10The mass difference is realized by terms such as χ2H2 after the electroweak (and U(1)D) symmetry
+breaking, where χ is a complex scalar field yielding χ1,2, and H is the Higgs doublet [13].
+See, e.g.,
+Refs. [36–39] for models realizing scalar inelastic DM.
+17
+
+10-3
+10-2
+10-1
+100
+10-16
+10-15
+10-14
+10-13
+10-12
+10-11
+10-10
+10-9
+10-8
+10-7
+mχ [GeV]
+y = ϵ2αD(mχ/mA')4
+Relic Target
+Belle II (extrapolated)
+LDMX
+ILC-250 10-year
+103
+102
+10
+SENSEI
+SuperCDMS
+SNOLAB
+ILC (recoil)
+(a) electron beam dump
+10-3
+10-2
+10-1
+100
+10-16
+10-15
+10-14
+10-13
+10-12
+10-11
+10-10
+10-9
+10-8
+10-7
+mχ [GeV]
+y = ϵ2αD(mχ/mA')4
+Relic Target
+Belle II (extrapolated)
+LDMX
+ILC-250 10-year
+103
+102
+10
+SENSEI
+SuperCDMS
+SNOLAB
+ILC (recoil)
+(b) positron beam dump
+Figure 9: The results for the Majorana DM. The prescription αD = 0.5, mA′ = 3mχ is
+adopted. Notation is the same as in Fig. 7.
+Similar to the scalar DM, the pair-annihilation rate of the DM into SM particles is sup-
+pressed by the DM velocity, and a sub-GeV Majorana fermion DM is safe from the CMB
+constraints. The results of our analysis for the Majorana DM are shown in Fig. 9, with the
+same notation as in the scalar elastic DM case (Fig. 7).
+4.5
+Existing constraints and projected sensitivities at other ex-
+periments
+Searches for sub-GeV DM are pursued by accelerator-based experiments and direct detec-
+tion experiments. In the accelerator-based experiments, the DM particles may be produced
+by beam-beam or beam-target collisions, and the searches are performed either by detect-
+ing their scattering off of the SM particles in the detectors, the visible decays of the excited
+DM states, or the missing energy/momentum events. In the direct detection experiments,
+the DM particles from the halo are searched for by detecting their scattering off of the SM
+particles in the detectors. We list the existing constraints included in Figs. 5-9 as follows:
+• E137 electron beam dump experiment — The light DM may be produced at the SLAC
+electron beam dump by injection of 20 GeV electron beams [40]. The produced DM
+particles may be detected by their scattering on electrons in the aluminum and plastic
+scintillator 400 m downstream from the beam dump. The visible decays of the heavier
+18
+
+dark sector particles are also detectable at the scintillator [29].
+• LSND proton beam dump experiment — The light DM may also be produced in the
+Los Alamos LSND proton beam dump experiment by injection of proton beams with
+kinetic energy of 800 MeV [29,32,33,41]. The DM particles are produced dominantly
+by π0 decays at the fixed target, and DM-electron scattering or visible decay events
+are detected at the downstream detector.
+• γ+missing energy search — The DM may be produced at B-factories by process
+e+e− → γ +A′(∗) → γχ¯χ. The mono-photon and missing-energy events at the BaBar
+set bounds on the DM models [32,42,43].
+• Direct detection experiments — The DM-SM non-relativistic scattering may be de-
+tected in direct detection experiments, such as XENON-10/100 and CRESST II [44–
+47]. For the DM-nuclear scattering process, the CRESST II experiment is sensitive
+to mχ ∼ 1 GeV because of the low nuclear recoil energy threshold. The sub-GeV
+DMs with MeV to GeV mass may also be explored using the DM-electron scattering
+processes.
+As the future DM searches, we included projected sensitivities at Belle II [12, 32],
+BDX [29], LDMX [32], SENSEI [32], and SuperCDMS [12, 32] in Figs. 5-9 to compare
+with the sensitivity reaches of the ILC-BDX.
+5
+Summary
+In this paper, we explored the potential of a search for sub-GeV dark matter at the ILC
+beam dumps using the ILC-BDX setup. Previous studies of beam dump experiments at
+the ILC focused on long-lived BSM particles decaying into visible SM final states. In this
+study, on the contrary, we consider DM models where a dark photon connects the stable
+DM to the SM sector, and focus on DM detection. Five DM models, i.e., pseudo-Dirac DM
+with small and large mass-splitting, scalar elastic and inelastic DM, and Majorana DM are
+considered as reference scenarios. At the ILC, electrons and positrons in electromagnetic
+shower are produced by the beam injection into the beam dumps. These particles produce
+dark photons through the bremsstrahlung and pair-annihilation processes, which in turn
+decay to pairs of DM particles. (Pair-production of DM via virtual dark photon exchange
+is also possible,) Produced DM particles propagate over a long distance (of order 100 m) to
+the detector and elastically scatter with electrons inside the detector, and then the recoil
+electrons are detected as a signature of the DM production. For the inelastic DM models,
+there is another signal produced by a visible decay of the heavier dark state into the lighter
+one and SM particles. The ILC-BDX setup includes a multi-layer tracker designed to search
+for this visible-decay signal.
+19
+
+In the DM search at the ILC-BDX, there are two kinds of backgrounds, i.e., the beam-
+induced and beam-unrelated BGs. The beam-induced BG is due to neutrinos mainly pro-
+duced by meson and lepton decays in the beam dumps. Elastic neutrino-electron scat-
+tering in the detector is nearly indistinguishable from DM-induced electron recoil events,
+providing irreducible BGs to this search. Inelastic neutrino-nucleus interactions can also
+mimic electron-recoil and visible-decay signals, though these backgrounds are potentially
+reducible. We evaluated the neutrino-induced irreducible and reducible BGs by using the
+neutrino fluxes at the detector calculated with Monte Carlo simulation. We also estimated
+the beam-unrelated BG, which comes predominantly from cosmic-ray muons. This BG
+component can be significantly suppressed by the deep underground location of the detec-
+tor and beam-coincidence time window.
+We evaluated the number of DM particles produced at the ILC electron and positron
+beam dumps and the rate of expected signal events, i.e., electron-DM recoils and visible-
+decays at the ILC-BDX, in our reference models. The predicted signal and background
+rates were then used to estimate the sensitivity reach of the ILC-BDX experiment with a
+data set corresponding to a 10-year ILC run at √s = 250 GeV. The results for each DM
+model are shown in Figs. 5-9. We found that in all five reference models considered here,
+the ILC-BDX experiment has sensitivity to regions of model parameter space well beyond
+the existing constraints. In many cases, the ILC-BDX can conclusively probe the region
+of parameter space where the DM has the thermal abundance consistent with observations
+(the “relic target”). We find that while electron and positron beam dumps have similar
+performances, the sensitivity of the latter is somewhat better because the primary positron
+beam contributes to the DM production through pair-annihilation on electrons in the dump.
+The sensitivity of the ILC-BDX is particularly impressive in models where the visible-decay
+signal is available, such as the pseudo-Dirac or scalar inelastic DM models with mass-
+splitting ∆ > 2me. In such models, the reach of the ILC-BDX significantly exceeds that of
+even the most ambitious proposed dedicated searches for sub-GeV DM, such as LDMX.
+In the discussion of future searches for sub-GeV dark matter, it is important to note
+the complementarity between the electron-recoil technique used by ILC-BDX and the miss-
+ing energy/momentum technique used by experiments such as LDMX. The signal rate at
+missing energy experiments is simply proportional to the total cross section of DM produc-
+tion by the beam electron interactions with the target material. On the other hand, the
+signal rate at electron-recoil experiments is proportional to the product of this production
+cross section and that of the elastic scattering of DM on target electrons. The relationship
+between the production and elastic scattering cross sections is model-dependent. Thus,
+analyzing data from both types of experiments will provide an opportunity to identify the
+underlying DM model, if a signal is observed with one or both approaches. This is an
+exciting possibility, and it strongly motivates pursuing both approaches in parallel in the
+future.
+20
+
+In our evaluation of the acceptance of the signal events, we adopted some simplifying
+assumptions, for instance, the approximated angular distribution of the heavy DM state
+in Eq. (2.5) and radial deviations in Eqs. (2.7) and (2.10). Likewise, several simplifying
+assumptions were made in the estimates of the background rates; see Section 3.
+As a
+next step, it would be important to design a realistic Monte Carlo model of the ILC-BDX
+detector, and use it to provide a more precise evaluation of both signal and background
+rates. Another important direction for future work is to design strategies for handling the
+reducible beam-induced BG, such as additional cuts or a veto system, and to evaluate such
+strategies quantitatively using Monte Carlo simulations.
+Acknowledgements
+We are grateful to the members of the ILC Task Force on fixed-target experiments and dark
+sectors, especially Claude Vallee, for discussions that initiated this work. KA is supported
+by JSPS KAKENHI Grant Number JP18H01210, JP21K20365, and MEXT KAKENHI
+Grant Number JP18H05543. YS is supported by JSPS KAKENHI JP21H05466. MP is
+supported by the NSF grant PHY-2014071.
+A
+Dark photon production cross sections
+In this appendix, we list the cross sections of dark photon production via pair-annihilation
+and bremsstrahlung.
+A.1
+Pair-annihilation production
+The cross section of the resonant annihilation process is given by [48]
+σ(e+e− → A′) = 12π
+m2
+A′
+Γ2
+A′/4
+(√s − mA′)2 + Γ2
+A′/4,
+(A.1)
+where s is the center-of-mass energy squared and ΓA′ is the total decay width of the dark
+photon. In this work, ΓA′ is assumed to be small enough that the narrow-width approxi-
+mation can be used:
+σ(e+e− → A′) ≃ 2π2αϵ2
+me
+δ
+�
+Ei − m2
+A′
+2me
++ me
+�
+,
+(A.2)
+where δ(x) denotes the Dirac delta function.
+21
+
+A.2
+Bremsstrahlung production
+The differential cross section of the bremsstrahlung process under the Weizs¨acker-Williams
+approximation [49–52] is given by [53,54]
+d2σ(iN → iA′N)
+dEA′ dθA′
+= ϵ2α3 1
+Ei
+sin θA′
+�
+E2
+A′ − m2
+A′
+�
+E2
+i − m2
+e
+1
+1 − x
+A2→2
+t=tWW
+min
+2tWW
+min
+χ,
+(A.3)
+where x = EA′/Ei, tWW
+min = ˜s2/4E2
+i , ˜s = −˜u/(1 − x), and
+˜u = −xE2
+i θ2
+A′ − m2
+A′ 1 − x
+x
+− m2
+ex.
+(A.4)
+The effective flux of photons, χ, is given by
+χ =
+� tmax
+tmin
+dt t − tmin
+t2
+G2(t)
+= Z2
+�
+1 − b
+a
+�−2 �
+− (a + b + 2abtmax)(tmax − tmin)
+(1 + atmax)(1 + btmax)
++ a + b + 2abtmin
+a − b
+ln
+�1 + atmax
+1 + btmin
+1 + btmin
+1 + atmin
+� �
+(A.5)
+with tmin = m4
+A′/4E2
+i , tmax = m2
+A′ + m2
+e, and
+G2(t) =
+��
+at
+1 + at
+� �
+1
+1 + bt
+�
+Z
+�2
+,
+a = 1122Z−2/3
+m2
+e
+,
+b =
+1
+0.164 GeV2A−2/3,
+(A.6)
+where Z is the atomic number and A is the mass number of the target atom. The amplitude
+under the the Weizs¨acker-Williams approximation is given by
+A2→2
+t=tWW
+min = 22 − 2x + x2
+1 − x
++ 4(m2
+A′ + 2m2
+e) ˜ux + m2
+A′(1 − x) + m2
+ex2
+˜u2
+.
+(A.7)
+B
+DM-electron recoil cross sections
+For the inelastic-fermion DM model and the scalar DM model, the cross section of the
+DM-electron recoil process is given by [30,55],
+dσ(χ1e → χ2e)
+dEe
+=
+me
+8πλ(s, m2
+e, m2
+χ1)|M|2,
+(B.1)
+where Ee is the energy of the recoil electron, λ(x, y, z) = (x − y − z)2 − 4yz, Ee = Eχ1 +
+me − Eχ2, and s = m2
+χ1 + m2
+e + 2meEχ1. The squared matrix elements are given by
+|M|2 =
+8me(ϵegD)2
+[2me(Eχ2 − Eχ1) − m2
+A′]2
+�
+me(E2
+χ1 + E2
+χ2)
+− ∆2
+2 (Eχ2 − Eχ1 + me) + m2
+e(Eχ2 − Eχ1) + m2
+χ1Eχ2 − m2
+χ2Eχ1
+�
+(B.2)
+22
+
+for the pseudo-Dirac inelastic DM model and
+|M|2 =
+8me(ϵegD)2
+[2me(Eχ2 − Eχ1) − m2
+A′]2
+�
+2meEχ1Eχ2 + m2
+χ1Eχ2 − m2
+χ2Eχ1
+�
+(B.3)
+for the scalar inelastic DM model, where Eχi (i = 1, 2) is the energy of χi measured in the
+laboratory frame. The kinematically allowed maximum (minimum) recoil energy E+
+e (E−
+e )
+is given by [55]
+E±
+e = s + m2
+e − m2
+χ2
+2√s
+Eχ1 + me
+√s
+±
+�
+λ(s, m2
+e, m2
+χ2)
+2√s
+pχ1
+√s,
+(B.4)
+where pχ1 = �E2
+χ1 − m2
+χ1.
+The recoil cross section for the other models are given by [30,40,55,56]
+dσ(χe → χe)
+dEe
+= 4πϵ2ααD
+2meE2
+χ − (2meEχ − meEe + m2
+χ + 2m2
+e)(Ee − me)
+(E2
+χ − m2
+χ)(m2
+A′ + 2meEe − 2m2
+e)2
+(B.5)
+for the pseudo-Dirac elastic DM model,
+dσ(χe → χe)
+dEe
+= 4πϵ2ααD
+2meE2
+χ − (2meEχ + m2
+χ)(Ee − me)
+(E2
+χ − m2
+χ)(m2
+A′ + 2meEe − 2m2
+e)2
+(B.6)
+for the scalar elastic DM model, and
+dσ(χe → χe)
+dEe
+= 4πϵ2ααD
+2me(E2
+χ − m2
+χ) + [m2
+χ − me(2Eχ − Ee + 2me)](Ee − me)
+(E2
+χ − m2
+χ)(m2
+A′ + 2meEe − 2m2
+e)2
+(B.7)
+for the Majorana DM model. The range of Ee is given by Eq. (B.4) with replacing χi by
+χ.
+C
+Neutrino-induced background
+We evaluate the number of neutrino-induced irreducible and reducible BG events from
+the neutrino flux of Fig. 2. The number of irreducible BG events does not depend on the
+details of the detector design and is evaluated in the same way as the signal rate calculation
+in Section 2. On the other hand, the number of reducible BG events highly depends on
+the detector design, so we conservatively take a pessimistic scenario without imposing the
+threshold energy of the electromagnetic showers#11. This section describes the detailed
+calculations of the number of BG events.
+23
+
+10-3
+10-2
+10-1
+100
+101
+0
+1
+2
+3
+4
+5
+6
+Emin [GeV]
+NBG
+10-3
+10-2
+10-1
+100
+101
+0
+10
+20
+Emin [GeV]
+N95 %
+ILC-250 
+1 yr. 
+10 yr. 
+Figure 10: Left panel: The number of irreducible BG events in a 1-year run as a function
+of the threshold Emin. Right panel: The number of signal events at 95% C.L. in a 1-year
+(blue) and 10-year (red) run as a function of the threshold Emin.
+C.1
+Irreducible background
+The number of neutrino-electron recoil events is expressed by the following formula:
+NBG = Ne±
+�
+i=νe,µ,τ,¯νe,µ,τ
+�
+dEi πr2
+det · dφi
+dEi
+· n(det)
+e−
+· ldet ·
+�
+Emin
+dER
+dσ(ie− → ie−)
+dER
+,
+(C.1)
+where φi is the neutrino flux at the detector position (Fig. 2 right) and σ(ie− → ie−)
+denotes the neutrino-electron elastic scattering cross section [57,58].
+In the left panel of Fig. 10, the number of neutrino-electron recoil events per year is
+shown as a function of the threshold Emin. The rate is reduced to N BG
+beam ∼ 1/year by
+imposing Emin = 1 GeV. By taking into account the irreducible BG events and assuming
+no signal events are observed, we estimate the expected upper bounds on the number of
+signal events at 95% C.L.#12, shown by the solid lines in the right panel of Fig. 10.
+#11The rate of quasi-elastic scattering and pion production processes would not change significantly by
+imposing the threshold Emin = 1 GeV because their cross sections are very small for neutrino energy
+smaller than 1 GeV.
+#12The Poisson 95% C.L. upper bound corresponds to three signal events in the absence of BG events.
+24
+
+C.2
+Reducible background
+The number of quasi-elastic scattering events is estimated by the following formula:
+NQE = Ne±
+�
+dEν πr2
+det · ldet ·
+�dφνe
+dEν
+n(det)
+n
+σ(νen → e−p) + dφ¯νe
+dEν
+n(det)
+p
+σ(¯νep → e+n)
+�
+,
+(C.2)
+where φνe and φ¯νe are the electron-neutrino and electron-antineutrino fluxes at the detector
+position (see the right panel of Fig. 2), n(det)
+n(p) is the neutron (proton) number density of
+the detector, and σ denotes the quasi-elastic scattering cross section [58]. In contrast to
+the estimation of the irreducible BG events, we do not impose the threshold energy of the
+electromagnetic shower; Eq. (C.2) is therefore a conservative estimate. From Eq. (C.2), the
+number of the quasi-elastic scattering events per year is estimated as
+NQE ∼ 2 × 102 ×
+�
+Ne±
+4 × 1021
+�� rdet
+2 m
+�2�
+ldet
+0.64 m
+�
+.
+(C.3)
+Similarly, the number of the single-π0 production events is given by
+Npion = Ne±
+�
+dEν πr2
+det · ldet
+×
+�
+dφνµ
+dEν
+�
+n(det)
+n
+�
+σ(νµn → µ−pπ0) + σ(νµn → νµnπ0)
+�
++ n(det)
+p
+σ(νµp → νµpπ0)
+�
++ dφν¯µ
+dEν
+�
+n(det)
+p
+�
+σ(¯νµp → µ+nπ0) + σ(¯νµp → ¯νµpπ0)
+�
++ n(det)
+n
+σ(¯νµn → ¯νµnπ0)
+��
+,
+(C.4)
+where the single-π0 production cross sections are listed in Ref. [58]. Combining the neutrino
+flux of the right panel of Fig. 2 and the cross sections in Ref. [58], the number of the pion
+production events per year is obtained as
+Npion ∼ 2 × 103 ×
+�
+Ne±
+4 × 1021
+�� rdet
+2 m
+�2�
+ldet
+0.64 m
+�
+.
+(C.5)
+As in the calculation of NQE, the threshold energy of the electromagnetic shower is not
+imposed and thus this estimate is conservative.
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new file mode 100644
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+page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' China  Abstract  Light dark matter particles may be produced in electron and positron beam dumps  of the International Linear Collider (ILC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We propose an experimental setup to  search for such events, the Beam-Dump eXperiment at the ILC (ILC-BDX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The  setup consists of a muon shield placed behind the beam dump, followed by a multi-  layer tracker and an electromagnetic calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The calorimeter can detect electron  recoils due to elastic scattering of dark matter particles produced in the dump, while  the tracker is sensitive to decays of excited dark-sector states into the dark matter  particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We study the production, decay and scattering of sub-GeV dark matter  particles in this setup in several models with a dark photon mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Taking into  account beam-related backgrounds due to neutrinos produced in the beam dump as  well as the cosmic-ray background, we evaluate the sensitivity reach of the ILC-BDX  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We find that the ILC-BDX will be able to probe interesting regions of  the model parameter space and, in many cases, reach well below the relic target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='03816v1  [hep-ph]  10 Jan 2023  Contents  1  Introduction  1  2  Beam dump experiment  4  3  Expected backgrounds  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='3  Scalar inelastic DM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='4  Majorana DM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='  17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5  Existing constraints and projected sensitivities at other experiments .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='  18  5  Summary  19  A Dark photon production cross sections  21  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1 Pair-annihilation production .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='2 Bremsstrahlung production  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  22  B DM-electron recoil cross sections  22  C Neutrino-induced background  23  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1 Irreducible background .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  24  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2 Reducible background  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  25  1  Introduction  The International Linear Collider (ILC) has been proposed as the next energy-frontier  facility in particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The physics program of the ILC is well established [1–3]: it in-  cludes measurements of the Higgs boson couplings with unprecedented precision, searches  1  for new physics in rare Higgs decays and other channels, and precise top-mass determina-  tion, among other topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Most studies of the physics potential of the ILC to date focus on  experiments with the detector situated at the main interaction point, where the electron  and positron beams collide head-on, yielding the maximum collision energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Recently, it was suggested that, in parallel with the experiments at the main interaction  point, the ILC can pursue a complementary physics program using its beam dumps [4–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Particle collisions within the beam dumps occur in fixed-target kinematics, with typical  effective energies of O(10) GeV or below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' At the same time, since every beam-electron  and beam-positron interacts with the dump, the beam-dump experiments can accumulate  enormous integrated luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' This makes the beam dump the ideal place to search for  relatively light new particles with very small couplings to the Standard Model (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The new physics targeted by the beam-dump experimental program is very well mo-  tivated theoretically [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Previously studied examples include visibly-decaying dark  photons, axion-like particles, vector bosons associated with gauged lepton flavor symme-  tries, and heavy neutral leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In all these cases, it was demonstrated that beam-dump  experiments at the ILC can probe model parameters well beyond the reach of the currently  available experiments and are competitive with proposed future dedicated facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  In this paper, we propose a new search for light dark matter using the ILC beam dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  It is well known that a stable particle with mass in the MeV–GeV range, interacting with  the SM via a dark photon mediator, is an attractive dark matter (DM) candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' With  reasonable model parameters, its thermal relic density matches the observed DM abundance  and it is consistent with all astrophysical and experimental constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The DM particles  can be pair-produced at the ILC beam dump via virtual dark-photon exchanges or in decays  of dark photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' An electromagnetic calorimeter placed ∼ 100 m downstream of the beam  dump, behind a lead shield is designed to search for the DM by detecting elastic scattering  of the DM particles on the electrons in the detector material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In addition, in many models  such as inelastic DM [13], the DM particle is the lowest-lying state of a multiplet with small  mass splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' A long-lived excited state in the same multiplet as the DM, produced at  the beam dump, may propagate through the lead shield and decay into the DM and charged  SM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The charged particles can then be detected by a tracker, providing a “visible  decay” signature of the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We will estimate the experimental reach in both electron-recoil  and visible-decay signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The proposed experimental setup is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The concept of this experiment  is the same as the Beam-Dump eXperiment (BDX) proposed at JLab [14], and we, therefore,  call our proposal “ILC-BDX.” The advantage of the ILC, compared to the original BDX  proposal, is the higher beam energy as well as availability of both electron and positron  beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='#1 We also note that the experimental setup used here is essentially identical to  #1An additional advantage is the availability of polarized beams at the ILC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We will not explore conse-  2  ldec  Beam dump  Muon shield  lsh  ldump  rdet  Detector  z  Lead  Concrete  Water  e±  ldet  Decay volume (Multi-layer tracker)  rdec  e+  e−  χ  χ  e±  Nucleus  e±  χ  χ  γ*  A′  A′  e−  e−  A′ *  χ  χ  A′ *  e−  e+  χ2  χ1  Figure 1: The ILC-BDX experimental setup consists of the main beam dump, a muon  shield, a decay volume, and a detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' A multi-layer tracker is placed in the decay volume  to measure the charged tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The DM particles may be produced at electron and positron  beam dumps via pair-annihilation of positron on an atomic electron, or via bremsstrahlung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The resulting DM particles can scatter off electrons in the detector and yield observable  electron-recoil events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In models where the DM particle is part of a multiplet, an excited  DM state may be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' It can then decay into a lighter DM state and SM particles in  the decay volume, producing a visible signature in the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  that proposed previously for searches for visible dark photons and axion-like particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (Detailed design of the detector may be different to optimize the sensitivity for various  searches, but this is outside the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=') The ILC-BDX can be pursued in  parallel to the previously discussed searches, adding to the rich physics program at the ILC  beam dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' It is important to emphasize that this program can be pursued in parallel  with the exploration of the Higgs and electroweak physics at the main IP, and can broaden  the capabilities of the ILC to search for physics beyond the Standard Model (BSM) at a  modest additional cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In Section 2, we introduce the setup  of our proposed experiment for sub-GeV DM search at the ILC beam dumps, and list  formulae used to calculate the number of expected signal events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In Section 3, we discuss  the two types of background (BG) events that occur in this setup, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=', beam-induced and  cosmic-ray backgrounds, and estimate the number of events expected from each BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In  Section 4, we introduce five DM models used as benchmarks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=', pseudo-Dirac DM with  small and large mass-splitting, scalar elastic and inelastic DM, and Majorana DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We  then present our estimates of the ILC-BDX sensitivity reach for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The main  findings of our analysis are summarized in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The Appendix contains the formulae  for cross sections of the relevant dark-photon production processes (Appendix A) and the  quences of polarization in this paper, but note that it can be particularly useful in characterizing the new  physics if a non-SM signal is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  3  DM-electron elastic scattering (Appendix B), as well as some details of our estimates of  neutrino-induced BG rates (Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  2  Beam dump experiment  We adopt the same experimental setup as that of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' [9], which is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  For both electron and positron beams, the ILC main beam dumps are planned as absorbers  consisting of water cylinders along the beam axes with the length of ldump = 11 m [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The proposed setup consists of a muon shield with the length of lsh = 70 m made of lead, a  cylindrical decay volume with ldec = 50 m and radius of rdec = 3 m with a multi-layer tracker  installed, and a cylindrical detector with the radius of rdet = 2 m and the length of ldet =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='64 m, behind either of the beam dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The multi-layer tracker is designed to detect  the visible-decay signal, while the cylindrical detector is assumed to be an electromagnetic  (EM) calorimeter made of CsI(Tl) scintillating crystals (n(det)  e−  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1 × 1024 cm−3) designed  to detect recoil electrons caused by incoming boosted DM particles (electron-recoil signals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  We focus on the 250 GeV ILC (ILC-250) with the beam energy of Ebeam = 125 GeV in  the lab frame, and the number of incident electrons and positrons into the beam dump of  Ne± = 4×1021/year [1,16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' As studied comprehensively in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' [5,6,9,20], this setup was  found very sensitive to long-lived BSM particles decaying into visible SM particles thanks  to its thick shield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We will see that, with the calorimeter as a recoil-electron detector, it is  also sensitive to DM particle production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  We study two types of signal events associated with different production mechanisms of  DM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' If the DM particles are produced in the water beam dump by a BSM interac-  tion (such as dark photons), they are highly boosted and, passing through the muon shield,  scatter off electrons in the calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Such electron recoils, detected by the calorimeter,  are a typical signal of DM production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' On the other hand, the DM particles may also be  produced in in-flight decays of heavier DM-sector particles if, for example, the DM is the  lowest-lying state of a multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The heavier particle, which we denote by χ2, can pass  through the muon shield and then eventually decay into a DM particle χ1 and SM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  If the decay happens in the decay volume, charged tracks may be observed in the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  This visible-decay signature is used as an additional signal of DM production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  In this work, we focus on five DM models with the dark photon mediator to provide a  benchmark study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' They are pseudo-Dirac-fermion DM with small or large mass-splitting,  scalar elastic DM, scalar inelastic DM, and Majorana-fermion DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The details of each  model are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Among the models, the pseudo-Dirac DM with small  mass-splitting, scalar DM, and Majorana DM can be observed as electron-recoil events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Meanwhile, the pseudo-Dirac DM with large mass-splitting can produce visible-decay events  in addition to electron-recoil signature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' specifically, due to the pseudo-Dirac nature, a  4  produced dark photon A′ decays mainly into χ2 ¯χ1 (or χ1 ¯χ2) and the heavier DM-sector  particle χ2 may decay visibly in the decay volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The number of signal events is schematically given by  Nsignal = Ne± × li × nj × σij→A′→DM × Acc,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1)  where we consider a particle i in the shower interacting with a particle j in the material  of the beam dump to produce DM particles through an on-shell dark photon A′ as the  mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The track length le± of a shower electron and positron is provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' [6],  nj is the number density of j, and Acc denotes the detector acceptance discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  More specifically#2,  N pair  signal = Ne±  �  dEe+ dle+  dEe+ · ne− · σ(e+e− → A′) · Br(A′ → χ¯χ) · Acc,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2)  N brems  signal = Ne±  �  i=e−,e+  �  dEi  dli  dEi  nN  �  dEA′  � π  0  dθA′ d2σ(iN → iA′N)  dEA′dθA′  Br(A′ → χ¯χ) · Acc  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='3)  for pair-annihilation e+e− → A′ and bremsstrahlung e±N → e±A′N with a target nucleus  N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The cross sections σ on the right-hand side are provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Here θA′ is the emission angle of A′ with respect to the direction of the e± beam in the lab  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In all the models we consider, it is assumed that dark photons decay exclusively  into the DM-sector particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=', Br(A′ → χ¯χ) = 1, with χ being a DM sector particle  (χ1 and/or χ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' This is justified since decays to SM final states are suppressed by a small  mixing parameter ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Let us estimate the acceptance for each type of signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' For a visible-decay event to be  observed, the visible decay of χ2 must occur in the decay volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Noting that A′ mainly  decays into a χ1-χ2 pair in the models we consider, we approximate the acceptance by  Acc(decay) =  � rdec/(ldump+lsh)  0  dθχ  � ldec  0  dzdPang  dθχ  dPdec  dz  Θ(rdec − rdec  ⊥ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4)  Here, we assume that the decay A′ → χ2 ¯χ1 (or χ1 ¯χ2) is immediate and χ2 exclusively  decays into electrons because of the small mass difference ∆ ≡ mχ2 − mχ1 < 2mµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The lab  frame distribution of the χ2 emission angle θχ is approximated by#3  dPang  dθχ  = sin θχ · 1  2  �  mA′  EA′ − pA′ cos θχ  �2  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5)  #2In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' [9], this calculation scheme is referred to as the coarse-grained integration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  #3The angular distribution in the CM-frame is given by dPang/d cos θCM  χ  = 1/2 when the polarization of  the dark photon is averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  5  and dPdec/dz denotes the probability of χ2 to decay at the position z (the horizontal axis  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=',  dPdec  dz  =  1  l(lab)  χ2  exp  �  −ldump + lsh + z  l(lab)  χ2  �  ,  l(lab)  χ2  = pχ2  mχ2  1  Γχ2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='6)  with l(lab)  χ2  denoting the lab frame decay length of χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The lab-frame momentum of χ2 is  approximated by pχ2 ≈ (EA′ + pA′ cos θχ)/2, where the mass-splitting is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The  momentum of A′ is obtained by pA′ =  �  E2  A′ − m2  A′, and Γχ2 (mχ2) is the total decay width  (the mass) of χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The Heaviside function Θ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4) governs the radial requirement on  the decay position of χ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=', the radial deviation  rdec  ⊥  ≈  �  θ2  e + θ2  A′ + θ2  χ · (ldump + lsh + z)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='7)  must be smaller than the radius rdec of the multi-layer tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='7), θe is the angle  of the beam-oriented e± with respect to the beam axis, θA′ is the production angle of A′  and is equal to 0 for pair-annihilation, and θχ is the emission angle of χ2 at the decay of  A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We estimate θe by Monte Carlo simulations [6] and use the mean value  θe = 16 mrad · GeV/Ee±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='8)  Similarly, we estimate the acceptance for electron-recoil signal as  Acc(recoil) =  � rdet/(ldump+lsh+ldec)  0  dθχ  dPang  dθχ  Θ(rdet − rrec  ⊥ ) · Precoil,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='9)  where the radial deviation is approximately given by  rrec  ⊥ =  �  θ2  e + θ2  A′ + θ2  χ · (ldump + lsh + ldec),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10)  and Precoil is the probability of electron recoil  Precoil = n(det)  e− ldet  � E+  e  E−  e  dER  dσrecoil  dER  Θ(ER − Emin)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='11)  given by the electron number density n(det)  e−  of the detector, the length ldet of detector,  kinematically allowed maximum (minimum) recoil energy E+  e (E−  e ) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4),  and the effective recoil cross section approximated by  dσrecoil  dER  ≈ dσ(χ1e− → χ2e−)  dER  + dσ(χ2e− → χ1e−)  dER  exp  �  −ldump + lsh + ldec  l(lab)  χ2  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='12)  for models with χ2 and  dσrecoil  dER  ≈ 2 × dσ(χe− → χe−)  dER  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='13)  for the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The analytical formulae for the differential cross section of the DM-  electron scattering are shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Here, the electron recoil energy ER is required  to be larger than the threshold Emin = 1 GeV to reduce BG events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='dE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='[1/cm2/incident-e]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='Figure 2: The beam-induced neutrino fluxes at the end of the muon shield (left) and at the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='EM calorimeter location (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  3  Expected backgrounds  Background (BG) events in this experiment are classified into two types: beam-induced  and beam-unrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The beam-induced BG events arise from the SM particles produced  by the injected beam at the main beam dump, while the beam-unrelated ones are mainly  due to cosmic-ray muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1  Beam-induced background  The beam injected into the main beam dumps produces SM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In addition to light  particles such as pions, neutrons, and muons, because of the initial high-energy beams,  tau-leptons and heavy mesons (D, B, and Bc) are also produced by the shower photon  hitting nuclei [9]#4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Both light and heavy particle decays can produce neutrinos, which  pass through the muon shield and reach the detectors to generate the beam-induced BG  events for both the electron-recoil and visible-decay signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The neutrino fluxes are calculated with Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We use PHITS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='25 [21]  for production and transport of SM particles other than heavy mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' For heavy meson  production, the differential production cross sections obtained by PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='3 [22] are  implemented into PHITS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' [9] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 2, we show the neutrino fluxes per  #4Heavy meson production through the electromagnetic-shower photons is taken into account, which is  the dominant channel of heavy meson production in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  7  electron injection at the end of the muon shield in the left panel and at the detector behind  the decay volume in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Tau-type neutrino fluxes are negligible, compared  to electron- and muon-types, because the beam energy is not large enough to produce a  considerable number of source particles such as tau-leptons and Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Due to the stopping  power of pions, many pions at rest are produced in the beam dump so that the neutrino flux  increases in the low-energy regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The electron-type neutrinos in a high-energy regime  mainly arise from the decays of Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  A source of irreducible BG events for the electron-recoil searches are neutrino-electron  scattering,#5  νe− → νe−,  ¯νe− → ¯νe−,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1)  where beam-originated neutrinos scatter on the atomic electrons in the detector material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The recoil electrons produce electromagnetic showers in the calorimeter, which are difficult  to distinguish from the DM-electron recoil events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' From the neutrino flux in the right panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 2, the number of neutrino-originated electron-recoil events is estimated to be around  1/year by imposing Emin = 1 GeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' see Appendix C for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Neutrino-nucleon scattering  is another source of beam-induced BG for the electron-recoil search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We list the possible  processes below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Note that, in contrast to the neutrino-electron scattering, these processes  provide recoil nucleons in the calorimeter, making this component of the BG potentially  reducible depending on the design of the calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Quasi-elastic scattering — Neutrinos produced at the beam dump or in the muon  shield may interact with material in the calorimeter through the following quasi-  elastic scattering processes:  CC :  νℓn → ℓ−p,  ¯νℓp → ℓ+n,  NC :  νp → νp,  ¯νp → ¯νp,  νn → νn,  ¯νn → ¯νn,  where CC (NC) denotes charged (neutral) current interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Among these, the  processes νen → e−p and ¯νep → e+n will produce an electromagnetic shower in the  calorimeter and thus potentially be misidentified as an electron-recoil signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The  number of such events is conservatively estimated as 2×102/year using the simulated  neutrino flux (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 2 right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' see Appendix C for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Other processes are without  electromagnetic showers, and thus we expect them to be distinguishable from the  signal events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Neutral pion production — In the detector, muon-type neutrinos may produce reso-  nant neutral pions (π0) decaying into photons:  CC :  νµn → µ−pπ0,  ¯νµp → µ+nπ0,  #5The processes νµe− → µνe and ντe− → τνe do not contribute because the neutrino flux above the  threshold energy, Eν ≥ (m2  µ − m2  e)/2me ≃ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='8 GeV and Eν ≥ (m2  τ − m2  e)/2me ≃ 3 TeV, is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  8  NC :  νµp → νµpπ0,  ¯νµp → ¯νµpπ0,  νµn → νµnπ0,  ¯νµn → ¯νµnπ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The electromagnetic showers from the photons, which is accompanied by a recoil  nucleon, may be misidentified as electron recoils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' As detailed in Appendix C, the  number of the single-π0 production is conservatively estimated as 2 × 103/year from  the flux (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 2 right), but this BG may be reducible depending on the calorimeter  design by, for example, using the information on the radius of the electromagnetic  shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Since these BG events are due to misidentification of electron recoils, we postpone quanti-  tative studies to the detector-design stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Instead, in our plots displaying the ILC-BDX  sensitivity of the electron-recoil searches, we show curves corresponding to 10, 100, and  1000 signal events in a 10-year ILC run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' These choices illustrate a range of plausible sce-  narios for the degree to which beam-induced background from neutrino-nucleon scattering  can be controlled in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Lastly, we consider the potential beam-induced BG events for the visible-decay signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The main source will be the following processes involving strange mesons:  CC :  νµn → µ−K0p,  νµn → µ−K0Σ+,  NC :  νµn → νµK0Λ0,  νµp → νµK0Σ+,  νµn → νµK0Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Neutral kaons produced at the end of the muon shield and the surrounding walls of the  decay volume can decay in the decay volume to yield charged tracks mimicking the visible-  decay signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' [23] of the SHiP experiment, this class of neutrino-induced BG events  is evaluated using Geant4 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We infer the number of charged tracks from neutral kaon  decays in the decay volume by comparing the number of neutrinos between the ILC beam  dump and the SHiP experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In the SHiP experiment with 2 × 1020 protons on target,  7 × 1017 neutrinos with momentum between 2 GeV and 100 GeV are expected, which  results in ∼ 104 pairs of charged tracks from neutral kaon decays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4% of them are  rejected by using the topology of oppositely-charged two tracks thanks to the fact that  the BG-originated tracks generally do not point to the beam-target interaction points [23],  and around 60 events are expected to remain as the BG#6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' At the ILC-BDX, the number  of neutrinos with momentum between 2 GeV and 100 GeV is estimated as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4 × 1017 per  10-year run (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 2 left), and the ILC-BDX expects 12 BG events with pairs of charged  tracks per 10-year run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Consequently, in the following analysis of ILC-BDX visible-decay  searches, we expect 12 BG events and show the expected exclusion limit sensitivity with  95% confidence-level (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' ), which corresponds to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4 signal events in a 10-year run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  #6The SHiP experiment has a veto system to reduce the BG further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We do not discuss it because the  reduction strongly depends on the detector setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='Figure 3: The cosmic-muon fluxes at sea level (dashed line) and the beam dump area (black  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The uncertainly in the latter flux is due to the indeterminate density of subsurface  materials in the Kitakami Mountains at the experimental site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The subsurface materials  are assumed to be soil and granite, with an average density of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4 g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2  Cosmic-ray background  The beam-unrelated BG is mainly due to cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Figure 3 shows the fluxes of cosmic  muons at sea level and the beam dump area evaluated by EXPACS [25–27] and PHITS [21]#7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The kinetic energy loss of the cosmic muon from the ground surface to a depth of ∼120  meters below the ground surface is estimated as ρ × ⟨dE/dx⟩ × 120 m ∼ 50 GeV with the  mass density of the ground ρ ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2 g/cm3 and the stopping power of muon ⟨dE/dx⟩ ∼  2 MeV · cm2/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Due to this large energy loss, the muon flux at the beam dump area is 200  times smaller than that on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Then, the number of the cosmic-muon BG events  for 10 years is estimated as  N BG  cos ∼ O(10) · ϵveto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2)  This estimate arises from the following factors:  O(10) ∼ 10 year  (operation time)  × 10−4 muon/cm2/s  (cosmic muon flux at beam dump area)  × 1002 cm2  (detector area from top view)  × 10−4/muon  (hit rate per cosmic muon)  × 1312 × 5 bunch/s  (bunch number per second)  × 100 ns/bunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (time window per bunch)  #7Muon-induced neutrons avoid the muon veto and may become a BG event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Evaluation of this effect is  left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  10  Time  1312 bunches  1312 bunches  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='73 ms  200 ms  100 ns  ~600 ns  time window  Figure 4: A schematic picture of the ILC beam [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The number of bunches per pulse  is 1312, the beam pulse length is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='73 ms, the pulse repetition rate is 1/200 ms−1 = 5 Hz,  and the bunch spacing is ∼600 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The number of the beam-unrelated BG events can be  reduced by imposing 100 ns time window per bunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The hit rate is the probability that a cosmic-ray muon on an iron block of size 10 cm ×  10 cm×10 cm will cause an energy deposition above the threshold value of 1 GeV, which is  evaluated in the Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Detectors with smaller cell sizes are sufficiently  realistic, and the above estimation of the hit rate is conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The time structure  of the ILC bunches [17, 18] is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The factor ϵveto is the reduction  factor by the cosmic-muon veto, which is typically much smaller than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The deep  underground location of the detector and coincidence time window significantly reduce the  beam-unrelated BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Consequently, we will neglect the cosmic-muon BG in the rest of this  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  4  Examples of detectable DM models  We evaluate the sensitivity of the ILC electron and positron beam dump experiments to DM  particles using the formulae in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Taking into account the beam-induced BG events  coming from the SM neutrinos, we show the prospects of the ILC-BDX for electron-recoil  and visible-decay searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' For the electron-recoil searches, we illustrate parameter spaces  in which more than 10, 102, and 103 signal events are expected at a 10-year ILC-BDX run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  For the visible-decay searches, regions with more than 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4 signal events in a 10-year run are  shown, which corresponds to a 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Throughout this work, we assume that  the beam-unrelated BG events are negligible, as indicated by the estimates in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  As benchmark models, we consider a class of DM models in which the DM field is  charged under a new “dark” gauge symmetry, U(1)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  At low energy, where U(1)D is  spontaneously broken and the dark photon A′ acquires a mass mA′, the relevant terms of  11  the Lagrangian are given by  L ⊃ −1  4FµνF µν − 1  4F ′  µνF ′µν + 1  2m2  A′A′  µA′µ − ϵ  2F ′  µνF µν − gDA′  µJµ  χ − eAµJµ  EM ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1)  where Fµν (F ′  µν) is the photon (dark photon) field strength, gD is the U(1)D gauge coupling,  Jµ  χ (Jµ  EM) is the DM (electromagnetic-matter) current, and ϵ parametrizes the kinetic mixing  between the photon and the dark photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' By the redefinition Aµ → Aµ − ϵA′  µ, the gauge  kinetic terms become canonical, and the interaction terms read  Lint = −gDA′  µJµ  χ + ϵeA′  µJµ  EM − eAµJµ  EM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2)  We consider five models for the nature of the DM: pseudo-Dirac-fermion DM with small  or large mass-splitting, scalar elastic DM, scalar inelastic DM, and Majorana-fermion DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The DM current Jµ  χ for each of these models is listed below, and the recoil profiles for  χe → χe scattering in the lab frame are summarized in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The DM models used in this study have four or five free parameters: the dark photon  mass mA′, the DM mass mχ1, the dark fine structure constant αD ≡ g2  D/(4π), the kinetic  mixing parameter ϵ, and the mass difference ∆ = mχ2 − mχ1 of DMs if χ2 is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The  DM particles remain in chemical equilibrium with the SM plasma in the early universe  before freezing out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The DM relic density is determined by the cross section of the pair-  annihilation process DM + DM ↔ SM + SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' For mχ ≪ mA′, the annihilation cross section  for these models can be parametrized as σv ∝ y/m2  χ with y ≡ ϵ2αD(mχ/mA′)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The cross  section for which the relic density matches the observed value defines the “relic target” in the  (mχ, y) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Following the common practice, we will use this two-dimensional parameter  space to illustrate the sensitivity of the ILC beam dump searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Note however that the  signal event rates at the ILC-BDX do not depend exclusively on these two parameters and  thus further assumptions are necessary to represent the reach on the (mχ, y) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' These  assumptions will be specified in the captions of each of our reach plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1  Pseudo-Dirac DM  Pseudo-Dirac inelastic DM#8 is described by a pair of two-component Weyl fermions (η, ξ)  that have opposite unit charge under U(1)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Both a U(1)D-conserving Dirac mass mD and  a U(1)D-breaking Majorana mass mM are present in the low-energy theory, since the U(1)D  symmetry is spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Namely, the Lagrangian has the mass terms  −L ⊃ mDηξ + 1  2mM(η2 + ξ2) + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='3)  #8If the DM χ is a Dirac fermion, the DM current is given by Jµ  χ ∝ ¯χγµχ, and it annihilates to SM  in s-wave two-to-two processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Constraints from the cosmic microwave background (CMB) power in-  jection have ruled out s-wave two-to-two annihilating DM lighter than O(10) GeV as the thermal DM  candidate [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' This motivates pseudo-Dirac DM as the simplest viable model of sub-GeV fermionic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  12  Here and below, we assume mD ≫ mM > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The mass eigenstates are then given by  χ1 =  i  √  2(η − ξ),  χ2 = 1  √  2(η + ξ)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4)  with masses mχ1,2 = mD ∓ mM, and the DM current becomes off-diagonal:  Jµ  χ = i¯χ2γµχ1 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5)  The ILC-BDX search strategy depends on the mass difference ∆ ≡ mχ2 − mχ1 = 2mM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  For ∆ > 2me (large mass-splitting), the heavier DM particle χ2 can decay into χ1e−e+  with the partial decay width [29,30]  Γ(χ2 → χ1e−e+) ≃ 4ϵ2ααD∆5  15πm4  A′  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='6)  which results in a visible-decay signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Otherwise, if ∆ < 2me (small mass-splitting),  χ2 does not decay inside the apparatus because of the small width of the main decay  channel χ2 → χ1 + 3γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' This region of parameter space is accessible only by searches for  the electron-recoil signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Figure 5 summarizes our results for the small mass-splitting case, where we take the limit  ∆ ≃ 0 and fix αD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5 and mA′ = 3mχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The three red solid lines show the sensitivity of the  ILC-BDX recoil-electron search with √s = 250 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The lines correspond to 10, 102, and  103 signal events with 10-year statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='#9 The three lines illustrate the range of plausible  scenarios for the level of reducible beam-induced backgrounds affecting the search, see  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The shaded regions are excluded by past experiments listed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5, and  the solid black lines are the relic target for pseudo-Dirac DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Figure 6 shows the result for larger mass-splitting, where we fix ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1mχ1 as well  as αD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1 and mA′ = 3mχ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Consequently, the visible-decay signal is expected for  mχ1 ≳ 1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' It is noteworthy that the search for visible decays of χ2 provides a unique  sensitivity for this model, well beyond that achievable at proposed dedicated future facilities  such as LDMX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' This strongly motivates the inclusion of a multi-layer tracker in the decay  volume (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 1) as a key component of the ILC-BDX design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Below, we discuss the physics that determines the structure of the reach curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 5  and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1  Small mass-splitting: ∆ < 2me  Both pair-annihilation and bremsstrahlung are included as the source of signal events,  where on-shell dark photons are produced and decay into a χ2-χ1 pair, and either χ2 or χ1  is detected as electron recoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  #9We also checked that the visible decay signals coming from Z decays at the Giga-Z program [35] of  the ILC are not significant for mχ ≲ 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content="y = ϵ2αD(mχ/mA')4  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='BaBar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Relic Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Belle II (extrapolated)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='LDMX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC-250 10-year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='BDX (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='(a) electron beam dump  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content="y = ϵ2αD(mχ/mA')4  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='BaBar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Relic Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Belle II (extrapolated)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='LDMX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC-250 10-year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='BDX (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='(b) positron beam dump  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Figure 5: Projected sensitivity reach of a 10-year ILC-250 run in the pseudo-Dirac DM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='model with small mass-splitting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' ∆ ≪ min(me,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' mχ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' It is assumed that αD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5 and  mA′ = 3mχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' On each panel, the three red solid lines show the sensitivity of the ILC-BDX  recoil-electron search, corresponding to 10, 102, and 103 signal events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The black solid line  shows DM relic targets [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The shaded grey region is excluded by the past experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The dashed lines show the sensitivity of the BDX recoil-electron search  (purple) [29], LDMX (blue) [31], and Belle II experiment (green) [12,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Pair-annihilation (recoil-electron) — The DM with mass less than ≃ 10−2 GeV cannot  be detected because of the threshold Emin = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' This is because decays of the  dark photon with mass less than √2meEmin cannot contribute to signal events since  the energy of the produced dark photon is Elab  A′ ≃ Ee+ ≃ m2  A′/2me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Also, the DM  with mass larger than ≃ 10−1 GeV cannot be produced from decays of the dark  photon because the dark photon mass cannot exceed √2meEbeam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In the positron  beam dump for the DM mass of √2meEbeam/3, primary positron beam production  dominates, and a peak structure arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Bremsstrahlung (recoil-electron) — For the DM mass smaller than ∼ 10−2 GeV, the  number of the recoil-electron events is suppressed by the threshold Emin = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Since the expected decay angle of the dark photon is (π/2) · (mA′/EA′), the dark  photon energy has to satisfy mA′ · (π/2) · (ldump + lsh + ldec)/rdet ≲ EA′ to obtain  sufficient angular acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' the minimum energy of the dark photon is  EA′ ∼ Emin = 1 GeV because of the threshold,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' and the signal events from the DM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='with mass smaller than ∼ 10−2 GeV are suppressed even if the angular acceptance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='mχ1 [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content="y = ϵ2αD(mχ1/mA')4  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Belle II (extrapolated)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='LDMX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='BDX(decay)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Relic Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC-250 10-year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (decay)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='(a) electron beam dump  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='mχ1 [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content="y = ϵ2αD(mχ1/mA')4  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Belle II (extrapolated)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='LDMX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Relic Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC-250 10-year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (decay)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='BDX(decay)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='(b) positron beam dump  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Figure 6: Projected sensitivity reach of a 10-year ILC-250 run in the pseudo-Dirac DM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='model with a large mass-splitting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1mχ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  It is assumed that αD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1 and  mA′ = 3mχ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The notation is similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' in addition, dot-dashed lines show the  sensitivity of the ILC-BDX decay-signal search (95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' exclusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Also shown are  expected sensitivities of the BDX visible-decay search (purple-dashed lines) [14, 29], the  LDMX (blue) [33], and Belle II (green) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' For the DM with mass larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1 GeV, the angular acceptance becomes  worse, and the sensitivity rapidly decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2  Large mass-splitting: ∆ > 2me  The new feature in this case is the availability of the visible-decay signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Here, we discuss  the sensitivity reach for this channel, highlighting each of the DM decay processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The  results of the electron recoil searches are similar to the case of the small mass-splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Pair-annihilation (visible-decay) — For ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1mχ1, the heavier DM with the mass  smaller than ≃ 10−2 GeV cannot decay, and the visible-decay signals do not arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Similar to the recoil-electron search, the DM with mass larger than 10−1 GeV can-  not be produced because the dark photon mass cannot exceed √2meEbeam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In the  positron beam dump experiment, for mχ1 = √2meEbeam/3, the peak structure arises  due to the primary positron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Bremsstrahlung (visible-decay) — Similar to the pair-annihilation process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' for ∆ =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content="y = ϵ2αD(mχ/mA')4  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Relic Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Belle II (extrapolated)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SuperCDMS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SNOLAB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SENSEI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='LDMX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC-250 10-year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='(a) electron beam dump  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content="y = ϵ2αD(mχ/mA')4  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Relic Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Belle II (extrapolated)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SuperCDMS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SNOLAB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SENSEI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='LDMX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC-250 10-year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='(b) positron beam dump  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Figure 7: The results for the scalar elastic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' It is assumed that αD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5 and mA′ =  3mχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The notation is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' in addition, the gray lines show the expected  sensitivities of SENSEI [32] and SuperCDMS [12,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1mχ1, the heavier DM with the mass smaller than ≃ 10−2 GeV cannot decay and  produce the visible-decay signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' For 10−1 GeV ≲ mχ, the number of signal events  rapidly decreases because the angular acceptance becomes worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2  Scalar elastic DM  The model with a scalar elastic DM is described by a complex scalar field χ that is SM-  singlet with unit U(1)D-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' It is safe from CMB bounds since the DM annihilation  is in p-wave and thus with a suppressed rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The DM current, originating in the kinetic  term |Dµχ|2, is given by  Jµ  χ = i(χ∗∂µχ − χ∂µχ∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='7)  The analysis results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 7 with the same notation as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' For the non-  relativistic DM-SM fermion elastic scattering cross section, there is no velocity suppression,  and the constraints from direct detection experiments are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content="y = ϵ2αD(mχ/mA')4  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Relic Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Belle II (extrapolated)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='LDMX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC-250 10-year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='(a) electron beam dump  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content="y = ϵ2αD(mχ/mA')4  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Relic Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Belle II (extrapolated)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='LDMX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC-250 10-year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='(b) positron beam dump  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Figure 8: The results for the scalar inelastic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The prescription αD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5, mA′ = 3mχ  is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Notation is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='3  Scalar inelastic DM  The scalar inelastic DM is characterized by the dark photon current  Jµ  χ = χ1∂µχ2 − χ2∂µχ1  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='8)  involving two real scalar fields χ1,2 that are quasi-degenerate but have non-zero mass dif-  ference ∆ = mχ2 − mχ1 > 0 after the spontaneous symmetry breaking of U(1)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='#10 We  focus only on the small mass-splitting case since the large mass-splitting case of the scalar  inelastic DM is similar to that of the pseudo-Dirac inelastic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The results of our analysis  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 8, with the same notation as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4  Majorana DM  For a Majorana fermion χ, the DM current can be an axial-vector as follows:  Jµ  χ = 1  2 ¯χγµγ5χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='9)  #10The mass difference is realized by terms such as χ2H2 after the electroweak (and U(1)D) symmetry  breaking, where χ is a complex scalar field yielding χ1,2, and H is the Higgs doublet [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=',  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' [36–39] for models realizing scalar inelastic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+page_content='10-9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content="y = ϵ2αD(mχ/mA')4  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Relic Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Belle II (extrapolated)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='LDMX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC-250 10-year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SENSEI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SuperCDMS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SNOLAB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='(a) electron beam dump  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10-7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content="y = ϵ2αD(mχ/mA')4  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Relic Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Belle II (extrapolated)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='LDMX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC-250 10-year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='103  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SENSEI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SuperCDMS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='SNOLAB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='ILC (recoil)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='(b) positron beam dump  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='Figure 9: The results for the Majorana DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The prescription αD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5, mA′ = 3mχ is  adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Notation is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Similar to the scalar DM, the pair-annihilation rate of the DM into SM particles is sup-  pressed by the DM velocity, and a sub-GeV Majorana fermion DM is safe from the CMB  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The results of our analysis for the Majorana DM are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 9, with the  same notation as in the scalar elastic DM case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5  Existing constraints and projected sensitivities at other ex-  periments  Searches for sub-GeV DM are pursued by accelerator-based experiments and direct detec-  tion experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In the accelerator-based experiments, the DM particles may be produced  by beam-beam or beam-target collisions, and the searches are performed either by detect-  ing their scattering off of the SM particles in the detectors, the visible decays of the excited  DM states, or the missing energy/momentum events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In the direct detection experiments,  the DM particles from the halo are searched for by detecting their scattering off of the SM  particles in the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We list the existing constraints included in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 5-9 as follows:  E137 electron beam dump experiment — The light DM may be produced at the SLAC  electron beam dump by injection of 20 GeV electron beams [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The produced DM  particles may be detected by their scattering on electrons in the aluminum and plastic  scintillator 400 m downstream from the beam dump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The visible decays of the heavier  18  dark sector particles are also detectable at the scintillator [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  LSND proton beam dump experiment — The light DM may also be produced in the  Los Alamos LSND proton beam dump experiment by injection of proton beams with  kinetic energy of 800 MeV [29,32,33,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The DM particles are produced dominantly  by π0 decays at the fixed target, and DM-electron scattering or visible decay events  are detected at the downstream detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  γ+missing energy search — The DM may be produced at B-factories by process  e+e− → γ +A′(∗) → γχ¯χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The mono-photon and missing-energy events at the BaBar  set bounds on the DM models [32,42,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Direct detection experiments — The DM-SM non-relativistic scattering may be de-  tected in direct detection experiments, such as XENON-10/100 and CRESST II [44–  47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' For the DM-nuclear scattering process, the CRESST II experiment is sensitive  to mχ ∼ 1 GeV because of the low nuclear recoil energy threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The sub-GeV  DMs with MeV to GeV mass may also be explored using the DM-electron scattering  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  As the future DM searches, we included projected sensitivities at Belle II [12, 32],  BDX [29], LDMX [32], SENSEI [32], and SuperCDMS [12, 32] in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 5-9 to compare  with the sensitivity reaches of the ILC-BDX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  5  Summary  In this paper, we explored the potential of a search for sub-GeV dark matter at the ILC  beam dumps using the ILC-BDX setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Previous studies of beam dump experiments at  the ILC focused on long-lived BSM particles decaying into visible SM final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In this  study, on the contrary, we consider DM models where a dark photon connects the stable  DM to the SM sector, and focus on DM detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Five DM models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=', pseudo-Dirac DM  with small and large mass-splitting, scalar elastic and inelastic DM, and Majorana DM are  considered as reference scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' At the ILC, electrons and positrons in electromagnetic  shower are produced by the beam injection into the beam dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' These particles produce  dark photons through the bremsstrahlung and pair-annihilation processes, which in turn  decay to pairs of DM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' (Pair-production of DM via virtual dark photon exchange  is also possible,) Produced DM particles propagate over a long distance (of order 100 m) to  the detector and elastically scatter with electrons inside the detector, and then the recoil  electrons are detected as a signature of the DM production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' For the inelastic DM models,  there is another signal produced by a visible decay of the heavier dark state into the lighter  one and SM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The ILC-BDX setup includes a multi-layer tracker designed to search  for this visible-decay signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  19  In the DM search at the ILC-BDX, there are two kinds of backgrounds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=', the beam-  induced and beam-unrelated BGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The beam-induced BG is due to neutrinos mainly pro-  duced by meson and lepton decays in the beam dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Elastic neutrino-electron scat-  tering in the detector is nearly indistinguishable from DM-induced electron recoil events,  providing irreducible BGs to this search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Inelastic neutrino-nucleus interactions can also  mimic electron-recoil and visible-decay signals, though these backgrounds are potentially  reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We evaluated the neutrino-induced irreducible and reducible BGs by using the  neutrino fluxes at the detector calculated with Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We also estimated  the beam-unrelated BG, which comes predominantly from cosmic-ray muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' This BG  component can be significantly suppressed by the deep underground location of the detec-  tor and beam-coincidence time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  We evaluated the number of DM particles produced at the ILC electron and positron  beam dumps and the rate of expected signal events, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=', electron-DM recoils and visible-  decays at the ILC-BDX, in our reference models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The predicted signal and background  rates were then used to estimate the sensitivity reach of the ILC-BDX experiment with a  data set corresponding to a 10-year ILC run at √s = 250 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The results for each DM  model are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 5-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We found that in all five reference models considered here,  the ILC-BDX experiment has sensitivity to regions of model parameter space well beyond  the existing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In many cases, the ILC-BDX can conclusively probe the region  of parameter space where the DM has the thermal abundance consistent with observations  (the “relic target”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' We find that while electron and positron beam dumps have similar  performances, the sensitivity of the latter is somewhat better because the primary positron  beam contributes to the DM production through pair-annihilation on electrons in the dump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The sensitivity of the ILC-BDX is particularly impressive in models where the visible-decay  signal is available, such as the pseudo-Dirac or scalar inelastic DM models with mass-  splitting ∆ > 2me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In such models, the reach of the ILC-BDX significantly exceeds that of  even the most ambitious proposed dedicated searches for sub-GeV DM, such as LDMX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  In the discussion of future searches for sub-GeV dark matter, it is important to note  the complementarity between the electron-recoil technique used by ILC-BDX and the miss-  ing energy/momentum technique used by experiments such as LDMX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The signal rate at  missing energy experiments is simply proportional to the total cross section of DM produc-  tion by the beam electron interactions with the target material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' On the other hand, the  signal rate at electron-recoil experiments is proportional to the product of this production  cross section and that of the elastic scattering of DM on target electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The relationship  between the production and elastic scattering cross sections is model-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Thus,  analyzing data from both types of experiments will provide an opportunity to identify the  underlying DM model, if a signal is observed with one or both approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' This is an  exciting possibility, and it strongly motivates pursuing both approaches in parallel in the  future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  20  In our evaluation of the acceptance of the signal events, we adopted some simplifying  assumptions, for instance, the approximated angular distribution of the heavy DM state  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5) and radial deviations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Likewise, several simplifying  assumptions were made in the estimates of the background rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  As a  next step, it would be important to design a realistic Monte Carlo model of the ILC-BDX  detector, and use it to provide a more precise evaluation of both signal and background  rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Another important direction for future work is to design strategies for handling the  reducible beam-induced BG, such as additional cuts or a veto system, and to evaluate such  strategies quantitatively using Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Acknowledgements  We are grateful to the members of the ILC Task Force on fixed-target experiments and dark  sectors, especially Claude Vallee, for discussions that initiated this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' KA is supported  by JSPS KAKENHI Grant Number JP18H01210, JP21K20365, and MEXT KAKENHI  Grant Number JP18H05543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' YS is supported by JSPS KAKENHI JP21H05466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' MP is  supported by the NSF grant PHY-2014071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  A  Dark photon production cross sections  In this appendix, we list the cross sections of dark photon production via pair-annihilation  and bremsstrahlung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1  Pair-annihilation production  The cross section of the resonant annihilation process is given by [48]  σ(e+e− → A′) = 12π  m2  A′  Γ2  A′/4  (√s − mA′)2 + Γ2  A′/4,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1)  where s is the center-of-mass energy squared and ΓA′ is the total decay width of the dark  photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In this work, ΓA′ is assumed to be small enough that the narrow-width approxi-  mation can be used:  σ(e+e− → A′) ≃ 2π2αϵ2  me  δ  �  Ei − m2  A′  2me  + me  �  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2)  where δ(x) denotes the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  21  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2  Bremsstrahlung production  The differential cross section of the bremsstrahlung process under the Weizs¨acker-Williams  approximation [49–52] is given by [53,54]  d2σ(iN → iA′N)  dEA′ dθA′  = ϵ2α3 1  Ei  sin θA′  �  E2  A′ − m2  A′  �  E2  i − m2  e  1  1 − x  A2→2  t=tWW  min  2tWW  min  χ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='3)  where x = EA′/Ei, tWW  min = ˜s2/4E2  i , ˜s = −˜u/(1 − x), and  ˜u = −xE2  i θ2  A′ − m2  A′ 1 − x  x  − m2  ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4)  The effective flux of photons, χ, is given by  χ =  � tmax  tmin  dt t − tmin  t2  G2(t)  = Z2  �  1 − b  a  �−2 �  − (a + b + 2abtmax)(tmax − tmin)  (1 + atmax)(1 + btmax)  + a + b + 2abtmin  a − b  ln  �1 + atmax  1 + btmin  1 + btmin  1 + atmin  � �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5)  with tmin = m4  A′/4E2  i , tmax = m2  A′ + m2  e, and  G2(t) =  ��  at  1 + at  � �  1  1 + bt  �  Z  �2  ,  a = 1122Z−2/3  m2  e  ,  b =  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='164 GeV2A−2/3,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='6)  where Z is the atomic number and A is the mass number of the target atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The amplitude  under the the Weizs¨acker-Williams approximation is given by  A2→2  t=tWW  min = 22 − 2x + x2  1 − x  + 4(m2  A′ + 2m2  e) ˜ux + m2  A′(1 − x) + m2  ex2  ˜u2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='7)  B  DM-electron recoil cross sections  For the inelastic-fermion DM model and the scalar DM model, the cross section of the  DM-electron recoil process is given by [30,55],  dσ(χ1e → χ2e)  dEe  =  me  8πλ(s, m2  e, m2  χ1)|M|2,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1)  where Ee is the energy of the recoil electron, λ(x, y, z) = (x − y − z)2 − 4yz, Ee = Eχ1 +  me − Eχ2, and s = m2  χ1 + m2  e + 2meEχ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The squared matrix elements are given by  |M|2 =  8me(ϵegD)2  [2me(Eχ2 − Eχ1) − m2  A′]2  �  me(E2  χ1 + E2  χ2)  − ∆2  2 (Eχ2 − Eχ1 + me) + m2  e(Eχ2 − Eχ1) + m2  χ1Eχ2 − m2  χ2Eχ1  �  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2)  22  for the pseudo-Dirac inelastic DM model and  |M|2 =  8me(ϵegD)2  [2me(Eχ2 − Eχ1) − m2  A′]2  �  2meEχ1Eχ2 + m2  χ1Eχ2 − m2  χ2Eχ1  �  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='3)  for the scalar inelastic DM model, where Eχi (i = 1, 2) is the energy of χi measured in the  laboratory frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The kinematically allowed maximum (minimum) recoil energy E+  e (E−  e )  is given by [55]  E±  e = s + m2  e − m2  χ2  2√s  Eχ1 + me  √s  ±  �  λ(s, m2  e, m2  χ2)  2√s  pχ1  √s,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4)  where pχ1 = �E2  χ1 − m2  χ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  The recoil cross section for the other models are given by [30,40,55,56]  dσ(χe → χe)  dEe  = 4πϵ2ααD  2meE2  χ − (2meEχ − meEe + m2  χ + 2m2  e)(Ee − me)  (E2  χ − m2  χ)(m2  A′ + 2meEe − 2m2  e)2  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5)  for the pseudo-Dirac elastic DM model,  dσ(χe → χe)  dEe  = 4πϵ2ααD  2meE2  χ − (2meEχ + m2  χ)(Ee − me)  (E2  χ − m2  χ)(m2  A′ + 2meEe − 2m2  e)2  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='6)  for the scalar elastic DM model, and  dσ(χe → χe)  dEe  = 4πϵ2ααD  2me(E2  χ − m2  χ) + [m2  χ − me(2Eχ − Ee + 2me)](Ee − me)  (E2  χ − m2  χ)(m2  A′ + 2meEe − 2m2  e)2  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='7)  for the Majorana DM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The range of Ee is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4) with replacing χi by  χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  C  Neutrino-induced background  We evaluate the number of neutrino-induced irreducible and reducible BG events from  the neutrino flux of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The number of irreducible BG events does not depend on the  details of the detector design and is evaluated in the same way as the signal rate calculation  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' On the other hand, the number of reducible BG events highly depends on  the detector design, so we conservatively take a pessimistic scenario without imposing the  threshold energy of the electromagnetic showers#11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' This section describes the detailed  calculations of the number of BG events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  23  10-3  10-2  10-1  100  101  0  1  2  3  4  5  6  Emin [GeV]  NBG  10-3  10-2  10-1  100  101  0  10  20  Emin [GeV]  N95 %  ILC-250  1 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  10 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  Figure 10: Left panel: The number of irreducible BG events in a 1-year run as a function  of the threshold Emin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Right panel: The number of signal events at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' in a 1-year  (blue) and 10-year (red) run as a function of the threshold Emin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1  Irreducible background  The number of neutrino-electron recoil events is expressed by the following formula:  NBG = Ne±  �  i=νe,µ,τ,¯νe,µ,τ  �  dEi πr2  det · dφi  dEi  n(det)  e−  ldet ·  �  Emin  dER  dσ(ie− → ie−)  dER  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='1)  where φi is the neutrino flux at the detector position (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 2 right) and σ(ie− → ie−)  denotes the neutrino-electron elastic scattering cross section [57,58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 10, the number of neutrino-electron recoil events per year is  shown as a function of the threshold Emin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' The rate is reduced to N BG  beam ∼ 1/year by  imposing Emin = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' By taking into account the irreducible BG events and assuming  no signal events are observed, we estimate the expected upper bounds on the number of  signal events at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='#12, shown by the solid lines in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  #11The rate of quasi-elastic scattering and pion production processes would not change significantly by  imposing the threshold Emin = 1 GeV because their cross sections are very small for neutrino energy  smaller than 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  #12The Poisson 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' upper bound corresponds to three signal events in the absence of BG events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  24  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2  Reducible background  The number of quasi-elastic scattering events is estimated by the following formula:  NQE = Ne±  �  dEν πr2  det · ldet ·  �dφνe  dEν  n(det)  n  σ(νen → e−p) + dφ¯νe  dEν  n(det)  p  σ(¯νep → e+n)  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2)  where φνe and φ¯νe are the electron-neutrino and electron-antineutrino fluxes at the detector  position (see the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 2), n(det)  n(p) is the neutron (proton) number density of  the detector, and σ denotes the quasi-elastic scattering cross section [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' In contrast to  the estimation of the irreducible BG events, we do not impose the threshold energy of the  electromagnetic shower;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2) is therefore a conservative estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='2), the  number of the quasi-elastic scattering events per year is estimated as  NQE ∼ 2 × 102 ×  �  Ne±  4 × 1021  �� rdet  2 m  �2�  ldet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='64 m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='3)  Similarly, the number of the single-π0 production events is given by  Npion = Ne±  �  dEν πr2  det · ldet  ×  �  dφνµ  dEν  �  n(det)  n  �  σ(νµn → µ−pπ0) + σ(νµn → νµnπ0)  �  + n(det)  p  σ(νµp → νµpπ0)  �  + dφν¯µ  dEν  �  n(det)  p  �  σ(¯νµp → µ+nπ0) + σ(¯νµp → ¯νµpπ0)  �  + n(det)  n  σ(¯νµn → ¯νµnπ0)  ��  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='4)  where the single-π0 production cross sections are listed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' Combining the neutrino  flux of the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' 2 and the cross sections in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content=' [58], the number of the pion  production events per year is obtained as  Npion ∼ 2 × 103 ×  �  Ne±  4 × 1021  �� rdet  2 m  �2�  ldet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='64 m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
+page_content='5)  As in the calculation of NQE, the threshold energy of the electromagnetic shower is not  imposed and thus this estimate is conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E2T4oBgHgl3EQfUwdx/content/2301.03816v1.pdf'}
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+ClimaBench: A Benchmark Dataset For Climate Change Text
+Understanding in English
+Tanmay Laud∗
+UCSD∗
+tlaud@ucsd.edu
+Daniel Spokoyny∗
+CMU
+dspokoyn@andrew.cmu.edu
+Tom Corringham
+SIO
+tcorringham@ucsd.edu
+Taylor Berg-Kirkpatrick
+UCSD
+tberg@eng.ucsd.edu
+Abstract
+The topic of Climate Change (CC) has re-
+ceived limited attention in NLP despite its real
+world urgency.
+Activists and policy-makers
+need NLP tools in order to effectively pro-
+cess the vast and rapidly growing textual data
+produced on CC. Their utility, however, pri-
+marily depends on whether the current state-
+of-the-art models can generalize across vari-
+ous tasks in the CC domain. In order to ad-
+dress this gap, we introduce Climate Change
+Benchmark (ClimaBench), a benchmark col-
+lection of existing disparate datasets for eval-
+uating model performance across a diverse set
+of CC NLU tasks systematically. Further, we
+enhance the benchmark by releasing two large-
+scale labelled text classification and question-
+answering datasets curated from publicly avail-
+able environmental disclosures.
+Lastly, we
+provide an analysis of several generic and
+CC-oriented models answering whether fine-
+tuning on domain text offers any improve-
+ments across these tasks. We hope this work
+provides a standard assessment tool for re-
+search on CC text data.
+1
+Introduction
+There is an ever growing body of text-based cli-
+mate documents that contain vital information on
+carbon emissions, impacts, and risks – for example,
+firms announce their emissions reduction targets
+or cities disclose their water risks and potential
+exposure to drought. These documents include
+corporate ESG (Environmental, Social, and Gov-
+ernance), climate assessments, climate legislation,
+regulatory filings, and are produced by corpora-
+tions, cities, states, and national governments. Al-
+though, there have been massive efforts by inter-
+national organizations to standardize climate re-
+porting and disclosures, the vast majority of data
+is still non machine-readable. Natural language
+∗ Equal Contribution
+processing has potential to help researchers and
+policy-makers search, extract and gain insights
+from climate texts. Several works have introduced
+small-scale climate-specific datasets for detecting
+relevance to climate (Leippold and Varini, 2020),
+identifying stance detection (Vaid et al., 2022b) and
+fact-checking (Leippold and Diggelmann, 2020) of
+social media claims. Other works have studied the
+effect of further pretraining of transformers using
+in-domain data such as news articles, academic
+papers, and climate reports (Luccioni et al., 2020;
+Webersinke et al., 2021).
+However, the use of proprietary datasets for both
+pretraining as well as evaluation makes it difficult
+to effectively track progress in this field. To help
+unify the disparate efforts of applying NLP tech-
+nologies to climate-related problems we propose
+CLIMABENCH, a benchmark for training and eval-
+uating NLP models on classification and question-
+answering tasks pertaining to climate texts. Bench-
+marks have been an effective tool for the general
+NLP community (Wang et al., 2018; Srivastava
+et al., 2022; Liang et al., 2020) as well as for spe-
+cific areas such as legal (Chalkidis et al., 2022) or
+biomedical (Lee et al., 2020) domains.
+We introduce two new large scale datasets,
+CLIMA-CDP and CLIMA-INSURANCE, which are
+based on publicly accessible climate questionnaires
+of cities, states and corporations. In contrast with
+existing datasets, these questionnaires have an or-
+der of magnitude more examples and the exam-
+ples themselves are longer. Further, CLIMA-QA
+uses a highly structured questionnaire: with hun-
+dreds of unique question types across a multi-
+tude of topics such as transportation, water se-
+curity, and governance. Additionally, we collate
+five existing climate datasets and, to the best of
+our knowledge, conduct the first evaluation on the
+SCIDCC (Mishra and Mittal, 2021) topic classifi-
+cation dataset.
+A common technique to adapt large language
+arXiv:2301.04253v1  [cs.CL]  11 Jan 2023
+
+Dataset
+Task Type
+Domain
+# Train
+# Dev
+# Test
+# Classes
+CLIMA-INSURANCE (Ours)
+Binary Classification
+NAIC
+13.8K
+1.6K
+1.7K
+2
+CLIMA-INSURANCE+ (Ours)
+Multi-class Classification
+NAIC
+13.7K
+1.7K
+1.7K
+8
+CLIMA-CDP (Ours)
+Topic Classification
+CDP Cities
+46.8K
+8.7K
+8.9K
+12
+CDP Cities
+48.2K
+8.5K
+9.3K
+294
+CLIMA-QA (Ours)
+QA Ranking
+CDP States
+8.7K
+0.9K
+1.1K
+132
+CDP Corporations
+34.5K
+3.6K
+4.9K
+43
+CLIMATEXT (Leippold and Varini, 2020)
+Binary Classification
+Wikipedia, 10-K
+6K
+0.3K
+1.6K
+2
+CLIMATESTANCE (Vaid et al., 2022a)
+Ternary Classification
+Twitter
+3K
+0.3K
+0.3K
+3
+CLIMATEENG (Vaid et al., 2022a)
+Multi-class Classification
+Twitter
+3K
+0.3K
+0.3K
+5
+CLIMATEFEVER (Leippold and Diggelmann, 2020)
+Fact-Checking
+Wikipedia
+-
+-
+1.5K
+3
+SCIDCC (Mishra and Mittal, 2021)
+Topic Classification
+Science Daily
+9.2K
+1.1K
+1.1K
+20
+Table 1: General Statistics of the CLIMABENCH datasets
+Text
+Class / Question
+CLIMA-
+INSURANCE+
+...Each year Aflac reports its US operations Scope 1 and Scope 2 emissions to the Carbon Disclosure
+Project. Since 2007, Aflac’s owned facilities in terms of square feet have increased by more than
+10% while total Scope 1 and 2 CO2e emissions have significantly decreased compared to 2007
+emissions...
+Question Type 1 / Yes
+CLIMA-CDP
+...These Plans must include management of CD&E waste, both through on-site recycling and re-use
+and on-site waste processing prior to disposal.Westminster will contribute to the London Plan target
+of net self-sufficiency (managing 100% of London’s waste within London) by 2026 by planning for
+Westminster’s apportionment targets...
+Governance
+and
+Data
+Management
+CLIMA-QA
+Flooding from sea level rise will damage building and roads in the coastal neighborhoods of the city.
+Flooding also represents a risk to major transportation hubs infrastructure in the region. Coastal
+flooding can have a long-term effect on major industrial and commercial activities along the coastal
+areas of the city as well as damage urban forestry and local natural biodiversity.
+Please describe the im-
+pacts experienced so far,
+and how you expect the
+hazard to impact in the fu-
+ture.
+SCIDCC
+...In the past, wave breaking has been tracked by so-called "whitecap coverage," in which still
+or video imagery was used to statically identify ocean whitecaps and the corresponding surface
+coverage by breaking waves. But those measurements suffered...
+Hurricanes and Cyclones
+Table 2: Examples (pairs of inputs and outputs) for some of the newly introduced datasets in CLIMABENCH
+models (LLM) to new tasks has been to pretrain
+on in-domain texts. In the CC domain, datasets
+of corporate sustainability reports, climate news
+articles, climate-related research abstracts and fi-
+nancial reports from EDGAR1 have been used for
+pretraining (Luccioni et al., 2020; Webersinke et al.,
+2021). However, certain alternate sources of cli-
+mate text data are proprietary and models trained
+on this data are generally unavailable for research.
+For example, the Global Reporting Initiative (GRI),
+a non-governmental organization, once hosted over
+63,000 corporate sustainability reports which are
+now inaccessible2.
+Our experiments on CLIMABENCH suggest that
+1U.S. Securities and Exchange Commission Electronic
+Data Gathering, Analysis, and Retrieval (EDGAR) Database
+of public company filings.
+2https://www.globalreporting.org/how-to-use-the-gri-
+standards/register-your-report/
+state-of-the-art LLMs provide satisfactory perfor-
+mance when finetuned on the downstream CC tasks,
+but there is still substantial room for improvement.
+Further, there was no single model that dominated
+the rest on all of the classification tasks. Beyond
+evaluation of existing models we also show that
+training models on our new benchmark can be di-
+rectly applied to a real-world climate focused down-
+stream task. Specifically, using the best performing
+model trained on our new benchmark, we predict
+relevant questions for text segments from U.S. State
+climate action plans which is a completely different
+type of CC document. Out of the 500 predictions
+judged by a climate change researcher 71% were
+relevant and 35.6% highly relevant.
+We hope that CLIMABENCH lowers the barrier
+for the NLP community to conduct research on cli-
+
+mate related text tasks. We release our benchmark3
+and open-source our trained models4 to encourage
+researchers to extend our existing datasets and con-
+tribute new ones.
+2
+CLIMABENCH Tasks and Datasets
+2.1
+New Datasets
+CDP (formerly the Carbon Disclosure Project5) is
+an international non-profit organisation that helps
+companies and cities disclose climate risks, low
+carbon opportunities and environmental impact. In
+2021, over 14,000 organizations disclosed their
+environmental information via CDP. The annual
+National Association Of Insurance Commissioners
+(NAIC)6 Climate Risk Disclosure Survey7 is a U.S.
+insurance regulation tool where insurers file non-
+confidential disclosures of their assessments and
+management of climate-related risks. The purpose
+of the survey is to enhance transparency about how
+insurers manage climate-related risks and opportu-
+nities and enable better-informed collaboration on
+climate-related issues. It consists of eight questions
+and the expected response is a Yes/No along with
+an explanation (see Table 2 for example).
+We release two new large-scale datasets with
+four tasks, based on publicly available surveys re-
+leased by CDP and NAIC. CLIMA-INSURANCE
+and CLIMA-CDP are for text classification tasks.
+CLIMA-QA is a question answering task based
+on the CLIMA-CDP dataset. Table 2 lists a few
+interesting examples from the newly introduced
+datasets.
+Practical Implications. Recently (2022), NAIC
+eliminated mandatory Insurance questionnaires to
+lessen the burden of compliance, allowing for free
+text report submissions. However, as the structured
+questionnaires provided an organized assessment
+of sector-wise exposure to climate risks, NAIC now
+invests substantial manual effort in mapping free
+text response segments to the original questions.
+A model trained on our new CLIMA-INSURANCE
+dataset could drastically help in this effort as this
+task very closely matches the real-world problem
+faced by NAIC. NAIC is just one organization out
+of many that touch on various aspects related to so-
+3https://github.com/climabench/
+climabench
+4https://huggingface.co/climabench/miniLM-cdp-all
+5https://www.cdp.net/
+6http://www.insurance.ca.gov/0250-insurers/0300-
+insurers/0100-applications/ClimateSurvey/
+7https://interactive.web.insurance.ca.gov
+cietal robustness to climate change. If we improve
+efficiency in processing similar reports in other do-
+mains affecting climate change (e.g. economic and
+ecological reporting), we can better inform policy
+decisions.
+Similarly, Climate Watch, a platform managed
+by the World Resources Institute, lets users analyze
+and compare the Nationally Determined Contri-
+butions (NDCs) under the Paris Agreement and
+discover how countries can leverage their climate
+goals to achieve their sustainable development ob-
+jectives. The data curation involves manually map-
+ping the NDCs to their corresponding Sustainable
+Development Goals, a task which has a strong over-
+lap with the categories in our CLIMA-CDP task.
+2.2
+Data Processing
+The overall statistics of each dataset are given in
+Table 1, the token length distribution is given in
+Appendix Table 9 and details are explained below.
+CLIMA-INSURANCE We create two tasks
+based on the responses to the NAIC survey,
+first being a binary Yes/No classification task
+(CLIMA-INSURANCE) and the other one being
+a question type classification problem (CLIMA-
+INSURANCE+). The data is web scraped from all
+the annual survey responses between 2012-2021
+giving a total of 17K labelled passages. We remove
+the first sentence in each response as it contains
+obvious markers (like "Yes, we do X." or "No, we
+do not participate in Y.") and create splits for train-
+ing, validation and testing. The splits of the two
+datasets are different since we stratify by the class
+label in order to fairly balance the classes across
+the splits.
+CLIMA-CDP The CDP survey responses fall
+into 3 buckets, namely Cities, States and Corpora-
+tions, each with a different questionnaire with vary-
+ing number of questions. We filter out non-English,
+short (less than 10 words) and duplicate responses.
+The Cities dataset is the largest by volume and
+consists of parent sections and subsections. We
+transform these sections into 12 broad categories
+with the help of our climate change researcher to
+curate the labelled data called CLIMA-CDP for the
+years 2018-2021 (Appendix Table 8). The goal is
+to classify CC data into relevant categories. The
+12-category mapping creates a more parsimonious
+set of labels which can be compared to other stud-
+ies and reduces noise in classification (The original
+CDP section labels have changed slightly in the
+
+period 2018-2021). The train, development and
+test splits are stratified by the organization so that
+some organization responses are not seen during
+training.
+CLIMA-QA Although the categories in CLIMA-
+CDP are valuable for CC topic classification, they
+restrict the downstream application to finite classes.
+Hence, we extend our focus on the actual question
+response pairs within the surveys to set up a more
+nuanced and challenging QA dataset called CLIMA-
+QA. Since there is no explicit output class, the
+hope is that models finetuned on this task should
+be able to generalise to unseen question-answer
+pairs, and we conduct extensive experiments to
+study this effect. The pre-processing steps are the
+same as that of CLIMA-CDP. We curate three dif-
+ferent subsets: CDP-CITIES, CDP-STATES and
+CDP-CORPORATION (Table 1) where the splits
+are stratified by the organizations. They have 294,
+132 and 43 unique questions respectively.
+2.3
+Collating existing datasets
+Further, we collate existing CC related text datasets,
+described in detail below.
+CLIMATEXT is a dataset for sentence-based cli-
+mate change topic detection (Leippold and Varini,
+2020). Each sentence is labelled indicating whether
+it talks about climate change or not. Sentences were
+collected from different sources: Wikipedia, the
+U.S. Securities and Exchange Commission (SEC)
+10Kfiles, which are annual regulatory filings in
+which listed companies in the US are required to
+self-identify climate-related risks that are material
+to their business, and a selection of climate-change
+claims collected from the web.
+CLIMATESTANCE and CLIMATEENG Vaid
+et al. (2022a) extracted Twitter data consisting of
+3777 tweets posted during the 2019 United Na-
+tions Framework Convention on Climate Change.
+Each tweet was labelled for two tasks, stance detec-
+tion and categorical classification. For the stance
+towards climate change prevention, the authors la-
+belled each tweet as In Favour, Against or Ambigu-
+ous. For categorical classification, the five classes
+are Disaster, Ocean/Water, Agriculture/Forestry,
+Politics, and General.
+CLIMATEFEVER (Leippold and Diggelmann,
+2020) adopts the methodology of FEVER, the
+largest dataset of artificially designed claims, for
+real-life claims on climate change collected on-
+line. It consists of 1,535 claims and each claim is
+mapped with five relevant evidence passages from
+Wikipedia giving a total of 7675 claim-evidence
+pairs. The labels for the evidences are SUPPORTS,
+REFUTES or NOT_ENOUGH_INFO. The dataset
+is also part of the recent BEIR benchmark for in-
+formation retrieval (Thakur et al., 2021).
+SCIDCC (Mishra and Mittal, 2021) The Sci-
+ence Daily Climate Change (SCIDCC) dataset is
+curated by web scraping news articles from the
+Science Daily (SD) website. It contains around
+11k news articles from 20 categories relevant to
+climate change, where each article comprises of
+a title, summary, and a body. Some of the major
+categories are Earthquakes, Pollution, Hurricanes
+and Cyclones. We propose to use this dataset for
+the first time as a category classification task for
+two reasons. Firstly, the SD news articles are rela-
+tively more scientific as compared to other online
+news. Secondly, the average document length is
+around 500-600 words with a maximum of roughly
+2.5k words, which is significantly longer than other
+existing public CC datasets.
+3
+Models
+We experiment with simple baselines (Majority
+class and Random class), Linear models and pre-
+trained Transformer-based classfiers. The Random
+classifier uniformly samples a label for each input
+while the Majority one always outputs the most
+frequent label.
+3.1
+Linear Models
+We assess Support Vector Machines (SVM) with
+linear kernel as one of the baseline models. The
+input features are the TF-IDF vectors with word
+n-grams in the range (1,3). The linear models are
+only trained on the text classification tasks.
+3.2
+Pretrained Transformers
+We examine Transformer-based (Vaswani et al.,
+2017) pre-trained language models like BERT (De-
+vlin et al., 2019) and RoBERTa (Liu et al., 2019).
+These models are first pretrained on very large unla-
+beled corpora on self-supervised learning tasks like
+masked language modeling or next sentence pre-
+diction. Then, they are fine-tuned on smaller task-
+specific labelled datasets, after adding task-oriented
+layers. We also examine distilled versions like
+DistilRoBERTa (Sanh et al., 2019), longer context
+models like Longformer (Beltagy et al., 2020), and
+domain specific models like ClimateBERT (We-
+
+bersinke et al., 2021) and SciBERT (Beltagy et al.,
+2019). This helps us contrast the effects of model
+architecture, input length and in-domain pretrain-
+ing on downstream tasks. Thus, we fine-tune and
+evaluate the performance of the publicly available
+models. We provide more details about models in
+Appendix Section A.3 and Table 10.
+MiniLM For QA (Wang et al., 2020) is an uncased
+distilled model released by Microsoft based on an
+in-house pre-trained UniLM v2 model in BERT-
+Base size. Reimers and Gurevych (2019) separately
+finetuned this model on the MSMARCO (Campos
+et al., 2016) QA ranking dataset and achieved state
+of the art performance with roughly 2.0x speedups
+over traditional BERT based rankers. We therefore
+consider this QA finetuned model (MSMARCO-
+MiniLM) as a strong baseline for our fine-tuning
+and transfer learning experiments on CLIMA-QA.
+3.3
+Task specific finetuning
+Text Classification For the text classification
+datasets like CLIMATEXT, CLIMATESTANCE, CLI-
+MATEENG, SCIDCC and CLIMA-CDP, we pass
+each document through the model and then apply
+linear head on top of the [CLS] token represen-
+tation of the model which is the defacto standard
+for finetuning transformer based language model
+classifiers. We apply a softmax layer on top of
+the linear head to get the probability distribution
+over classes. For SCIDCC, we concatenate the text
+fields and provide a train, validation and test split
+(80%, 10%, 10%) stratified by the categories.
+Fact Checking For CLIMATEFEVER, we convert
+each data point (the claim and 5 evidences with
+labels) into five input pairs. Each pair consists of
+the claim and one of the five candidate evidences,
+separated by the special [SEP] delimiter token.
+The output representation of the [CLS] token of
+each pair is passed to a linear layer followed by
+a softmax layer to obtain the output probabilities.
+We randomly sample train, dev and test samples in
+8:1:1 ratio, stratified by the labels.
+Question Answering For CLIMA-QA, we ran-
+domly sample 5 negative QA pairs for each positive
+QA pair in the dataset (1:5 ratio). Similar to the
+fact-checking setup, we pass each pair through the
+pretrained model, separated by the [SEP] token.
+This formulation more closely aligns with ques-
+tion answer ranking setup trained on datasets like
+MSMARCO (Campos et al., 2016). We specif-
+ically consider the early-interaction encoder (or
+cross-encoder) setup to simultaneously model the
+context of the question and answer, allowing for the
+self-attention to interact between the question and
+answer throughout the entire model. We narrow
+down to two models, the MSMARCO finetuned
+MiniLM, and the ClimateBERT model to study the
+effects of fine-tuning and transfer learning on the
+three subdomains: CDP-CITIES, CDP-STATES
+and CDP-CORPORATION.
+4
+Experiments
+4.1
+Setup
+We use the Scikit-learn API (Pedregosa et al., 2011)
+for the simple classifiers (Random and Majority
+class) and TFIDF-based linear SVM models. We
+grid-search the hyper parameters for SVM with 5-
+fold validation (Table 12). For all the pre-trained
+models, we use publicly available Hugging Face
+(Wolf et al., 2020) checkpoints.8 We use Adam op-
+timizer (Kingma and Ba, 2015) with learning rate
+of 5e-5 (linear warmup ratio of 0.1, weight decay
+of 0.01) for 10 epochs with early stopping based
+on performance on development data (macro-F1
+score). We use mixed precision (fp16), gradient
+checkpointing and gradient accumulation steps of
+2 to train models efficiently on the limited compute
+(Appendix A.1). For the Longformer, we use the
+default settings (windows of 512 tokens and a sin-
+gle global [CLS] token). The training batch size
+is set to 32. We use weighted cross-entropy loss
+for all the pretrained transformer models with class
+balanced weights. We truncate the input text when
+it exceeds the maximum input length of the model
+and pad otherwise.
+4.2
+Evaluation
+The classification models are evaluated using the
+macro-averaged F1 score on the development and
+test sets. We use macro-average since the datasets
+are imbalanced and all classes are equally impor-
+tant. We do not evaluate linear models on fact-
+checking or QA as the heterogeneity of the input
+in these tasks do not align with the linear setup.
+For the ranking task, we consider the Mean Re-
+ciprocal Rank at k (MRR@k) scores for the top k
+8We use the *-base configuration of each pre-trained
+model, i.e., 12 Transformer blocks, 768 hidden units, and
+12 attention heads. For ClimateBERT we report scores for the
+F variant model on Huggingface. For the QA Cross-encoder,
+we use the MiniLM (12 layer, 384 hidden-unit) finetuned on
+MSMARCO available at https://huggingface.co/
+cross-encoder/ms-marco-MiniLM-L-12-v2
+
+CLIMA-
+CLIMA-
+CLIMA-
+CLIMA-
+CLIMATE-
+CLIMATE-
+CLIMATE-
+Models
+INSURANCE
+INSURANCE+
+CDP
+TEXT
+STANCE
+ENG
+SCIDCC
+FEVER
+Average
+Majority Classifier
+0.4058
+0.0411
+0.0365
+0.4208
+0.2968
+0.1383
+0.0079
+0.2608
+0.2010
+Random Classifier
+0.4940
+0.1214
+0.0645
+0.4686
+0.2552
+0.1671
+0.0505
+0.3062
+0.2409
+TFIDF + Linear SVM
+0.8150
+0.8600
+0.5834
+0.8339
+0.4292
+0.5181
+0.4802
+-
+-
+BERT
+0.8396†
+0.8457
+0.6464†
+0.8704†
+0.5537†
+0.7178
+0.5474†
+0.6247†
+0.7057†
+RoBERTa
+0.8445
+0.8561†
+0.6522
+0.8597
+0.5969
+0.7458
+0.5290
+0.6074
+0.7114
+DistilRoBERTa
+0.8263
+0.8438
+0.6361
+0.8606
+0.5251
+0.7233†
+0.5113
+0.6154
+0.6927
+Longformer
+0.8301
+0.8435
+0.6403
+0.8780
+0.3468
+0.7228
+0.5479
+0.6082
+0.6772
+SciBERT
+0.8260
+0.8443
+0.6362
+0.8329
+0.4867
+0.7050
+0.5183
+0.6268
+0.6845
+ClimateBERT
+0.8214
+0.8480
+0.6424
+0.8514
+0.5284
+0.7183
+0.5297
+0.6154
+0.6944
+Table 3: Macro F1 Scores on the Classification Datasets. Bold and † indicate first and second highest performing
+model respectively. RoBERTa scores the best on average followed by BERT and ClimateBERT.
+items returned by a model. MRR, a popular metric
+used in the Information Retrieval field, is the aver-
+age of the reciprocal ranks of results for a sample
+of queries where the relevance grading is binary
+(Yes/No). One notable distinction is that we re-
+trieve the top questions for a given document as
+opposed to traditional search where the top answers
+are returned. We explain the rationale behind this
+peculiar design choice in Section 6 where we con-
+duct a pilot study with our climate expert. First, we
+construct the baseline with BM25 (Robertson and
+Zaragoza, 2009) and MSMARCO-MiniLM, where
+we evaluate on the three test sets CDP-CITIES,
+CDP-STATES, CDP-CORPORATION without any
+finetuning. Next, we finetune on the largest of the
+three corpora, CDP-CITIES, and then evaluate on
+all three datasets to analyse the transfer learning ca-
+pability on States and Corporations. Similarly, we
+train and test on States and Corporations. Lastly,
+we train and test the best model on all three datasets
+combined. We also fine-tune ClimateBERT as a
+second baseline.
+4.2.1
+Efficiency
+Further, we report the compute efficiency statis-
+tics like average training time and average training
+samples per second for each model in Appendix Ta-
+ble 11. These numbers are hardware dependent but
+provide a relative efficiency comparison between
+the models. The details about the compute and ap-
+proximate CO2 Emission usage is provided in A.2.
+5
+Results
+5.1
+Classification and Fact-Checking
+Table 3 reports the F1-macro on the test sets
+for the classification and fact-checking tasks in
+CLIMABENCH. We also give the average of the
+scores on all tasks for each model. There is no
+single model that does the best across the board,
+but RoBERTa is a clear winner as it beats the other
+baselines on four out of eight tasks. BERT falls
+second in line when we look at average scores.
+Longformer scores the highest on SCIDCC, per-
+haps due to the model being able to attend to more
+number of tokens available in the longer documents
+in this task. To our surprise, it also does well on
+CLIMATEXT which has comparatively shorter text.
+One reason for this could be that the sparse at-
+tention within Longformer may be having a reg-
+ularizing effect. SciBERT ranks on the top for
+CLIMATEFEVER and this could be because the
+evidences in the task have structure and style sim-
+ilar to the text in scientific papers used for pre-
+training the model. ClimateBERT and the model it
+was warm-started from, DistilRoBERTa, are very
+similar in performance. DistilRoBERTa beats Cli-
+mateBERT on CLIMA-INSURANCE, CLIMATEXT,
+and CLIMATEENG. Although this raises concerns
+on the quality of pretraining data used for Climate-
+BERT, we cannot say for sure, since neither the
+data nor the code is publicly available for diagnosis.
+Overall, the transformer models have significantly
+better gains over linear ones except on CLIMA-
+
+CDP-CITIES
+CDP-STATES
+CDP-CORPORATION
+Model
+MRR@10
+MRR@100
+MRR@200
+MRR@All
+MRR@10
+MRR@100
+MRR@All
+MRR@10
+MRR@All
+No Finetuning on CDP
+BM25
+0.055
+0.075
+0.076
+0.077
+0.084
+0.104
+0.105
+0.153
+0.180
+MiniLM
+0.099
+0.116
+0.117
+0.118
+0.120
+0.141
+0.142
+0.320
+0.342
+Finetuned on CDP
+In-Domain
+In-Domain
+In-Domain
+ClimateBERT
+0.331
+0.344
+0.344
+0.344
+0.422
+0.431
+0.431
+0.753
+0.754
+MiniLM
+0.366
+0.378
+0.378
+0.378
+0.482
+0.491
+0.491
+0.755
+0.757
+Best Model Finetuned on all
+MiniLM
+0.352
+0.364
+0.364
+0.364
+0.489
+0.497
+0.497
+0.745
+0.747
+Table 4: MRR@k scores for BM25, ClimateBERT and MSMARCO-MiniLM on the three subsets of CLIMA-QA.
+Models finetuned and evaluated on same subset fall under In-Domain.
+CDP-STATES
+CDP-CORPORATION
+Model
+MRR@10
+MRR@100
+MRR@All
+MRR@10
+MRR@All
+No Finetuning
+BM25
+0.084
+0.104
+0.105
+0.153
+0.180
+MiniLM
+0.120
+0.141
+0.142
+0.320
+0.342
+Finetuned on CDP-CITIES
+Transfer
+Transfer
+ClimateBERT
+0.298
+0.314
+0.314
+0.465
+0.477
+MiniLM
+0.353
+0.366
+0.366
+0.489
+0.500
+Table 5: MRR@k scores for BM25, ClimateBERT
+and MSMARCO-MiniLM on the Transfer experiments.
+Models are finetuned on CDP-CITIES and evaluated on
+States and Corporations.
+INSURANCE+ where the TFIDF+SVM model is
+superior. It shows that simple word co-occurence
+statistics are enough for certain tasks and deep lan-
+guage models might not be the right solution in
+such cases.
+5.2
+Question Answer Ranking
+We analyze and compare the results of our in-
+domain and transfer experiments on CLIMA-QA.
+Table 4 and Table 5 summarize the MRR@k results
+for the in-domain and transfer experiments respec-
+tively. MSMARCO-MiniLM beats ClimateBERT
+on all the subsets, for both in-domain and trans-
+fer learning. Both models do significantly better
+when fine-tuned on CDP-CITIES and evaluated on
+States and Corporations. This indicates that models
+trained on this dataset possess good transfer learn-
+ing capabilities. The in-domain training boosts per-
+formance further as we see for States and Corpora-
+tions. Lastly, the best performing model, MiniLM,
+when finetuned on all three subsets, achieves com-
+parable performance on Cities and Corporations
+while ranking highest on States.
+6
+Extrinsic Evaluation
+Beyond CDP. States and Companies have been uti-
+lizing sustainability reporting frameworks to help
+guide their non-financial reporting efforts. These
+frameworks are useful given the lack of uniform
+regulations on nonfinancial reporting. However, the
+variety of such frameworks can be daunting. For
+example, CDP aims to capture companies’ environ-
+mental performance data, whereas the Task Force
+on Climate-related Financial Disclosures9(TCFD)
+guidelines are intended to encourage companies
+to align their climate-related risk disclosures with
+investors’ needs. Automatic alignment of text to
+relevant sections of the disclosures reduces the ef-
+fort significantly. To motivate this usecase of the
+CLIMA-QA task, we conduct an experiment that
+involves populating the CDP survey using Climate
+Action Plans (CAPS).
+Utilizing CAPS. Disclosure of carbon accounting,
+climate risks and green targets is vital information
+for researchers and policy-makers to craft well-
+informed climate policy rules and regulations. In
+addition to these reports, cities, states, and corpo-
+rations may publish their CAP or sustainability
+reports. The CAPS include quantitative data, such
+as emission values or renewable electricity genera-
+tion capacity, and qualitative data such as specific
+policy interventions across different sectors. Popu-
+lating CDP surveys from new CAPs allows for con-
+sistent comparisons of CAPs to existing datasets
+which could be used to compare strategies, identify
+gaps, or rank jurisdictions on the content and level
+of ambition in their stated plans. We consider the
+scenario where the State’s CAP is available but
+their CDP report is not. The researcher’s aim is to
+9https://www.fsb-tcfd.org
+
+prec@1
+prec@2
+prec@3
+prec@4
+prec@5
+Relevant
+63.0
+67.0
+68.6
+69.5
+71.0
+Highly Relevant
+30.0
+32.0
+32.3
+32.5
+35.6
+Table 6: Precision@K: We report the fraction of items
+in the top K ranked retrievals that are either marked as
+highly relevant, or at least relevant, averaged across text
+examples. Relevance judgements for were performed
+manually by an expert annotator.
+pull together the relevant portions of the CAP to
+construct the missing CDP. Typically the CAPS
+are much longer (∼100 pages) and more compre-
+hensive than any particular disclosure report. Our
+aim is to measure the effectiveness of our model to
+assist in this task.
+6.1
+Task Setup
+Unlike typical search problems, the climate re-
+searchers need to map every data point in the CAP
+to the corresponding CDP question/section. We
+have observed that climate researchers annotate
+documents in full, similar to how Climate Watch10
+has been constructed. For this reason, it is criti-
+cal to evaluate the answer-to-question mapping. If
+we search for answers, then many of the questions
+will be answered by the summary section of a CAP,
+which does not help their goals. For example, a
+CDP question will ask to disclose all climate risks
+that a corporation may be exposed to. Our aim is
+to then identify all relevant sections.
+Therefore, our procedure is as follows: First,
+the expert (climate policy researcher on our team
+and co-author) selected 5 pages at random from a
+collection of 20 State CAPS and then selected a
+random paragraph from each page. For this experi-
+ment, we use the MSMARCO-MiniLM finetuned
+on all 3 subsets CDP-CITIES, CDP-STATES and
+CDP-CORPORATION (our best performing model
+in the intrinsic evaluation) on each text segment and
+select the top five scoring CDP-STATES questions.
+We then presented each segment along with the
+retrieved questions to the expert and had them an-
+notate the relevance for each question-answer pair
+on a three point scale: No Relevance, Relevant,
+Highly Relevant.11
+6.2
+Results
+Table 6 shows the climate change researcher’s eval-
+uation metrics for our model. Overall, 71.0% of
+the 500 questions retrieved were judged Relevant
+and 35.6% rated Highly Relevant. One pitfall of
+our model is that there were more Highly Relevant
+predictions ranked fifth than first. One possible ex-
+planation for this is that the top retrieved questions
+were often more general while the questions that
+were ranked lower were more specific and easier to
+match (see Table 13 in the Appendix.)
+We show some examples in Table 7. The first
+four examples show high degrees of success. In
+example 1, our model correctly identifies the state
+CAP text as impact-related and captures the spe-
+cific discussion of compound risks. However, ex-
+ample 6 appears to highlight a gap in the CDP
+questionnaire related to the topic of environmental
+justice, a result in itself of considerable interest.
+Although our study is limited, it shows the promise
+of using our benchmark. Similar extrinsic studies
+could be conducted on different climate reports
+such as corporate ESGs or national Paris Agree-
+ment plans.
+7
+Related Work
+7.1
+Benchmarks in NLP
+With the ever growing inventory of deep learning
+based models on a plethora of NLP tasks, the com-
+munity is working towards general purpose mod-
+els that perform well at multiple tasks. Conse-
+quently, various benchmarks that evaluate general-
+purpose understanding have been proposed. For
+example, GLUE (Wang et al., 2018) and the sub-
+sequently more challenging SuperGLUE (Wang
+et al., 2019) datasets were introduced to test lan-
+guage understanding capabilities. However, do-
+ing well on general purpose benchmarks does not
+translate well to domain-specific tasks like legal
+reports, biomedical files or numerical problems.
+This led to the emergence of benchmarks that cater
+to a particular field of expertise.
+For instance,
+LexGLUE (Chalkidis et al., 2022) for legal text
+and NumGLUE (Mishra et al., 2022) for mathe-
+matical reasoning. CLIMABENCH follows on this
+chain of thought to provide a unified way to evalu-
+ate models on CC specific problems.
+10https://www.climatewatchdata.org/
+11By construction, in our rating there may be multiple rele-
+vant questions found for each text segment.
+
+Ex.
+Text Segment from State Climate Action Plans
+Top Questions
+1
+Sea level rise will inundate some nearby coastal areas, and related
+salt-water intrusion, coupled with increased drought stress may impact
+water supplies.
+Q1: Please describe the current and/or anticipated impacts of climate
+change.
+Q3: Please detail any compounding factors that may worsen the impacts
+of climate change in your region.
+2
+The afforestation goal is to increase the area of forested lands in the state
+by 50,000 acres annually through 2025.
+Q1: Please provide the details of your region’s target(s).
+Q2: Please provide details of your climate actions in the Land use sector.
+3
+By a majority vote, the ICCAC presents a policy option that, if deemed
+necessary, would build one new 1200-megawatt nuclear power plant in
+Iowa by January 1, 2020.
+Q3: Please provide details of your renewable energy or electricity tar-
+get(s).
+Q4: Please provide details of your climate actions in the Energy sector.
+4
+California maintains a GHG inventory that is consistent with IPCC
+practices ... Reports from facilities and entities that emit more than
+25,000 MTCO2e are verified by a CARB-accredited third-party
+verification body.
+Q1: Please give the name of the primary protocol, standard, or methodol-
+ogy you have used to calculate your government’s GHG emissions.
+Q3: Please provide the following information about the emissions verifi-
+cation process.
+5
+A leading driver of these high emissions is the fact that the District’s
+daytime population swells by 400,000 workers every workday, which is
+the largest percentage increase in daytime population of any large city in
+the nation.
+Q4: Please indicate if your region-wide emissions have increased, de-
+creased, or stayed the same since your last emissions inventory, and
+please describe why.
+Q5: Please report your region-wide base year emissions in the table
+below.
+6
+State law defines environmental justice as the fair treatment of people of
+all races, cultures, and incomes with respect to the development,
+adoption, implementation, and enforcement of environmental laws,
+regulations, and policies.
+Q1: Please explain why you do not have policies on deforestation and/or
+forest degradation.
+Q4: Please provide details of your climate actions in the Governance
+sector.
+Table 7: Examples from our human pilot study in which our climate expert has evaluated the relevance of CDP
+questions linked to selected text from state climate action plans. A fragment of the matched text is presented with
+two illustrative questions from the set of five question matches generated by our model.
+7.2
+Climate NLP
+CLIMATEXT (Leippold and Varini, 2020) and CLI-
+MATEFEVER (Leippold and Diggelmann, 2020)
+extracted and filtered documents from Wikipedia
+and other sources to curate CC corpus that was fur-
+ther annotated by humans. Due to the challenging
+and evolving nature of this field, the labelling pro-
+cess is not only time consuming but prone to noise
+due to poor inter-annotator agreements. Unlike
+these small annotated datasets, we are able to ef-
+fectively utilize existing semi-structured disclosure
+forms for a much larger set of supervised data.
+Berrang-Ford et al. (2021) use machine learn-
+ing to select scientific literature on climate adap-
+tation for the task of systematic evidence map-
+ping.
+Luccioni et al. (2020) use Task Force
+on Climate-related Financial Disclosures (TCFD)
+questionnaires in a QA setup similar to ours while
+Corringham et al. (2021) use document headers
+for course-grained classification on Paris Climate
+Agreements.
+8
+Limitations and Future Work
+One current limitation of our benchmark is that the
+datasets are English only, thus restricting evalua-
+tion to English trained models. Although CDP-
+STATES has disclosures in other languages it rep-
+resents a small portion of the reports. We plan
+to include relevant CC datasets from the multilin-
+gual European Union Public Data Catalog12 in the
+future, while encouraging contributions from the
+broader community. Another limitation is that ex-
+pert human evaluation is missing for our new tasks.
+We conduct an analysis involving a climate expert
+for one of our best-performing models on the QA
+task. But, a general human evaluation of the other
+classification tasks is not conducted.
+We do not thoroughly investigate the efficiency
+of the Transformer models in this work. But, the
+compute and training efficiency statistics we pro-
+vide in A.2 and Table 11 are a step in this direction.
+We encourage future work on CLIMABENCH to
+leverage models that are both performant and effi-
+cient.
+Finally, there exists more types of carbon disclo-
+sures (TCFD, SBTi) as well as publicly accessible
+corporate sustainability reports that we wish to in-
+clude but require more difficult scraping and data
+preprocessing.
+9
+Acknowledgement
+This project was funded by the Climate Change
+AI Innovation Grants program, hosted by Climate
+Change AI with the support of the Quadrature Cli-
+mate Foundation, Schmidt Futures, and the Canada
+Hub of Future Earth.
+12data.europa.edu
+
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+pretrained language model for climate-related text.
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+A
+Appendix
+A.1
+Compute Details
+We used a 24 core AMD Ryzen CPU machine with
+128 GB RAM for data processing. For training and
+inference of the deep learning models, we utilize
+4 Nvidia RTX 2080Ti GPUs with 11GB memory
+each. Each model was trained on a single GPU at a
+time.
+A.2
+CO2 Emission Related to Experiments
+A cumulative of 338 hours of computation was
+performed on hardware of type RTX 2080 Ti (TDP
+of 250W). Total emissions are estimated to be 36.5
+kgCO2eq. Estimations were conducted using the
+MachineLearning Impact calculator presented in
+(Lacoste et al., 2019).
+A.3
+Pretrained Transformer Models
+BERT (Devlin et al., 2019) is a popular
+Transformer-based language model pre-trained on
+masked language modeling and next sentence pre-
+diction tasks. It makes use of WordPiece tokeniza-
+tion algorithm that breaks a word into several sub-
+words, such that commonly seen subwords can also
+be represented by the model.
+RoBERTa (Liu et al., 2019) uses dynamic masking
+and eliminates the next sentence prediction pre-
+training task, while using a larger vocabulary and
+pre-training on much larger corpora compared to
+BERT. Another notable difference is the use of byte
+pair encoding compared to wordPiece in BERT.
+DistilRoBERTa (Sanh et al., 2019) leverages
+knowledge distillation during the pre-training
+phase reducing the size of the RoBERTa model
+by 40%, while retaining 97% of its language under-
+standing capabilities and being 60% faster. Sanh
+et al. (2019) originally distilled the BERT model
+but we utilize the better performing RoBERTa ver-
+sion in our experiments.
+Longformer (Beltagy et al., 2020) extends
+Transformer-based models to support longer se-
+quences with the help of sparse-attention. It uses a
+combination of local attention and global attention
+
+Section
+Category/Label
+Hazards: Adaptation
+Adaptation
+Adaptation
+Adaptation
+Buildings
+Buildings
+Hazards: Climate Hazards
+Climate Hazards
+Hazards: Social Risks
+Climate Hazards
+Climate Hazards
+Climate Hazards
+Climate Hazards and Vulnerability
+Climate Hazards
+Climate Hazards & Vulnerability
+Climate Hazards
+City-wide Emissions
+Emissions
+Emissions Reduction
+Emissions
+GHG Emissions Data
+Emissions
+Local Government Emissions
+Emissions
+Emissions Reduction: City-wide
+Emissions
+City Wide Emissions
+Emissions
+Emissions Reduction: Local Government
+Emissions
+Local Government Operations GHG Emissions Data
+Emissions
+Energy Data
+Energy
+Energy
+Energy
+Food
+Food
+Governance and Data Management
+Governance and Data Management
+Opportunities
+Opportunities
+Strategy
+Strategy
+Urban Planning
+Strategy
+Transport
+Transport
+Waste
+Waste
+Water
+Water
+Water Security
+Water
+Table 8: The Sections in the CDP Cities Questionnaire and the corresponding Label assigned by climate expert.
+Task
+Average
+Max
+Min
+Std
+CLIMA-INSURANCE
+203
+4588
+11
+326
+CLIMA-INSURANCE+
+206
+4588
+11
+335
+CLIMA-CDP
+73
+801
+11
+83
+CLIMA-QA
+105
+834
+15
+88
+CLIMATEXT
+23
+124
+11
+10
+CLIMATESTANCE
+30
+98
+11
+12
+CLIMATEENG
+30
+98
+11
+12
+CLIMATEFEVER
+47
+311
+11
+19
+SCIDCC
+580
+2014
+13
+223
+Table 9: Statistics for the number of tokens in each task
+of CLIMABENCH
+mechanism that allows for linear attention com-
+plexity and thus makes it feasible to run on longer
+documents (max 4096 tokens). It however takes
+much longer to train than the shorter context (512
+tokens) models.
+SciBERT (Beltagy et al., 2019), a pretrained lan-
+guage model based on BERT, leverages unsuper-
+vised pretraining on a large multi-domain corpus of
+scientific publications to improve performance on
+downstream scientific NLP tasks. It was evaluated
+on tasks like sequence tagging, sentence classifi-
+cation and dependency parsing with datasets from
+scientific domains. SciBERT gives significant im-
+provements over BERT on these datasets.
+ClimateBERT (Webersinke et al., 2021) was
+warm-started from the DistilRoBERTa model and
+pretrained on text corpora from climate-related re-
+search paper abstracts, corporate and general news
+and reports from companies that were not publicly
+released with the model. It was evaluated on tasks
+like sentiment analysis (using a private dataset),
+and public datasets like CLIMATEXT and CLI-
+MATEFEVER. In this paper, we evaluate and com-
+pare the performance of ClimateBERT on diverse
+CC tasks for the first time, providing a comprehen-
+sive, publicly available and reproducible evalua-
+tion.
+
+Model
+Source
+# Params
+Vocab Size
+Max Length
+BERT
+(Devlin et al., 2019)
+110M
+30K
+512
+RoBERTa
+(Liu et al., 2019)
+125M
+50K
+512
+DistilRoBERTa
+(Sanh et al., 2019)
+82M
+50K
+512
+Longformer
+(Beltagy et al., 2020)
+149M
+50K
+4096
+SciBERT
+(Beltagy et al., 2019)
+110M
+30K
+512
+ClimateBERT
+(Webersinke et al., 2021)
+82M
+50K
+512
+Table 10: Pretrained Transformer Language Models used for Classification tasks
+Model
+Avg. Runtime (in hours)
+Avg. Train Samples/Second
+Avg. Train Steps/Second
+ClimateBERT
+0.40
+104.83
+1.64
+DistilRoBERTa
+0.40
+101.04
+1.58
+SciBERT
+0.70
+53.86
+0.84
+RoBERTa
+0.80
+50.46
+0.79
+BERT
+0.85
+49.32
+0.77
+Longformer
+14.95
+13.82
+0.76
+Table 11: Compute Efficiency Metrics for the Pretrained Transformer models for the experiments conducted on
+CLIMABENCH. Models based on the DistilRoBERTa architecture are the most efficient due to smaller model size.
+Parameter
+Values
+loss
+hinge, squared_hinge
+C
+0.01, 0.1, 1, 10
+class_weight
+none, balanced
+Table 12: For the linear SVM, we grid search over the parameters with 5-fold validation to get the best fit out of
+80 candidates (16 values * 5 folds) with F1 Macro as the scoring mechanism
+
+Question
+MRR@132
+Please provide details of your climate actions in the Agriculture sector.
+0.870
+Please provide details of your climate actions in the Waste sector.
+0.789
+Please provide details of your climate actions in the Transport sector.
+0.774
+Please provide details of your climate actions in the Buildings & Lighting sector.
+0.597
+Please describe these current and/or anticipated impacts of climate change.
+0.492
+Please complete the table below.
+0.487
+Please indicate the opportunities and describe how the region is positioning itself to take
+advantage of them.
+0.445
+Please provide details of your climate actions in the Energy sector.
+0.397
+Please describe the adaptation actions you are taking to reduce the vulnerability of your
+region’s citizens, businesses and infrastructure to the impacts of climate change identified in
+6.6a.
+0.378
+Please describe these current and/or future risks due to climate change.
+0.327
+List any emission reduction, adaptation, water related or resilience projects that you have
+planned within your region for which you hope to attract financing, and provide details on the
+estimated costs and status of the project. If your region does not have any relevant projects,
+please select “No relevant projects” under Project Area.
+0.319
+Please provide details of your climate actions in the Land use sector.
+0.286
+Please provide the details of your region-wide base year emissions reduction target(s). You
+may add rows to provide the details of your sector-specific targets by selecting the relevant
+sector in the sector field.
+0.252
+Please describe the adaptation actions you are taking to reduce the vulnerability of your
+region’s citizens, businesses and infrastructure to the risks due to climate change identified
+in 5.4a.
+0.247
+Table 13: Question difficulty evaluated on the test set of CDP-STATES ranked from best performing to worst
+performing. Filtered to only questions that appeared at least twenty times.
+
diff --git a/JtE3T4oBgHgl3EQfAAk2/content/tmp_files/load_file.txt b/JtE3T4oBgHgl3EQfAAk2/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf,len=940
+page_content='ClimaBench: A Benchmark Dataset For Climate Change Text  Understanding in English  Tanmay Laud∗  UCSD∗  tlaud@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='edu  Daniel Spokoyny∗  CMU  dspokoyn@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='edu  Tom Corringham  SIO  tcorringham@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='edu  Taylor Berg-Kirkpatrick  UCSD  tberg@eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='edu  Abstract  The topic of Climate Change (CC) has re-  ceived limited attention in NLP despite its real  world urgency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Activists and policy-makers  need NLP tools in order to effectively pro-  cess the vast and rapidly growing textual data  produced on CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Their utility, however, pri-  marily depends on whether the current state-  of-the-art models can generalize across vari-  ous tasks in the CC domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In order to ad-  dress this gap, we introduce Climate Change  Benchmark (ClimaBench), a benchmark col-  lection of existing disparate datasets for eval-  uating model performance across a diverse set  of CC NLU tasks systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Further, we  enhance the benchmark by releasing two large-  scale labelled text classification and question-  answering datasets curated from publicly avail-  able environmental disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Lastly, we  provide an analysis of several generic and  CC-oriented models answering whether fine-  tuning on domain text offers any improve-  ments across these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We hope this work  provides a standard assessment tool for re-  search on CC text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  1  Introduction  There is an ever growing body of text-based cli-  mate documents that contain vital information on  carbon emissions, impacts, and risks – for example,  firms announce their emissions reduction targets  or cities disclose their water risks and potential  exposure to drought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' These documents include  corporate ESG (Environmental, Social, and Gov-  ernance), climate assessments, climate legislation,  regulatory filings, and are produced by corpora-  tions, cities, states, and national governments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Al-  though, there have been massive efforts by inter-  national organizations to standardize climate re-  porting and disclosures, the vast majority of data  is still non machine-readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Natural language  ∗ Equal Contribution  processing has potential to help researchers and  policy-makers search, extract and gain insights  from climate texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Several works have introduced  small-scale climate-specific datasets for detecting  relevance to climate (Leippold and Varini, 2020),  identifying stance detection (Vaid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2022b) and  fact-checking (Leippold and Diggelmann, 2020) of  social media claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Other works have studied the  effect of further pretraining of transformers using  in-domain data such as news articles, academic  papers, and climate reports (Luccioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Webersinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  However, the use of proprietary datasets for both  pretraining as well as evaluation makes it difficult  to effectively track progress in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' To help  unify the disparate efforts of applying NLP tech-  nologies to climate-related problems we propose  CLIMABENCH, a benchmark for training and eval-  uating NLP models on classification and question-  answering tasks pertaining to climate texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Bench-  marks have been an effective tool for the general  NLP community (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Srivastava  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2020) as well as for spe-  cific areas such as legal (Chalkidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2022) or  biomedical (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2020) domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  We introduce two new large scale datasets,  CLIMA-CDP and CLIMA-INSURANCE, which are  based on publicly accessible climate questionnaires  of cities, states and corporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In contrast with  existing datasets, these questionnaires have an or-  der of magnitude more examples and the exam-  ples themselves are longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Further, CLIMA-QA  uses a highly structured questionnaire: with hun-  dreds of unique question types across a multi-  tude of topics such as transportation, water se-  curity, and governance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Additionally, we collate  five existing climate datasets and, to the best of  our knowledge, conduct the first evaluation on the  SCIDCC (Mishra and Mittal, 2021) topic classifi-  cation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  A common technique to adapt large language  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='CL]  11 Jan 2023  Dataset  Task Type  Domain  # Train  # Dev  # Test  # Classes  CLIMA-INSURANCE (Ours)  Binary Classification  NAIC  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='2K  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1K  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1K  20  Table 1: General Statistics of the CLIMABENCH datasets  Text  Class / Question  CLIMA-  INSURANCE+  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Each year Aflac reports its US operations Scope 1 and Scope 2 emissions to the Carbon Disclosure  Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Since 2007, Aflac’s owned facilities in terms of square feet have increased by more than  10% while total Scope 1 and 2 CO2e emissions have significantly decreased compared to 2007  emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Question Type 1 / Yes  CLIMA-CDP  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='These Plans must include management of CD&E waste, both through on-site recycling and re-use  and on-site waste processing prior to disposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Westminster will contribute to the London Plan target  of net self-sufficiency (managing 100% of London’s waste within London) by 2026 by planning for  Westminster’s apportionment targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Governance  and  Data  Management  CLIMA-QA  Flooding from sea level rise will damage building and roads in the coastal neighborhoods of the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Flooding also represents a risk to major transportation hubs infrastructure in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Coastal  flooding can have a long-term effect on major industrial and commercial activities along the coastal  areas of the city as well as damage urban forestry and local natural biodiversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Please describe the im-  pacts experienced so far,  and how you expect the  hazard to impact in the fu-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  SCIDCC  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='In the past, wave breaking has been tracked by so-called "whitecap coverage," in which still  or video imagery was used to statically identify ocean whitecaps and the corresponding surface  coverage by breaking waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' But those measurements suffered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Hurricanes and Cyclones  Table 2: Examples (pairs of inputs and outputs) for some of the newly introduced datasets in CLIMABENCH  models (LLM) to new tasks has been to pretrain  on in-domain texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In the CC domain, datasets  of corporate sustainability reports, climate news  articles, climate-related research abstracts and fi-  nancial reports from EDGAR1 have been used for  pretraining (Luccioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Webersinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' However, certain alternate sources of cli-  mate text data are proprietary and models trained  on this data are generally unavailable for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  For example, the Global Reporting Initiative (GRI),  a non-governmental organization, once hosted over  63,000 corporate sustainability reports which are  now inaccessible2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Our experiments on CLIMABENCH suggest that  1U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Securities and Exchange Commission Electronic  Data Gathering, Analysis, and Retrieval (EDGAR) Database  of public company filings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='globalreporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='org/how-to-use-the-gri-  standards/register-your-report/  state-of-the-art LLMs provide satisfactory perfor-  mance when finetuned on the downstream CC tasks,  but there is still substantial room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Further, there was no single model that dominated  the rest on all of the classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Beyond  evaluation of existing models we also show that  training models on our new benchmark can be di-  rectly applied to a real-world climate focused down-  stream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Specifically, using the best performing  model trained on our new benchmark, we predict  relevant questions for text segments from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' State  climate action plans which is a completely different  type of CC document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Out of the 500 predictions  judged by a climate change researcher 71% were  relevant and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='6% highly relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  We hope that CLIMABENCH lowers the barrier  for the NLP community to conduct research on cli-  mate related text tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We release our benchmark3  and open-source our trained models4 to encourage  researchers to extend our existing datasets and con-  tribute new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  2  CLIMABENCH Tasks and Datasets  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1  New Datasets  CDP (formerly the Carbon Disclosure Project5) is  an international non-profit organisation that helps  companies and cities disclose climate risks, low  carbon opportunities and environmental impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In  2021, over 14,000 organizations disclosed their  environmental information via CDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The annual  National Association Of Insurance Commissioners  (NAIC)6 Climate Risk Disclosure Survey7 is a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  insurance regulation tool where insurers file non-  confidential disclosures of their assessments and  management of climate-related risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The purpose  of the survey is to enhance transparency about how  insurers manage climate-related risks and opportu-  nities and enable better-informed collaboration on  climate-related issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' It consists of eight questions  and the expected response is a Yes/No along with  an explanation (see Table 2 for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  We release two new large-scale datasets with  four tasks, based on publicly available surveys re-  leased by CDP and NAIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' CLIMA-INSURANCE  and CLIMA-CDP are for text classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  CLIMA-QA is a question answering task based  on the CLIMA-CDP dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Table 2 lists a few  interesting examples from the newly introduced  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Practical Implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Recently (2022), NAIC  eliminated mandatory Insurance questionnaires to  lessen the burden of compliance, allowing for free  text report submissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' However, as the structured  questionnaires provided an organized assessment  of sector-wise exposure to climate risks, NAIC now  invests substantial manual effort in mapping free  text response segments to the original questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  A model trained on our new CLIMA-INSURANCE  dataset could drastically help in this effort as this  task very closely matches the real-world problem  faced by NAIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' NAIC is just one organization out  of many that touch on various aspects related to so-  3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='com/climabench/  climabench  4https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='co/climabench/miniLM-cdp-all  5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='cdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='net/  6http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='gov/0250-insurers/0300-  insurers/0100-applications/ClimateSurvey/  7https://interactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='gov  cietal robustness to climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' If we improve  efficiency in processing similar reports in other do-  mains affecting climate change (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' economic and  ecological reporting), we can better inform policy  decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Similarly, Climate Watch, a platform managed  by the World Resources Institute, lets users analyze  and compare the Nationally Determined Contri-  butions (NDCs) under the Paris Agreement and  discover how countries can leverage their climate  goals to achieve their sustainable development ob-  jectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The data curation involves manually map-  ping the NDCs to their corresponding Sustainable  Development Goals, a task which has a strong over-  lap with the categories in our CLIMA-CDP task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2  Data Processing  The overall statistics of each dataset are given in  Table 1, the token length distribution is given in  Appendix Table 9 and details are explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  CLIMA-INSURANCE We create two tasks  based on the responses to the NAIC survey,  first being a binary Yes/No classification task  (CLIMA-INSURANCE) and the other one being  a question type classification problem (CLIMA-  INSURANCE+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The data is web scraped from all  the annual survey responses between 2012-2021  giving a total of 17K labelled passages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We remove  the first sentence in each response as it contains  obvious markers (like "Yes, we do X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='" or "No, we  do not participate in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='") and create splits for train-  ing, validation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The splits of the two  datasets are different since we stratify by the class  label in order to fairly balance the classes across  the splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  CLIMA-CDP The CDP survey responses fall  into 3 buckets, namely Cities, States and Corpora-  tions, each with a different questionnaire with vary-  ing number of questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We filter out non-English,  short (less than 10 words) and duplicate responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  The Cities dataset is the largest by volume and  consists of parent sections and subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We  transform these sections into 12 broad categories  with the help of our climate change researcher to  curate the labelled data called CLIMA-CDP for the  years 2018-2021 (Appendix Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The goal is  to classify CC data into relevant categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The  12-category mapping creates a more parsimonious  set of labels which can be compared to other stud-  ies and reduces noise in classification (The original  CDP section labels have changed slightly in the  period 2018-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The train, development and  test splits are stratified by the organization so that  some organization responses are not seen during  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  CLIMA-QA Although the categories in CLIMA-  CDP are valuable for CC topic classification, they  restrict the downstream application to finite classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Hence, we extend our focus on the actual question  response pairs within the surveys to set up a more  nuanced and challenging QA dataset called CLIMA-  QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Since there is no explicit output class, the  hope is that models finetuned on this task should  be able to generalise to unseen question-answer  pairs, and we conduct extensive experiments to  study this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The pre-processing steps are the  same as that of CLIMA-CDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We curate three dif-  ferent subsets: CDP-CITIES, CDP-STATES and  CDP-CORPORATION (Table 1) where the splits  are stratified by the organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' They have 294,  132 and 43 unique questions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='3  Collating existing datasets  Further, we collate existing CC related text datasets,  described in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  CLIMATEXT is a dataset for sentence-based cli-  mate change topic detection (Leippold and Varini,  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Each sentence is labelled indicating whether  it talks about climate change or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Sentences were  collected from different sources: Wikipedia, the  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Securities and Exchange Commission (SEC)  10Kfiles, which are annual regulatory filings in  which listed companies in the US are required to  self-identify climate-related risks that are material  to their business, and a selection of climate-change  claims collected from the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  CLIMATESTANCE and CLIMATEENG Vaid  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' (2022a) extracted Twitter data consisting of  3777 tweets posted during the 2019 United Na-  tions Framework Convention on Climate Change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Each tweet was labelled for two tasks, stance detec-  tion and categorical classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For the stance  towards climate change prevention, the authors la-  belled each tweet as In Favour, Against or Ambigu-  ous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For categorical classification, the five classes  are Disaster, Ocean/Water, Agriculture/Forestry,  Politics, and General.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  CLIMATEFEVER (Leippold and Diggelmann,  2020) adopts the methodology of FEVER, the  largest dataset of artificially designed claims, for  real-life claims on climate change collected on-  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' It consists of 1,535 claims and each claim is  mapped with five relevant evidence passages from  Wikipedia giving a total of 7675 claim-evidence  pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The labels for the evidences are SUPPORTS,  REFUTES or NOT_ENOUGH_INFO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The dataset  is also part of the recent BEIR benchmark for in-  formation retrieval (Thakur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  SCIDCC (Mishra and Mittal, 2021) The Sci-  ence Daily Climate Change (SCIDCC) dataset is  curated by web scraping news articles from the  Science Daily (SD) website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' It contains around  11k news articles from 20 categories relevant to  climate change, where each article comprises of  a title, summary, and a body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Some of the major  categories are Earthquakes, Pollution, Hurricanes  and Cyclones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We propose to use this dataset for  the first time as a category classification task for  two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Firstly, the SD news articles are rela-  tively more scientific as compared to other online  news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Secondly, the average document length is  around 500-600 words with a maximum of roughly  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='5k words, which is significantly longer than other  existing public CC datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  3  Models  We experiment with simple baselines (Majority  class and Random class), Linear models and pre-  trained Transformer-based classfiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The Random  classifier uniformly samples a label for each input  while the Majority one always outputs the most  frequent label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1  Linear Models  We assess Support Vector Machines (SVM) with  linear kernel as one of the baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The  input features are the TF-IDF vectors with word  n-grams in the range (1,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The linear models are  only trained on the text classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2  Pretrained Transformers  We examine Transformer-based (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=',  2017) pre-trained language models like BERT (De-  vlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019) and RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  These models are first pretrained on very large unla-  beled corpora on self-supervised learning tasks like  masked language modeling or next sentence pre-  diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Then, they are fine-tuned on smaller task-  specific labelled datasets, after adding task-oriented  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We also examine distilled versions like  DistilRoBERTa (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019), longer context  models like Longformer (Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2020), and  domain specific models like ClimateBERT (We-  bersinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2021) and SciBERT (Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' This helps us contrast the effects of model  architecture, input length and in-domain pretrain-  ing on downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Thus, we fine-tune and  evaluate the performance of the publicly available  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We provide more details about models in  Appendix Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='3 and Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  MiniLM For QA (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2020) is an uncased  distilled model released by Microsoft based on an  in-house pre-trained UniLM v2 model in BERT-  Base size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Reimers and Gurevych (2019) separately  finetuned this model on the MSMARCO (Campos  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2016) QA ranking dataset and achieved state  of the art performance with roughly 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='0x speedups  over traditional BERT based rankers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We therefore  consider this QA finetuned model (MSMARCO-  MiniLM) as a strong baseline for our fine-tuning  and transfer learning experiments on CLIMA-QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='3  Task specific finetuning  Text Classification For the text classification  datasets like CLIMATEXT, CLIMATESTANCE, CLI-  MATEENG, SCIDCC and CLIMA-CDP, we pass  each document through the model and then apply  linear head on top of the [CLS] token represen-  tation of the model which is the defacto standard  for finetuning transformer based language model  classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We apply a softmax layer on top of  the linear head to get the probability distribution  over classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For SCIDCC, we concatenate the text  fields and provide a train, validation and test split  (80%, 10%, 10%) stratified by the categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Fact Checking For CLIMATEFEVER, we convert  each data point (the claim and 5 evidences with  labels) into five input pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Each pair consists of  the claim and one of the five candidate evidences,  separated by the special [SEP] delimiter token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  The output representation of the [CLS] token of  each pair is passed to a linear layer followed by  a softmax layer to obtain the output probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  We randomly sample train, dev and test samples in  8:1:1 ratio, stratified by the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Question Answering For CLIMA-QA, we ran-  domly sample 5 negative QA pairs for each positive  QA pair in the dataset (1:5 ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Similar to the  fact-checking setup, we pass each pair through the  pretrained model, separated by the [SEP] token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  This formulation more closely aligns with ques-  tion answer ranking setup trained on datasets like  MSMARCO (Campos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We specif-  ically consider the early-interaction encoder (or  cross-encoder) setup to simultaneously model the  context of the question and answer, allowing for the  self-attention to interact between the question and  answer throughout the entire model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We narrow  down to two models, the MSMARCO finetuned  MiniLM, and the ClimateBERT model to study the  effects of fine-tuning and transfer learning on the  three subdomains: CDP-CITIES, CDP-STATES  and CDP-CORPORATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  4  Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1  Setup  We use the Scikit-learn API (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2011)  for the simple classifiers (Random and Majority  class) and TFIDF-based linear SVM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We  grid-search the hyper parameters for SVM with 5-  fold validation (Table 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For all the pre-trained  models, we use publicly available Hugging Face  (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2020) checkpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='8 We use Adam op-  timizer (Kingma and Ba, 2015) with learning rate  of 5e-5 (linear warmup ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1, weight decay  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='01) for 10 epochs with early stopping based  on performance on development data (macro-F1  score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We use mixed precision (fp16), gradient  checkpointing and gradient accumulation steps of  2 to train models efficiently on the limited compute  (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For the Longformer, we use the  default settings (windows of 512 tokens and a sin-  gle global [CLS] token).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The training batch size  is set to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We use weighted cross-entropy loss  for all the pretrained transformer models with class  balanced weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We truncate the input text when  it exceeds the maximum input length of the model  and pad otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2  Evaluation  The classification models are evaluated using the  macro-averaged F1 score on the development and  test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We use macro-average since the datasets  are imbalanced and all classes are equally impor-  tant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We do not evaluate linear models on fact-  checking or QA as the heterogeneity of the input  in these tasks do not align with the linear setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  For the ranking task, we consider the Mean Re-  ciprocal Rank at k (MRR@k) scores for the top k  8We use the *-base configuration of each pre-trained  model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 12 Transformer blocks, 768 hidden units, and  12 attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='6944  Table 3: Macro F1 Scores on the Classification Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Bold and † indicate first and second highest performing  model respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' RoBERTa scores the best on average followed by BERT and ClimateBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  items returned by a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' MRR, a popular metric  used in the Information Retrieval field, is the aver-  age of the reciprocal ranks of results for a sample  of queries where the relevance grading is binary  (Yes/No).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' One notable distinction is that we re-  trieve the top questions for a given document as  opposed to traditional search where the top answers  are returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We explain the rationale behind this  peculiar design choice in Section 6 where we con-  duct a pilot study with our climate expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' First, we  construct the baseline with BM25 (Robertson and  Zaragoza, 2009) and MSMARCO-MiniLM, where  we evaluate on the three test sets CDP-CITIES,  CDP-STATES, CDP-CORPORATION without any  finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Next, we finetune on the largest of the  three corpora, CDP-CITIES, and then evaluate on  all three datasets to analyse the transfer learning ca-  pability on States and Corporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Similarly, we  train and test on States and Corporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Lastly,  we train and test the best model on all three datasets  combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We also fine-tune ClimateBERT as a  second baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1  Efficiency  Further, we report the compute efficiency statis-  tics like average training time and average training  samples per second for each model in Appendix Ta-  ble 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' These numbers are hardware dependent but  provide a relative efficiency comparison between  the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The details about the compute and ap-  proximate CO2 Emission usage is provided in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  5  Results  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1  Classification and Fact-Checking  Table 3 reports the F1-macro on the test sets  for the classification and fact-checking tasks in  CLIMABENCH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We also give the average of the  scores on all tasks for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' There is no  single model that does the best across the board,  but RoBERTa is a clear winner as it beats the other  baselines on four out of eight tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' BERT falls  second in line when we look at average scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Longformer scores the highest on SCIDCC, per-  haps due to the model being able to attend to more  number of tokens available in the longer documents  in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' To our surprise, it also does well on  CLIMATEXT which has comparatively shorter text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  One reason for this could be that the sparse at-  tention within Longformer may be having a reg-  ularizing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' SciBERT ranks on the top for  CLIMATEFEVER and this could be because the  evidences in the task have structure and style sim-  ilar to the text in scientific papers used for pre-  training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' ClimateBERT and the model it  was warm-started from, DistilRoBERTa, are very  similar in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' DistilRoBERTa beats Cli-  mateBERT on CLIMA-INSURANCE, CLIMATEXT,  and CLIMATEENG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Although this raises concerns  on the quality of pretraining data used for Climate-  BERT, we cannot say for sure, since neither the  data nor the code is publicly available for diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Overall, the transformer models have significantly  better gains over linear ones except on CLIMA-  CDP-CITIES  CDP-STATES  CDP-CORPORATION  Model  MRR@10  MRR@100  MRR@200  MRR@All  MRR@10  MRR@100  MRR@All  MRR@10  MRR@All  No Finetuning on CDP  BM25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='747  Table 4: MRR@k scores for BM25, ClimateBERT and MSMARCO-MiniLM on the three subsets of CLIMA-QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Models finetuned and evaluated on same subset fall under In-Domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  CDP-STATES  CDP-CORPORATION  Model  MRR@10  MRR@100  MRR@All  MRR@10  MRR@All  No Finetuning  BM25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='180  MiniLM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='342  Finetuned on CDP-CITIES  Transfer  Transfer  ClimateBERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='477  MiniLM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='500  Table 5: MRR@k scores for BM25, ClimateBERT  and MSMARCO-MiniLM on the Transfer experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Models are finetuned on CDP-CITIES and evaluated on  States and Corporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  INSURANCE+ where the TFIDF+SVM model is  superior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' It shows that simple word co-occurence  statistics are enough for certain tasks and deep lan-  guage models might not be the right solution in  such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2  Question Answer Ranking  We analyze and compare the results of our in-  domain and transfer experiments on CLIMA-QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Table 4 and Table 5 summarize the MRR@k results  for the in-domain and transfer experiments respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' MSMARCO-MiniLM beats ClimateBERT  on all the subsets, for both in-domain and trans-  fer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Both models do significantly better  when fine-tuned on CDP-CITIES and evaluated on  States and Corporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' This indicates that models  trained on this dataset possess good transfer learn-  ing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The in-domain training boosts per-  formance further as we see for States and Corpora-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Lastly, the best performing model, MiniLM,  when finetuned on all three subsets, achieves com-  parable performance on Cities and Corporations  while ranking highest on States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  6  Extrinsic Evaluation  Beyond CDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' States and Companies have been uti-  lizing sustainability reporting frameworks to help  guide their non-financial reporting efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' These  frameworks are useful given the lack of uniform  regulations on nonfinancial reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' However, the  variety of such frameworks can be daunting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For  example, CDP aims to capture companies’ environ-  mental performance data, whereas the Task Force  on Climate-related Financial Disclosures9(TCFD)  guidelines are intended to encourage companies  to align their climate-related risk disclosures with  investors’ needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Automatic alignment of text to  relevant sections of the disclosures reduces the ef-  fort significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' To motivate this usecase of the  CLIMA-QA task, we conduct an experiment that  involves populating the CDP survey using Climate  Action Plans (CAPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Utilizing CAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Disclosure of carbon accounting,  climate risks and green targets is vital information  for researchers and policy-makers to craft well-  informed climate policy rules and regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In  addition to these reports, cities, states, and corpo-  rations may publish their CAP or sustainability  reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The CAPS include quantitative data, such  as emission values or renewable electricity genera-  tion capacity, and qualitative data such as specific  policy interventions across different sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Popu-  lating CDP surveys from new CAPs allows for con-  sistent comparisons of CAPs to existing datasets  which could be used to compare strategies, identify  gaps, or rank jurisdictions on the content and level  of ambition in their stated plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We consider the  scenario where the State’s CAP is available but  their CDP report is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The researcher’s aim is to  9https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='fsb-tcfd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='org  prec@1  prec@2  prec@3  prec@4  prec@5  Relevant  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='0  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='0  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='6  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='5  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='0  Highly Relevant  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='0  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='0  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='3  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='5  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='6  Table 6: Precision@K: We report the fraction of items  in the top K ranked retrievals that are either marked as  highly relevant, or at least relevant, averaged across text  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Relevance judgements for were performed  manually by an expert annotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  pull together the relevant portions of the CAP to  construct the missing CDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Typically the CAPS  are much longer (∼100 pages) and more compre-  hensive than any particular disclosure report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Our  aim is to measure the effectiveness of our model to  assist in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1  Task Setup  Unlike typical search problems, the climate re-  searchers need to map every data point in the CAP  to the corresponding CDP question/section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We  have observed that climate researchers annotate  documents in full, similar to how Climate Watch10  has been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For this reason, it is criti-  cal to evaluate the answer-to-question mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' If  we search for answers, then many of the questions  will be answered by the summary section of a CAP,  which does not help their goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For example, a  CDP question will ask to disclose all climate risks  that a corporation may be exposed to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Our aim is  to then identify all relevant sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Therefore, our procedure is as follows: First,  the expert (climate policy researcher on our team  and co-author) selected 5 pages at random from a  collection of 20 State CAPS and then selected a  random paragraph from each page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For this experi-  ment, we use the MSMARCO-MiniLM finetuned  on all 3 subsets CDP-CITIES, CDP-STATES and  CDP-CORPORATION (our best performing model  in the intrinsic evaluation) on each text segment and  select the top five scoring CDP-STATES questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  We then presented each segment along with the  retrieved questions to the expert and had them an-  notate the relevance for each question-answer pair  on a three point scale: No Relevance, Relevant,  Highly Relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='11  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2  Results  Table 6 shows the climate change researcher’s eval-  uation metrics for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Overall, 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='0% of  the 500 questions retrieved were judged Relevant  and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='6% rated Highly Relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' One pitfall of  our model is that there were more Highly Relevant  predictions ranked fifth than first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' One possible ex-  planation for this is that the top retrieved questions  were often more general while the questions that  were ranked lower were more specific and easier to  match (see Table 13 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=')  We show some examples in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' The first  four examples show high degrees of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In  example 1, our model correctly identifies the state  CAP text as impact-related and captures the spe-  cific discussion of compound risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' However, ex-  ample 6 appears to highlight a gap in the CDP  questionnaire related to the topic of environmental  justice, a result in itself of considerable interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Although our study is limited, it shows the promise  of using our benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Similar extrinsic studies  could be conducted on different climate reports  such as corporate ESGs or national Paris Agree-  ment plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  7  Related Work  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1  Benchmarks in NLP  With the ever growing inventory of deep learning  based models on a plethora of NLP tasks, the com-  munity is working towards general purpose mod-  els that perform well at multiple tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Conse-  quently, various benchmarks that evaluate general-  purpose understanding have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For  example, GLUE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2018) and the sub-  sequently more challenging SuperGLUE (Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019) datasets were introduced to test lan-  guage understanding capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' However, do-  ing well on general purpose benchmarks does not  translate well to domain-specific tasks like legal  reports, biomedical files or numerical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  This led to the emergence of benchmarks that cater  to a particular field of expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  For instance,  LexGLUE (Chalkidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2022) for legal text  and NumGLUE (Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2022) for mathe-  matical reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' CLIMABENCH follows on this  chain of thought to provide a unified way to evalu-  ate models on CC specific problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  10https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='climatewatchdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='org/  11By construction, in our rating there may be multiple rele-  vant questions found for each text segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Text Segment from State Climate Action Plans  Top Questions  1  Sea level rise will inundate some nearby coastal areas, and related  salt-water intrusion, coupled with increased drought stress may impact  water supplies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q1: Please describe the current and/or anticipated impacts of climate  change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q3: Please detail any compounding factors that may worsen the impacts  of climate change in your region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  2  The afforestation goal is to increase the area of forested lands in the state  by 50,000 acres annually through 2025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q1: Please provide the details of your region’s target(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q2: Please provide details of your climate actions in the Land use sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  3  By a majority vote, the ICCAC presents a policy option that, if deemed  necessary, would build one new 1200-megawatt nuclear power plant in  Iowa by January 1, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q3: Please provide details of your renewable energy or electricity tar-  get(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q4: Please provide details of your climate actions in the Energy sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  4  California maintains a GHG inventory that is consistent with IPCC  practices .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Reports from facilities and entities that emit more than  25,000 MTCO2e are verified by a CARB-accredited third-party  verification body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q1: Please give the name of the primary protocol, standard, or methodol-  ogy you have used to calculate your government’s GHG emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q3: Please provide the following information about the emissions verifi-  cation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  5  A leading driver of these high emissions is the fact that the District’s  daytime population swells by 400,000 workers every workday, which is  the largest percentage increase in daytime population of any large city in  the nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q4: Please indicate if your region-wide emissions have increased, de-  creased, or stayed the same since your last emissions inventory, and  please describe why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q5: Please report your region-wide base year emissions in the table  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  6  State law defines environmental justice as the fair treatment of people of  all races, cultures, and incomes with respect to the development,  adoption, implementation, and enforcement of environmental laws,  regulations, and policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q1: Please explain why you do not have policies on deforestation and/or  forest degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Q4: Please provide details of your climate actions in the Governance  sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Table 7: Examples from our human pilot study in which our climate expert has evaluated the relevance of CDP  questions linked to selected text from state climate action plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' A fragment of the matched text is presented with  two illustrative questions from the set of five question matches generated by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2  Climate NLP  CLIMATEXT (Leippold and Varini, 2020) and CLI-  MATEFEVER (Leippold and Diggelmann, 2020)  extracted and filtered documents from Wikipedia  and other sources to curate CC corpus that was fur-  ther annotated by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Due to the challenging  and evolving nature of this field, the labelling pro-  cess is not only time consuming but prone to noise  due to poor inter-annotator agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Unlike  these small annotated datasets, we are able to ef-  fectively utilize existing semi-structured disclosure  forms for a much larger set of supervised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Berrang-Ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' (2021) use machine learn-  ing to select scientific literature on climate adap-  tation for the task of systematic evidence map-  ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Luccioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' (2020) use Task Force  on Climate-related Financial Disclosures (TCFD)  questionnaires in a QA setup similar to ours while  Corringham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' (2021) use document headers  for course-grained classification on Paris Climate  Agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  8  Limitations and Future Work  One current limitation of our benchmark is that the  datasets are English only, thus restricting evalua-  tion to English trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Although CDP-  STATES has disclosures in other languages it rep-  resents a small portion of the reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' We plan  to include relevant CC datasets from the multilin-  gual European Union Public Data Catalog12 in the  future, while encouraging contributions from the  broader community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Another limitation is that ex-  pert human evaluation is missing for our new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  We conduct an analysis involving a climate expert  for one of our best-performing models on the QA  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' But, a general human evaluation of the other  classification tasks is not conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  We do not thoroughly investigate the efficiency  of the Transformer models in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' But, the  compute and training efficiency statistics we pro-  vide in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2 and Table 11 are a step in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  We encourage future work on CLIMABENCH to  leverage models that are both performant and effi-  cient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Finally, there exists more types of carbon disclo-  sures (TCFD, SBTi) as well as publicly accessible  corporate sustainability reports that we wish to in-  clude but require more difficult scraping and data  preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  9  Acknowledgement  This project was funded by the Climate Change  AI Innovation Grants program, hosted by Climate  Change AI with the support of the Quadrature Cli-  mate Foundation, Schmidt Futures, and the Canada  Hub of Future Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content=' Brucher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Perrot, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Duch-  esnay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Scikit-learn:  Machine learning in  Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Journal of Machine Learning Research,  12:2825–2830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Nils Reimers and Iryna Gurevych.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Sentence-  bert:  Sentence embeddings using siamese bert-  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In Proceedings of the 2019 Conference on  Empirical Methods in Natural Language Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Stephen Robertson and Hugo Zaragoza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  The  probabilistic relevance framework: Bm25 and be-  yond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Trends Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Retr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 3(4):333–389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Victor Sanh, Lysandre Debut, Julien Chaumond, and  Thomas Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Distilbert, a distilled version  of bert: smaller, faster, cheaper and lighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' ArXiv,  abs/1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='01108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Aarohi Srivastava, Abhinav Rastogi, Abhishek B Rao,  Abu Awal Md Shoeb, Abubakar Abid, Adam Fisch,  Adam R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Brown, Adam Santoro, Aditya Gupta,  Adrià Garriga-Alonso, Agnieszka Kluska, Aitor  Lewkowycz, Akshat Agarwal, Alethea Power, Alex  Ray, Alex Warstadt, Alexander W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Kocurek, Ali  Safaya, Ali Tazarv, Alice Xiang, Alicia Parrish,  Allen Nie, Aman Hussain, and Amanda Askell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Beyond the imitation game: Quantifying and  extrapolating the capabilities of language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  ArXiv, abs/2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='04615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Nandan Thakur, Nils Reimers, Andreas Ruckl’e, Ab-  hishek Srivastava, and Iryna Gurevych.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Beir:  A heterogenous benchmark for zero-shot evalu-  ation of information retrieval models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  ArXiv,  abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='08663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Roopal Vaid, Kartikey Pant, and Manish Shrivastava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Towards fine-grained classification of cli-  mate change related social media text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In Proceed-  ings of the 60th Annual Meeting of the Association  for Computational Linguistics: Student Research  Workshop, pages 434–443, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Associa-  tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Roopal Vaid, Kartikey Pant, and Manish Shrivastava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Towards fine-grained classification of cli-  mate change related social media text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob  Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz  Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Attention is all  you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In Advances in Neural Information Pro-  cessing Systems, volume 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Alex Wang,  Yada Pruksachatkun,  Nikita Nangia,  Amanpreet Singh, Julian Michael, Felix Hill, Omer  Levy, and Samuel Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Superglue: A  stickier benchmark for general-purpose language un-  derstanding systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In Advances in Neural Infor-  mation Processing Systems, volume 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Curran As-  sociates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Alex Wang, Amanpreet Singh, Julian Michael, Fe-  lix Hill, Omer Levy, and Samuel Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  GLUE: A multi-task benchmark and analysis plat-  form for natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  In Pro-  ceedings of the 2018 EMNLP Workshop Black-  boxNLP: Analyzing and Interpreting Neural Net-  works for NLP, pages 353–355, Brussels, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Wenhui Wang, Furu Wei, Li Dong, Hangbo Bao, Nan  Yang, and Ming Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Minilm: Deep self-  attention distillation for task-agnostic compression  of pre-trained transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Nicolas Webersinke, Mathias Kraus, Julia Anna Bin-  gler, and Markus Leippold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Climatebert: A  pretrained language model for climate-related text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  ArXiv, abs/2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='12010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Thomas Wolf, Lysandre Debut, Victor Sanh, Julien  Chaumond, Clement Delangue, Anthony Moi, Pier-  ric Cistac, Tim Rault, Rémi Louf, Morgan Funtow-  icz, Joe Davison, Sam Shleifer, Patrick von Platen,  Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu,  Teven Le Scao, Sylvain Gugger, Mariama Drame,  Quentin Lhoest, and Alexander M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Transformers: State-of-the-art natural language pro-  cessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In Proceedings of the 2020 Conference on  Empirical Methods in Natural Language Processing:  System Demonstrations, pages 38–45, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Asso-  ciation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  A  Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1  Compute Details  We used a 24 core AMD Ryzen CPU machine with  128 GB RAM for data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' For training and  inference of the deep learning models, we utilize  4 Nvidia RTX 2080Ti GPUs with 11GB memory  each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Each model was trained on a single GPU at a  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2  CO2 Emission Related to Experiments  A cumulative of 338 hours of computation was  performed on hardware of type RTX 2080 Ti (TDP  of 250W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Total emissions are estimated to be 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='5  kgCO2eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Estimations were conducted using the  MachineLearning Impact calculator presented in  (Lacoste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='3  Pretrained Transformer Models  BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019) is a popular  Transformer-based language model pre-trained on  masked language modeling and next sentence pre-  diction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' It makes use of WordPiece tokeniza-  tion algorithm that breaks a word into several sub-  words, such that commonly seen subwords can also  be represented by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019) uses dynamic masking  and eliminates the next sentence prediction pre-  training task, while using a larger vocabulary and  pre-training on much larger corpora compared to  BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Another notable difference is the use of byte  pair encoding compared to wordPiece in BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  DistilRoBERTa (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019) leverages  knowledge distillation during the pre-training  phase reducing the size of the RoBERTa model  by 40%, while retaining 97% of its language under-  standing capabilities and being 60% faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Sanh  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' (2019) originally distilled the BERT model  but we utilize the better performing RoBERTa ver-  sion in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Longformer (Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2020) extends  Transformer-based models to support longer se-  quences with the help of sparse-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' It uses a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='combination of local attention and global attention  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Section  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Category/Label  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Hazards: Adaptation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Adaptation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Adaptation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Adaptation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='Hazards: Climate Hazards  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='Climate Hazards and Vulnerability  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='City-wide Emissions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Emissions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Emissions Reduction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Emissions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='GHG Emissions Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Emissions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Local Government Emissions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Emissions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Emissions Reduction: City-wide  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='City Wide Emissions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Emissions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Emissions Reduction: Local Government  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='Local Government Operations GHG Emissions Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Emissions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Energy Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Energy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Energy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='Food  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Food  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Governance and Data Management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Governance and Data Management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Opportunities  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='Strategy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Strategy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Urban Planning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Strategy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Transport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Transport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Waste  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+page_content='Water  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Water  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Water Security  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Water  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Table 8: The Sections in the CDP Cities Questionnaire and the corresponding Label assigned by climate expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Task  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Average  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Max  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Min  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Std  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='CLIMA-INSURANCE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='203  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='4588  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='326  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='CLIMA-INSURANCE+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='206  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='4588  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='335  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='CLIMA-CDP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='73  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='801  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='83  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='CLIMA-QA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='105  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='834  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='88  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='CLIMATEXT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='124  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='CLIMATESTANCE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='98  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='CLIMATEENG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='98  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='CLIMATEFEVER  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='47  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='311  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='SCIDCC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='580  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='2014  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='223  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='Table 9: Statistics for the number of tokens in each task  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='of CLIMABENCH  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='mechanism that allows for linear attention com-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='plexity and thus makes it feasible to run on longer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='documents (max 4096 tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' It however takes  much longer to train than the shorter context (512  tokens) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  SciBERT (Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019), a pretrained lan-  guage model based on BERT, leverages unsuper-  vised pretraining on a large multi-domain corpus of  scientific publications to improve performance on  downstream scientific NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' It was evaluated  on tasks like sequence tagging, sentence classifi-  cation and dependency parsing with datasets from  scientific domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' SciBERT gives significant im-  provements over BERT on these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  ClimateBERT (Webersinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2021) was  warm-started from the DistilRoBERTa model and  pretrained on text corpora from climate-related re-  search paper abstracts, corporate and general news  and reports from companies that were not publicly  released with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' It was evaluated on tasks  like sentiment analysis (using a private dataset),  and public datasets like CLIMATEXT and CLI-  MATEFEVER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' In this paper, we evaluate and com-  pare the performance of ClimateBERT on diverse  CC tasks for the first time, providing a comprehen-  sive, publicly available and reproducible evalua-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Model  Source  # Params  Vocab Size  Max Length  BERT  (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019)  110M  30K  512  RoBERTa  (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019)  125M  50K  512  DistilRoBERTa  (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019)  82M  50K  512  Longformer  (Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2020)  149M  50K  4096  SciBERT  (Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2019)  110M  30K  512  ClimateBERT  (Webersinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=', 2021)  82M  50K  512  Table 10: Pretrained Transformer Language Models used for Classification tasks  Model  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Runtime (in hours)  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Train Samples/Second  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Train Steps/Second  ClimateBERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='40  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='83  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='64  DistilRoBERTa  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='40  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='58  SciBERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='70  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='84  RoBERTa  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='80  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='79  BERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='85  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='77  Longformer  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='95  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='76  Table 11: Compute Efficiency Metrics for the Pretrained Transformer models for the experiments conducted on  CLIMABENCH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Models based on the DistilRoBERTa architecture are the most efficient due to smaller model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  Parameter  Values  loss  hinge, squared_hinge  C  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='1, 1, 10  class_weight  none, balanced  Table 12: For the linear SVM, we grid search over the parameters with 5-fold validation to get the best fit out of  80 candidates (16 values * 5 folds) with F1 Macro as the scoring mechanism  Question  MRR@132  Please provide details of your climate actions in the Agriculture sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='870  Please provide details of your climate actions in the Waste sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='789  Please provide details of your climate actions in the Transport sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='774  Please provide details of your climate actions in the Buildings & Lighting sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='597  Please describe these current and/or anticipated impacts of climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='492  Please complete the table below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='487  Please indicate the opportunities and describe how the region is positioning itself to take  advantage of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='445  Please provide details of your climate actions in the Energy sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='397  Please describe the adaptation actions you are taking to reduce the vulnerability of your  region’s citizens, businesses and infrastructure to the impacts of climate change identified in  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='378  Please describe these current and/or future risks due to climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='327  List any emission reduction, adaptation, water related or resilience projects that you have  planned within your region for which you hope to attract financing, and provide details on the  estimated costs and status of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' If your region does not have any relevant projects,  please select “No relevant projects” under Project Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='319  Please provide details of your climate actions in the Land use sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='286  Please provide the details of your region-wide base year emissions reduction target(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' You  may add rows to provide the details of your sector-specific targets by selecting the relevant  sector in the sector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='252  Please describe the adaptation actions you are taking to reduce the vulnerability of your  region’s citizens, businesses and infrastructure to the risks due to climate change identified  in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content='247  Table 13: Question difficulty evaluated on the test set of CDP-STATES ranked from best performing to worst  performing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
+page_content=' Filtered to only questions that appeared at least twenty times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfAAk2/content/2301.04253v1.pdf'}
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+Cybersecurity Threat Hunting and Vulnerability Analysis Using a
+Neo4j Graph Database of Open Source Intelligence
+Elijah Pelofske∗1, Lorie M. Liebrock1, and Vincent Urias2
+1New Mexico Cybersecurity Center of Excellence, New Mexico Tech
+2Sandia National Laboratory
+Abstract
+Open source intelligence is a powerful tool for cybersecurity analysts to gather information both for analysis
+of discovered vulnerabilities and for detecting novel cybersecurity threats and exploits. However the scale of
+information that is relevant for information security on the internet is always increasing, and is intractable for
+analysts to parse comprehensively. Therefore methods of condensing the available open source intelligence, and
+automatically developing connections between disparate sources of information, is incredibly valuable. In this
+research, we present a system which constructs a Neo4j graph database formed by shared connections between
+open source intelligence text including blogs, cybersecurity bulletins, news sites, antivirus scans, social media
+posts (e.g., Reddit and Twitter), and threat reports. These connections are comprised of possible indicators
+of compromise (e.g., IP addresses, domains, hashes, email addresses, phone numbers), information on known
+exploits and techniques (e.g., CVEs and MITRE ATT&CK Technique ID’s), and potential sources of information
+on cybersecurity exploits such as twitter usernames.
+The construction of the database of potential IoCs is
+detailed, including the addition of machine learning and metadata which can be used for filtering of the data
+for a specific domain (for example a specific natural language) when needed. Examples of utilizing the graph
+database for querying connections between known malicious IoCs and open source intelligence documents,
+including threat reports, are shown. We show that this type of relationship querying can allow for more effective
+use of open source intelligence for threat hunting, malware family clustering, and vulnerability analysis.
+1
+Introduction
+Open source intelligence offers an extraordinary amount of information that a cybersecurity analyst can use for
+threat detection, mitigation, and analysis [1–6].
+However, open source intelligence contains a large amount of
+noise (i.e., irrelevant information) and most importantly the scale of the data is too large to be useful in its
+raw form.
+To this end, automating the process of finding indicators of compromise and relevant relationships
+between the indicators of compromise, has become increasingly important [5, 7, 8]. The central idea utilized in this
+article is forming a network of associations between open source intelligence documents and potential indicators of
+compromise (IoCs) that exist in the open source intelligence text. The term potential IoC is important because it
+specifies that the text is potentially useful and quite unstructured, but there is a pattern match in the data that
+does fit a particular form (for example, an IP address, or a CVE). Graphs are a natural way to express these types
+of higher order connections and are used in a variety of cybersecurity contexts [9, 10]. These types of networks,
+when they are intended for providing semantic meaning between heterogeneous data types are also referred to as
+knowledge graphs [11–13]. In order to create a graph representation of a large amount of open source intelligence,
+we utilize Neo4j which provides a visual interface to search the graph, allows a number of users to interact with the
+data, and also provides an efficient query time for the database to interact with other analysis systems or to simply
+query and display the raw OSint text documents that are connected to a relevant exploit or IoC. Neo4j graph
+databases have been used in other domains for the purpose of storing and querying data structures with complex
+networks [14–17] including social network analysis [18] and typhoon disaster knowledge [19]. Utilizing graphs in
+order to better evaluate potentially relevant connections that exist in a large dataset is a subject of considerable
+interest in the literature [20–22].
+In this article we outline a methodology which consists of collecting and aggregating open source intelligence,
+in conjunction with antivirus scan results, and presenting the information contained in this data in the form of
+a network, or a graph. Specifically, edges represent a connection between a potential IoC and a document. The
+∗Email: elijahpelofske@gmail.com
+1
+arXiv:2301.12013v1  [cs.CR]  27 Jan 2023
+
+Neo4J Graph Database 
+of potential cybersecurity 
+IOC's
+Arbitrarely 
+structured 
+text files
+OSint web crawler 
+(aka Spider) 
+structured data. 
+Includes metadata 
+such as time stamps
+File antivirus 
+scan 
+structured 
+data
+Execute CTC to 
+determine if the natural 
+language is likely 
+discussing 
+cybersecurity or not
+Execute Tram to 
+determine possible 
+Mitre Attack Technique 
+ID's
+Execute rcATT to 
+determine possible Mitre 
+Attack Techniques or 
+Tactics
+Extract metadata 
+and clean natural 
+language text when 
+required. 
+Deduplicate against 
+all previously 
+processed 
+documents.
+regex or search for 
+potential IOC's in the 
+natural language
+Package documents 
+and metadata with 
+edges to their 
+potential IOC's in the 
+form of a Neo4J 
+transaction and add 
+to the database
+Determine likely natural 
+languages being used
+Executed 
+in parallel
+Figure 1: IoC Neo4j graph database construction workflow diagram.
+challenge is that most open source intelligence text from the web is unstructured therefore analyzing the text for
+relevant pattern matches is a means to extract the potential IoCs.
+Here a document is simply a collection of
+natural language or data - for example a single tweet from Twitter could be a document. Furthermore, metadata
+is included with the node document when constructing the database - several machine learning algorithms identify
+whether the text is discussing cybersecurity and specifically what type of exploit techniques are being discussed.
+This type of metadata is important because it can can serve as a signal for the quality of the given node document
+- meaning that a piece of open source intelligence text gathered from social media will have a higher confidence of
+being relevant if the machine learning algorithm tells us it is likely discussing a cybersecurity topic. In Section 2
+we detail the construction of this type of cybersecurity intelligence graph, instantiated as a Neo4j graph database.
+In Section 3 we give specific examples of where this type of open source intelligence graph database can provide
+useful information and analysis in cybersecurity. We conclude in Section 4 with a discussion of what this type of
+approach to open source intelligence analysis can be used for and future research directions.
+2
+Methods
+In this section we outline the pipeline used to construct the Neo4j database.
+In Section 2.1 the open source
+intelligence data sources are defined, in Section 2.2 the database construction is outlined.
+2.1
+Open source intelligence data sources
+The open source intelligence documents that are fed into the system broadly fall into three categories; the three
+green starting blocks in Figure 1 show these categories.
+The first type of data is entirely unstructured text -
+typically this type of data is simply a threat report or a list of indicators or vulnerabilities. This option allows any
+collection of text sourced from any internet discussion to be analyzed for potential IoCs and to be added into the
+Neo4j database. Figure 1 was created in Lucidchart1.
+The second type of data comes from a system of web crawlers (the details of this system are outlined in ref.
+[23]). The text found by the web crawlers is typically similarly unstructured and can originate from a variety of
+sources such as Reddit, Twitter, blogs, cybersecurity bulletins, and news sites. However, the web crawler data also
+contains metadata including links that the crawler followed to this current site, time and date information, and
+1www.lucidchart.com
+2
+
+cybersecurity keywords that were found when parsing the site. Therefore, this data is slightly more structured and
+is therefore parsed differently from arbitrary text; specifically the metadata is kept separate from the text that will
+be analyzed for potential IoCs, which is eventually added into the node information in Neo4j.
+The third type of data is structured antivirus scans of files that are potentially malware. This data is entirely
+machine readable and does not contain natural language information. Therefore, this data is also parsed differently
+to the other two data sources. In particular, these antivirus scans include hashes of files and the names of those
+files. Therefore, those specific fields (hashes and file names) that are present in the structured data are parsed for
+creating edges in Neo4j, but no other natural language analysis or pattern matching for potential IoCs is performed.
+2.2
+Neo4j database construction
+Node type
+Count of unique nodes
+Count of edges
+Document
+2128992
+MD5 Hash
+394826
+944671
+SHA1 Hash
+323321
+990817
+SHA256 Hash
+642535
+1941248
+SHA512 Hash
+18339
+28112
+Malware name
+365
+117528
+APT name
+457
+165741
+Email
+85396
+953757
+CVE ID
+174668
+1313206
+Twitter username
+174143
+402899
+Phone number
+23756
+144397
+IP address
+119699
+705386
+domain
+214720
+3116508
+File name
+351507
+1326480
+MITRE ATT&CK Technique ID
+445
+21018
+Table 1: Neo4j database statistics. The “Count of edges” column in the table is the number of edges between
+the listed node type and Document nodes. For each non-document node type, the edges between document nodes
+and the unique nodes for that indicator are always labelled the same as the indicator; for example CVE nodes are
+connected to document nodes by edge labelled as CVE. Those edge types are the counts displayed in the third
+column. Therefore, there are no edges between document nodes which is why the edge count in that cell is empty.
+Figure 1 details the high level workflow that constructs the Neo4j database of potential IoCs. This entire system
+is operating continuously by reading in new data from the independent web crawler infrastructure and then adding
+relevant information into the Neo4j database. First, the data is parsed depending on its source, as outlined in
+Section 2.1. This input data is referred to as a document because this collection of text is all originating from the
+same source and is therefore logically linked, which could be helpful for identifying relationships when searching
+the database. As an example, a single document could be a piece of text from social media, such as a Reddit
+post or a tweet from Twitter. Any metadata that is associated with the input text is parsed at this point, to be
+added into the Neo4j database if a node representing this document is created. Importantly, once the raw text
+input has been extracted (this includes the JSON structured antivirus file scan data), a SHA256 checksum of the
+data is created and this checksum is then checked against the checksums of all of the documents that are in the
+Neo4j database; and if the document is a duplicate, it is not added to the database. Next, several processes begin
+executing in parallel, all of which are designed to extract useful meaning from the natural language input. To
+this end, the antivirus file scan data is not parsed using the machine learning algorithms or language detection;
+however the potential IoCs such as filenames and hashes are extracted and used to create edge relationships in the
+database. The Cybersecurity Topic Classification (CTC) tool, which is compromised of multiple machine learning
+algorithms trained to detect cybersecurity vs non-cybersecurity discussions from social media and developer forum
+English text sources [24], is executed on the text. The output of this algorithm is simply three states - either it is
+likely cybersecurity related text, or not, or there was not enough data (e.g., English words) to make a decision.
+The Reports Classification by Adversarial Tactics and Techniques (rcATT) machine learning python tool [25]
+is also executed on the text. This tool gives a list of likely MITRE ATT&CK tactics and techniques 2 that were
+2https://attack.mitre.org/
+3
+
+mentioned in the text. The MITRE ATT&CK framework provides a consistent basis for tracking cybersecurity
+techniques [26–31]. The rcATT tool was trained on threat report text and therefore the error rates on non-threat
+report documents is expected to be high. The tool is applied uniformly to all documents because it is not necessarily
+known a-priori what the exact semantic content of the document is (e.g., whether it is a threat report, or a news
+report, or entirely non-cybersecurity). It is generally expected that based on the training data used to create the
+machine learning models in the tool, cybersecurity content (e.g., documents where CTC returned True) will have
+higher accuracy results in regards to detecting discussions of specific MITRE ATT&CK tactics and techniques.
+The Threat Report ATT&CK Mapper (TRAM) machine learning python tool3 is also executed on the document.
+TRAM, similar to rcATT, returns likely MITRE ATT&CK techniques that were mentioned in threat reports.
+Therefore similarly to rcATT, this data will likely have high error rates for natural language text that is quite
+different from threat reports, but could be more accurate for cybersecurity related text.
+Especially for the data coming from the web crawlers, the natural language text could be non-English. Having
+some signal to indicate when this occurs in the Neo4j database could be useful (for example if one just wants to
+query documents that are only English or Spanish text). The other reason that this signal is important is because
+all of the natural language machine learning algorithms vectorize the input text using a very broadly defined English
+dictionary - meaning that other languages are not used in these models, which means that their results will be
+very inaccurate and should not be used. Therefore, the python tool langdetect is also executed on the text and the
+resulting language detection information is included in the node metadata.
+Lastly, pattern matches for all potential IoCs are performed on the text. With the exception of the structured
+antivirus file scans, where the potential IoCs that can be extracted can be done automatically, all of the other
+input text can be entirely unstructured. Therefore, simple pattern matching procedures are performed in order
+to find potential IoCs. Hashes (md5, sha1, sha256, sha512) are found by searching for high entropy hexadecimal
+text that fits within the required character length. File names are found by matching tokenized words (NLTK
+[32] was used for most of the tokenization procedures) that have a file ending that matches some standard file
+ending (for example .py for python). Advanced Persistent Threat (APT) group names and malware names are
+all simply pattern matched against tokenized words in the text. Phone numbers, Email addresses, IP addresses,
+twitter usernames, and domain names are all found by pattern match searching for the known standard formats
+that they appear in. Some simple checks are used to rule out pattern matches which do not fit the expected format
+of the data type and in the case of domain names, the top 1 million (Alexa top 1 million list) most searched domains
+are removed in order to reduce noise in the graph. Common Vulnerabilities and Exposures (CVEs) [33] and the
+unique ID numbers of MITRE ATT&CK techniques [34] are also pattern matched for; both of these data types also
+follow a standard format which can be identified. Each of these pattern matches will correspond to an edge (e.g.,
+a connection) between the document node that contained this data and a node representing that unique pattern
+match. This unique pattern match we broadly call a potential IoC, but it can also simply be a unique ID to track
+a known vulnerability (for example a CVE), or it could be a potentially useful piece of information for connecting
+two document nodes but not be a malicious IoC.
+Once all of this pattern matching and data processing has been completed, a number of Neo4j transactions are
+created in order to add the new data into the database. All of the processed documents have unique nodes created
+which contain several different components. The most important part of the node is the original raw text that
+was in the node document. Next, web crawler metadata is in the nodes which originated from web crawling - here
+the metadata includes the link, parent link, time stamps, keywords, and potentially other data such as checksums.
+Next, some document nodes are antivirus scans - in these cases there is a large amount of metadata about the
+scan that was performed, but typically the data is not natural language. All nodes also have language detection
+metadata, if there was natural language text that could be processed. The natural language detection is useful
+for filtering nodes which contain only a specific language of interest, however as expected most of the document
+nodes are English text. The pattern matches that were found in the text that form the edges in the graph are also
+included as a segment in the node data. Lastly, the machine learning results from CTC, rcATT, and TRAM are also
+included as a separate data segment (if there is applicable English text that could be fed to these ML algorithms).
+Each of these components of node data can be used to filter for specific nodes which have specific attributes that
+are relevant for a specific task. Any documents which have no pattern matches are not added to the database
+as they would simply be degree 0 nodes. Next, for all unique pattern matches found in this set of documents,
+check if the nodes already exist in the database. If they do exist, do nothing, but if they do not exist, then create
+them. This is to avoid creating duplicate potential IoC or pattern match nodes. Lastly, edges are formed between
+the document nodes and the nodes representing the pattern matches found in those documents. Table 1 shows
+the counts of nodes and edges in the database as of the writing of this paper. Therefore, the constructed graph
+3https://github.com/center-for-threat-informed-defense/tram/
+4
+
+database is always bipartite where one partition is the node documents (i.e., the sources of information) and the
+other partition is the set of nodes representing different potential IoCs or vulnerability IDs that exist in the node
+documents. This structure allows users, or algorithms, to query for relationships based on potential IoCs (e.g., a
+path connecting two potential IoCs or the network of neighbors associated with a potential IoC) and then examine
+the sources of the information within the node documents, including machine learning and language metadata.
+3
+IoC connections in the graph database
+In this section some specific visual examples of the structure of the database indicates what is available to the user
+when searching for IoCs or vulnerability IDs. The hard part of cybersecurity analysis is aggregating information,
+in this case open source unstructured text, into a meaningful form that can provide a network of related documents
+and potential indicators of compromise. In Section 3.1 we show the graph connectivity around an md5 hash of a
+known malware file from the WannaCry ransomware. In Section 3.2 we show the graph connectivity surrounding
+an IP address known to be a command and control server for the Qakbot trojan. In Section 3.3 we show the graph
+connectivity around a CVE that is known to be exploited in the wild.
+Section 3.4 shows a specific case where a sha256 hash of a contained resource was in several antivirus scans of
+different portable executable malware. This shared contained resource is reasonably unique, and indicates software
+re-use during the development of otherwise seemingly unconnected malware samples.
+In Section 3.5 we show the distribution of CVE node degrees in the Noe4j database compared to their Common
+Vulnerability Scoring System (CVSS) scores, and show that for reputable open source cybersecurity sources (e.g.,
+threat reports), there is a weak-to-median linear relationship between the two. Lastly, in Section 3.6 the Neo4j
+Graph Data Science implementation of PageRank is applied to the graph database, allowing a ranking of the most
+influential CVEs across the entire database.
+Figure 2: Node and edge coloring legend for potential IoCs.
+For the visual graph examples we query the database with the Neo4j browser using the Cypher query language4.
+The node and edge coloring’s encode the following information. Document nodes are large cyan nodes, SHA1 nodes
+are magenta, SHA256 nodes are lime, SHA512 nodes are grey, MD5 nodes are purple, file name nodes are teal,
+email address nodes are yellow, IP address nodes are orange, malware name nodes are brown, Twitter usernames
+are maroon, APT name nodes are lavender, domain name nodes are green, phone number nodes are blue, CVE
+nodes are red, and MITRE ATT&CK technique IDs are beige. The edge coloring matches the node coloring; for
+example an edge connecting a CVE to a document node where it was mentioned will also be colored red. Figure 2
+shows the node and edge coloring legend. In order to reduce visual clutter, if there are near duplicate documents
+from the web crawlers which have the same connections to a group of nodes, we manually remove all but one of
+the duplicate document nodes.
+Figure 3: Degree 1 connections associated with the md5 hash 84c82835a5d21bbcf75a61706d8ab549
+4https://neo4j.com/developer/cypher/
+5
+
+node_CVE
+node_IP
+node_phone_number
+node_sha1
+node_MitreAttack id
+node_sha256
+node_sha512
+node_document
+node_domain
+node_twitter_username
+node_email
+node_file_name
+node_APT_name
+node_malware_name
+node_md5Figure
+4:
+Expanded
+connections
+from
+Figure
+3;
+degree
+2
+connections
+out
+from
+the
+md5
+hash
+84c82835a5d21bbcf75a61706d8ab549
+3.1
+Visual analysis: md5 malware hash
+In order to show a specific malware hash subgraph of the database, we will perform a query to the Neo4j database
+for the md5 hash 84c82835a5d21bbcf75a61706d8ab549. The Cypher language query used to search for this
+single md5 node is:
+MATCH p=(find:node md5 {name:
+‘84c82835a5d21bbcf75a61706d8ab549’}) RETURN p
+In order to illustrate the utility of the database, we have selected this hash because it is known to be a hash of
+a piece of malware. The query results show that there are three unique document nodes that reference this specific
+hash. Figure 3 shows this connectivity graph. The content of these three document nodes contains useful context
+information. The lower left hand node document is a cybersecurity blog-style website that is detailing a network
+and security analysis tool, which used the example of this md5 hash for detecting malware using antivirus software.
+The lower right hand node document is a manalyzer report on this md5 hash5. The top node document is a 4chan
+thread for which a user linked to this manalyzer report within a long discussion thread.
+Next, we can query the neighbors of those three document nodes - the connectivity graph for this result is
+shown in Figure 4. Notably, there is a SHA1 and a SHA256 hash which were both in two of the node documents.
+These these two hashes and the md5 hash are all checksums of the same file.
+Next, we expand the neighbor relationships for these two SHA1 and SHA256 hashes that were both contained
+in the two threat report nodes in order to see if there are relevant connections we can investigate further. This
+neighborhood expansion is shown in Figure 5. We see that there are two node documents that are linked to the
+SHA256 hash. These two nodes are threat reports. One of these threat reports is also linked to the earlier manalyzer
+report node by a common filename. However, this filename is simply the command prompt executable (cmd.exe).
+These two nodes represent slightly different versions of a threat report titled The Lazarus Constellation, authored
+by Avisa Partners.
+This specific malware hash is associated with the WannaCry ransomware.
+WannaCry is
+ransomware [35–39] that propagated across the world in 2017 targeting computers running Windows OS primarily
+using an exploit known as EternalBlue. Figure 6 shows the neighborhood of connections that the The Lazarus
+Constellation threat reports contain, which includes a large number of hashes, CVEs, domains, and filenames that
+5https://manalyzer.org/report/84c82835a5d21bbcf75a61706d8ab549
+6
+
+Figure 5: Expanding the neighbors for two of the hash node connections from Figure 4.
+are all associated with this APT group. This malware hash example shows how the graph database can retrieve
+relevant information for an IoC - here we were able to find two other hashes of the same file, a place where the
+hash was mentioned on a 4chan board, and finally threat reports which give the larger context of why these hashes
+are relevant.
+3.2
+Visual analysis: Qakbot IP address
+Next we will consider the connectivity graph around a known malicious IP address 89.101.97.139, which is the
+IP address for a command and control server for the malware called Qakbot. The connections showing where
+this IP address was mentioned in the database are shown in Figure 7. There were three document nodes which
+mentioned this IP address. Two of these documents were from Github repositories where known IoCs on Qakbot
+were published, and the third document is from a pastebin page which also posted Qakbot IoCs. Interestingly,
+there are several IP addresses that are common to all, or a subset, of these documents. Note that the malware
+name (brown colored degree three node near the center of the graph) was the name Qakbot. This graph shows
+the relevant context around this IP address - namely that it is a command and control server for Qakbot and the
+connectivity graph shows many additional IoCs that should be monitored in association with Qakbot. Importantly,
+these graph connections take advantage of multiple (separated) data sources - where if we were to have read only a
+single one of these documents we would not have the aggregated data which includes many more IP addresses and
+hash IoCs. It is clear from this graph that this specific IP address is not a lone command and cotnrol server for
+Qakbot, rather it is one of many IP addresses and domains that are connected to this malware. Figure 15 in the
+Appendix B shows significantly larger graph renderings for the degree 3 and 4 connections in the database from
+this IP address node.
+3.3
+Visual analysis: known exploited CVE
+Here we show the connectivity graph for all neighboring nodes up to degree 2 away from a CVE that is known to be
+exploited in the wild6; CVE-2014-4404. CVE-2014-4404 is a remote code execution exploit on Apple OS X, earlier
+6https://www.cisa.gov/known-exploited-vulnerabilities-catalog
+7
+
+Figure 6: Expanding the two threat report node connections that were connected to the two shared hash nodes in
+Figure 5
+iOS versions and Apple TV. The neighborhood graph is shown in Figure 8. The neighboring nodes connected to this
+specific node are showing the halo of data that is connected to this CVE. In particular, these neighboring nodes show
+filenames, other CVEs, twitter usernames, domain names, hashes etc., that were mentioned along with this CVE
+in various documents. Two of these nodes were threat reports which described the state of various cyberattacks
+in 2021 - therefore they broadly discussed a number of different exploits, APT groups, twitter usernames, and
+malware families with CVE-2014-4404 being one of the more exploited Apple software CVEs. As one would expect,
+the CTC tool labelled each of these text documents that are in this halo as being cybersecurity related. This ML
+metadata can be used to filter for a subgraph of the graph database which contains only cybersecurity related
+(English) natural language.
+3.4
+Malware hash files: code and resource re-use
+The static analysis antivirus [40] scan hash file data nodes that are formed in Neo4j include hashes of extracted
+files, functions, or strings contained within the malware (as well as the malware itself). What these additional
+hashes allow within the graph database is detection of shared code components, for example shared header files,
+strings, or functions. Of course some of these may not be incredibly useful indicators - for example they may be
+standard Portable Executable manifest files. However, if there are unique code artifacts in some subset of antivirus
+file scans which are known to be malicious, then those could serve as indicators of shared code re-use from a group.
+These could even serve as malware signatures for antivirus products. The graph database is good for investigating
+these shared resource hashes because it is easy to find where a set of antivirus scans have a shared resource hash
+node.
+Here we provide a specific example - that is a SHA256 hash which we can search for in the Neo4j graph database
+with the Cypher syntax of:
+MATCH p=(find:node sha256 {name:
+’84f7c54dc015637a28f06867607c2e0b
+dd225d10debb1390ff212d91cd2d042b’}) RETURN p
+Using FileScan7, which contains references to this hash, we can find that this is a hash of the following English
+7https://www.filescan.io
+8
+
+Figure 7: Degree 2 halo of connected nodes to the Qakbot 89.101.97.139 IP address, where the node representing
+the IP of interesting is the orange degree 3 node near the bottom document node.
+ASCII text: BundleInstall BundleInstall. Note that this representation is not necessarily capturing the full
+hexadecimal data present in the data segment. The terms bundle install suggest that the source code language is
+Ruby, and this text artifact could be a result of packaging Ruby source code into a portable executable.
+Within the current Neo4j database, this SHA256 hash node has a degree of 5 (meaning that is referenced in 5
+node documents). Two of these nodes are actually effectively duplicates - they are scans of the same file, meaning
+that they have the same file name and associated extracted hashes (such as the hashes of contained resources in
+the PE), but were scanned at different times and therefore have slightly different data so the direct de-duplication
+did not remove one of them. Therefore as with the other Neo4j figures in order to reduce clutter, one of these
+duplicates is removed in the displayed figures. The graph renderings of these nodes and their degree 1 neighborhood
+connections are shown in Figures 9.
+For the bottom graph in Figure 9 we can examine what some of the important information is that the nodes
+have, such as the malware filename and the hash of the file. These four nodes are known malware samples (this
+statement is based on the high proportion of antivirus scans indicating malware) and additional information on
+them is available on VirusTotal.
+Top document node in Figure 9: the filename is rkinstaller.exe and the full sha256 hash is given in the
+virustotal link8.
+Right document node in Figure 9: no associated filename and the full sha256 hash is given in the virustotal
+link9.
+8https://www.virustotal.com/gui/file/5577ce9aa4e4ec2735247c5769f0e84db599825f2d95159b0102f3b30e80b6bb/details
+9https://www.virustotal.com/gui/file/06f11f4a555a4891c93f13f82dc06e8bcedda2a71c8a5e6aa5c18da871f41238/details
+9
+
+Figure 8: All nodes and edges connected to the node for CVE-2014-4404 up to two degrees away. The degree 11
+red node approximately in the center of the graph is the CVE-2014-4404 node.
+Bottom document node in Figure 9: the filename is rkinstaller364.exe and the full sha256 hash is given in
+the virustotal link10.
+Left document node in Figure 9: the filename is poinstaller257.exe and the full sha256 hash is given in the
+virustotal link11.
+The notable observation from this data is that this shared portable executable resource hash is a reasonably
+unique artifact (meaning that it is not a commonly re-used ASCII text segment in PE development) and was found
+across a small subset of malware samples, some of which also have other shared characteristics such as similar (or
+identical) file names. This suggests, with reasonable confidence, that the development of these pieces of malware
+are linked in a meaningful way, for example the same developer could have created these portable executables. This
+demonstrates where the graph database construction of OSint allows a user to link together pieces of information
+in order to group together seemingly unconnected documents and other potential indicators.
+Portable executable visualizations for three other pieces of known malware which contain this specific PE
+resource sha256 hash are shown in Figure 14 in Appendix A.
+3.5
+CVE degree and CVSS score
+Because this database is constructed in a largely unsupervised manner - i.e., pattern matches are automatically
+generated and new data is added to the database without human review - a natural question that arises is whether
+the graph structure of the data represents the real world properties of the vulnerabilities or potential IoCs. An easy
+example of this that we can numerically compute is the degree of CVE nodes in the database (which corresponds to
+how many times that CVE was mentioned in the text from the different data sources) and compare that against the
+CVSS scores of CVEs. CVSS scores are intended to approximately represent the overall severity of the vulnerability
+[41–44] where a CVSS score of 0 is the lowest severity and 10 is the maximum severity. There are two CVSS score
+versions that we will compare - CVSS version 2 and version 3. Version 3 is the newest CVSS scoring method, which
+is intended to be a more accurate rating scale for modern cybersecurity threats.
+The relevant question is whether there exists a relationship of increasing degree of the CVE nodes in the graph
+database with respect to CVE CVSS score. Intuitively, if the severity of a CVE corresponds to how frequently that
+CVE is mentioned in social media, news, and threat reports, then higher CVSS scores will correspond to higher
+10https://www.virustotal.com/gui/file/f8d11b1e3e027355a11163049b530de4fd67183abd08a691d5d18744653ef575/details
+11https://www.virustotal.com/gui/file/f3efcfc7121f2348deb6f3b5ffde60878d978c25281e67defdc288feaef8b38c/details
+10
+
+Figure 9: Graph rendering of the neighboring nodes connected to the sha256 node hash of interest, which is the
+degree four green-yellow node in the center of the graph. The neighboring connections of these document nodes
+have also been displayed in order to determine if there are any other connections of interest. Each of the document
+nodes are antivirus scans of portable executables. The degree 1 expanded neighbors show that there is actually
+another sha256 hash node that is shared by three out of the four document nodes, and there are no other edges
+connecting the associated hashes and file names. That sha256 hash is a standard manifest for creating PEs and is
+common to a large number of PE samples (both benign-ware and malware) and is therefore not unique enough to
+attribute a meaningful connection.
+degree CVE nodes in the Neo4j graph database. The CVSS scores are retrieved from the NIST NVD dataset 12.
+To this end, we compute the Pearson correlation coefficient between CVE node degrees and their CVSS score using
+scipy in python [45–48]. Some CVEs do not have a version 2 or a version 3 score, and therefore are not able to
+plotted in this dataset.
+Figure 11 plots all CVE CVSS scores against CVE node degrees in the Neo4j graph database, which shows there
+is not a positive or linear correlation between the CVE node degrees and CVSS scores. This is notable because it
+shows that across all of the cybersecurity mentioned natural text that was gathered, there is not a strong correlation
+between the rate of CVE mentions (e.g. CVE popularity) and CVSS scores. However, it could be the case that
+more focused cybersecurity documents have a higher CVE CVSS score and Neo4j node degree correlation. It could
+also be the case there is not a strong signal of correlation for CVE nodes which are not commonly mentioned in
+news, threat reports, and cybersecurity bulletins - and therefore only considering nodes which have at least a degree
+of 2 could remove some noise in the dataset. Lastly, due to popularity of new CVEs, necessarily the web scraping
+will have a temporal bias for the times during which the spiders were operating.
+Figure 12 shows the correlation plot for CVE node degrees vs CVSS score, but the Neo4j node degrees are
+computed using a restricted set of document sources namely reputable blogs, information sites, and threat reports
+- specifically fireeye13, all threat reports, proofpoint14, exploitdb15, the Hackernews16. Additionally, CVE
+nodes with degree 1 are not included in this computation since low degree nodes do not necessarily provide a
+strong signal in regards to how referenced that CVE is. Lastly, the data in Figure 12 are restricted to the years
+during which the OSint crawler system was operational [23] because there is an inherent temporal bias in regards
+to what web pages are mentioned and scraped. Because only a subset of the CVEs had CVSS v3 scores, there
+12https://nvd.nist.gov
+13https://www.trellix.com/en-us/about/newsroom/stories/threat-labs.html
+14https://www.proofpoint.com/us/blog/threat-insight
+15https://www.exploit-db.com/
+16https://thehackernews.com/
+11
+
+Figure 10: This graph is a continuation of Figure 9 where further mentions of the filenames and hashes from the
+antivirus scans are displayed. One of the sha256 nodes in the bottom document node is also contained in many
+other node documents, however it is a hash of a standard DLL manifest file and therefore its neighbors are not
+included in this figure to reduce visual clutter. The only other node that was referenced outside of this small
+connected portion of the graph was the filename in the top document node, which was rkinstaller.exe. This
+filename was also mentioned in a site that listed a large number of filenames known to be malware - that node
+document along with the halo of its connected potential IoCs (most of which are other filenames) are shown in the
+upper left hand portion of the figure.
+0
+2
+4
+6
+8
+10
+CVSS v2 score
+0
+200
+400
+600
+800
+CVE degree in Neo4j
+Pearson correlation coefficient = 0.032
+2
+4
+6
+8
+10
+CVSS v3 score
+0
+1000
+2000
+3000
+4000
+5000
+CVE degree in Neo4j
+Pearson correlation coefficient = 0.007
+Figure 11: CVSS score (x-axis) version 2 (left) and version 3 (right) vs CVE node degree in the Neo4j indicator
+database (y-axis). Right hand figure contains 97489 datapoints, and the left hand figure contains 150940 datapoints.
+The pearson correlation coefficient for each dataset is shown in the plot titles. The outlier CVE node in the right
+hand plot, which has a degree of 4926, is CVE-2021-44228 (also known as Log4j).
+were fewer data points available for those plots. In Figure 12 we observe that there is a weak to median linear
+positive correlation between CVSS score and Neo4j degree for the documents that are cybersecurity domain focused.
+Interestingly, there is a slightly higher Pearson correlation (0.196) for CVSS version 2 compared to version 3 (0.23).
+This shows that taking into account the source of the data, temporal bias, and not considering degree 1 nodes
+shows that there is a CVSS and CVE node degree correlation, as opposed to these correlations shown in the entire
+dataset in Figure 11. There are two potential reasons why the CVSS score - node degree correlations are only
+12
+
+.:2
+4
+6
+8
+10
+CVSS v2 score
+0
+20
+40
+60
+80
+CVE degree in Neo4j
+Pearson correlation coefficient = 0.189
+2
+3
+4
+5
+6
+7
+8
+9
+10
+CVSS v3 score
+0
+20
+40
+60
+80
+CVE degree in Neo4j
+Pearson correlation coefficient = 0.211
+Figure 12: CVSS score (x-axis) version 2 (left) and version 3 (right) vs CVE node degree in the Neo4j indicator
+database (y-axis). The datapoints plotted is based on the degree of the CVE nodes connected to a subset of the
+document nodes which are more reputable for the domain of cybersecurity (e.g., threat reports), and CVE nodes
+with degree 1 are not considered. Additionally, only CVEs which were released in the time period of the web
+scraping upon which the graph database is built, are plotted in order to remove temporal bias which have existed
+in Figure 11.
+The pearson correlation coefficient is shown in the plot titles.
+Right hand figure contains 1898
+datapoints, and the left hand figure contains 1666 datapoints.
+weakly linear in Figure 12:
+1. The inherent bias present in the documents from the web crawlers - i.e., a CVE could be more commonly
+discussed on social media or news sites because of a reason other than its significance for cybersecurity. This
+reason seems to be the most prevalent due to the presence of social media and news content. In essence, the
+frequency of mentions of a CVE are more related to what catches attention than severity of the vulnerability.
+2. The CVSS score does not perfectly reflect the real world severity of a given CVE.
+2
+3
+4
+5
+6
+7
+8
+9
+10
+CVSS v2 score
+10
+20
+30
+40
+50
+CVE node PageRank score in Neo4j
+Pearson correlation coefficient = 0.149
+2
+3
+4
+5
+6
+7
+8
+9
+10
+CVSS v3 score
+0
+100
+200
+300
+400
+500
+600
+700
+CVE node PageRank score in Neo4j
+Pearson correlation coefficient = 0.102
+Figure 13: CVSS score (x-axis) version 2 (left) and version 3 (right) vs CVE node PageRank score in the Neo4j
+indicator database (y-axis).
+Right hand figure contains 715 datapoints, and the left hand figure contains 567
+datapoints. The pearson correlation coefficient is shown in the plot titles.
+3.6
+CVE Page Rank
+Neo4j has a library called Graph Data Science (GDS) which contains various graph algorithms that can be executed
+on a graph that is stored in Neo4j. These graphs algorithms could be useful for identifying patterns and clusters
+in this specific potential IoC graph database. As a simple example of how the structure of the graph yields node
+rankings based only on the reference to potential IOCs and vulnerability IDs, Table 2 details the top 11 highest
+13
+
+CVE ID
+CVE node
+Page Rank
+score
+in
+Neo4j
+Vulnerability name and description
+In
+CISA
+known
+exploited
+vulner-
+ability
+catalog
+CVSS
+score
+v2
+CVSS
+score
+v3
+CVE-2021-44228
+758.1
+Apache Log4j2 Remote Code Execution
+Vulnerability
+Yes
+9.2
+10.0
+CVE-2021-45046
+113.06
+It was found that the fix to address CVE-
+2021-44228 in Apache Log4j 2.15.0 was in-
+complete in certain non-default configura-
+tions.
+No
+5.1
+9.0
+CVE-2021-34527
+106.21
+“PrintNightmare” - Microsoft Windows
+Print Spooler Remote Code Execution Vul-
+nerability
+Yes
+9.0
+8.8
+CVE-2017-11882
+95.64
+Microsoft Office memory corruption vul-
+nerability
+Yes
+9.3
+7.8
+CVE-2012-0158
+86.71
+Microsoft
+MSCOMCTL.OCX
+Remote
+Code Execution Vulnerability
+Yes
+9.3
+N/A
+CVE-2014-0160
+81.13
+OpenSSL Information Disclosure Vulnera-
+bility. “heartbleed”
+Yes
+5.0
+7.5
+CVE-2021-34481
+73.47
+Windows Print Spooler Elevation of Privi-
+lege Vulnerability
+No
+4.6
+7.8
+CVE-2021-45105
+70.92
+Apache Log4j2 versions 2.0-alpha1 through
+2.16.0 (excluding 2.12.3 and 2.3.1) did not
+protect from uncontrolled recursion from
+self-referential lookups.
+No
+4.3
+5.9
+CVE-2021-1675
+64.43
+Microsoft Windows Print Spooler Remote
+Code Execution Vulnerability
+Yes
+9.3
+8.8
+CVE-2021-40444
+59.45
+Microsoft MSHTML Remote Code Execu-
+tion Vulnerability
+Yes
+6.8
+7.8
+CVE-2017-0199
+56.32
+Microsoft Office/WordPad Remote Code
+Execution
+Vulnerability
+with
+Windows
+API
+Yes
+9.3
+7.8
+Table 2: The top 11 most referenced CVEs ranked by their PageRank score in the Neo4j database in descending
+order. The PageRank computation was performed on all edges, regardless of type, in the database. PageRank
+scores are rounded to two decimal places. Note that the CISA known exploits catalog is continuously updated -
+this information is correct at the time this paper is written, but may change in the future. The PageRank score
+ranking is the most important part of the table, but the numerical page rank scores do approximately correspond
+to the degree of popularity and relevance of the CVE id. For example, it is clear that in the current dataset
+CVE-2021-44228 is significantly more referenced than every other CVE in the graph database. The intention of
+including the CVSS scores and whether the vulnerability is currently in the catalog is to give additional context
+on the severity of the vulnerability.
+ranked CVEs in the (undirected) graph database according to the PageRank algorithm. PageRank [49, 50] is an
+algorithm, originally designed for search engine ranking, which can be applied to any network data structure to
+determine which nodes are the most influential and referenced (here a reference is simply an edge in the network).
+All parameters were set to default for the PageRank computation with the exception of maxIterations which was
+set to 300 and dampingFactor which was set to 0.75 instead of the typical 0.85. The reasoning for selecting a
+smaller damping factor than what is typically used in search engines is that in this specific graph we are interested
+in potentially longer range influences on the relevance of nodes.
+The PageRank scores in Table 2 clearly show that the most referenced CVE nodes in the graph database are high
+impact and well known CVEs including the heartbleed, Log4j, and PrintNightmare vulnerabilities. The only two
+CVEs which are not currently in the CISA known exploited catalog are follow up CVEs to the Log4j vulnerability.
+14
+
+All of these CVEs not only are high impact and typically rank high on CVSS scores, but they are also notable
+because of how widespread their discussion was throughout the various social media channels, in large part because
+of the wide user base of the software these vulnerabilities exploit, that this graph database is constructed from.
+Another natural question that arises from the PageRank scores, is how those scores relate to the CVE CVSS
+score. Table 2 shows that there does seem to be at least some correlation, where the high PageRank score CVEs
+have high CVSS scores. Similar to the degree correlation plots in Figures 11 and 12, in Figure 13 CVE node
+PageRank scores are plotted against CVSS scores. Because the majority of the OSint crawler data is temporally
+biased towards the more recent years of data gathering, the datapoints plotted in Figure 13 are restricted to the
+years during which the OSint crawler system was operational [23], the same as in Figure 12. Additionally, in order
+to filter only for nodes which have a robust distribution of neighbors in the graph, only points with a PageRank
+score of 4.5 or over is plotted. Figure 13 shows that there is a low to medium linear correlation for the relevance,
+i.e. the PageRank score, against the CVE CVSS scores. Interestingly, the CVSS v3 scores are more correlated with
+the PageRank scores. This could indicate that the CVSS v3 scores more accurately represent the severity of the
+CVE’s compare to CVSS v2.
+4
+Discussion and Conclusion
+There are primarily two technical challenges that still need to be improved in this data gathering and analysis
+pipeline:
+1. There are many instances of near duplicate content from the web crawling system. For example, the content
+of a web page could be slightly altered from day to day (for example even if the web page content includes
+the current date and time); and if the web crawlers end up at that same web page multiple times direct
+de-duplication will not remove the content for being a near duplicate. Another example of near duplicate
+content that occurs often is that the web-crawling will catch a social media conversation (for example a Reddit
+thread) while it is occurring. We want the crawlers to catch this type of conversation because they could save
+content that is removed at some point in the future, but it also leads to potentially a large number of near
+duplicates of the same social media thread as it evolves over time. The challenge is in quantifying how close
+a document is to being a duplicate in order to remove it; in particular computing a distance metric pairwise
+between all documents can be very computationally intensive. A reasonable solution could be to not remove
+near duplicates, but instead to take the union of near duplicate documents, and to compute similarity metrics
+among close clusters of documents (for example determined by their relationship in the Neo4j potential IoC
+graph) for finding near duplicates.
+2. Reduce the amount of noise in the data set.
+It is difficult to know a-priori which content is relevant,
+and therefore in this work we tend to gather more information rather than remove potentially not useful
+information. This allows the database to be able to catch interesting edge cases and atypical cybersecurity
+content, but it comes at the cost of increased noise. However, there are some consistent sources of noise
+which are almost always not relevant which can be manually filtered out. For example localhost IP addresses
+or standard file names from popular programs. Another example of noise in the data set is software version
+numbers being identified as IP addresses due to the similarity in their format in text. It is not clear how
+to reduce the noise in the data set uniformly across all document types of potential IoCs, however, machine
+learning algorithms which more closely identify relevant information in a piece of text could be used to better
+filter the data.
+There are additional potential indicators and vulnerability tracking ids that could be extracted from natural
+language text in future work, such as CWE’s (Common Weakness Enumeration).
+Aggregating and condensing open source intelligence into a human readable and easily searchable form is an
+important task given the scale of the data that is available in the form of social media, news, blogs and threat
+reports in the cybersecurity space.
+Here we present one possible way to address this problem by parsing and
+transforming the open source data into a graph structure where each document can be associated with potential
+cybersecurity indicators of compromise, other infrastructure, CVEs, or MITRE ATT&CK Techniques. Querying
+this database based on the indicators then allows analysts to find open source intelligence documents connected
+to that indicator and review their content; this graph database thus reduces the overhead required in searching a
+massive amount of open source documents to a succinct cloud of relevant documents.
+15
+
+5
+Acknowledgements
+Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and
+Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S.
+Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.
+The New
+Mexico Cybersecurity Center of Excellence (NMCCoE) is a statewide Research and Public Service Project sup-
+ported center for economic development, education, and research. The authors would like to thank both SNL and
+NMCCoE for funding and computing system access and support.
+A
+Portable Executable malware visualization
+Figure
+14:
+Visualization
+of
+three
+different
+PE32
+malware
+samples
+byteplot,
+entropy,
+and
+struc-
+ture.
+Each
+of
+these
+contain
+at
+least
+one
+common
+resource
+file
+(detailed
+in
+Section
+3.4),
+which
+was
+sufficiently
+unique
+to
+link
+these
+pieces
+of
+malware.
+From
+left
+to
+right
+the
+sha256
+check-
+sums
+of
+these
+samples
+are
+34aa24656d5527a5ff1f7eb4ce4e782085618ded3766730c81f8f16a15d7e0ce,
+4f0d5a81b8a5bc3f998a0ac7a37db5bad49e1a22173251ed916363375360d5a4,
+8b53201f1914764f384c6ec5a7a
+5c5ab2924afaf382d2bbe79f68e43e5dfa3ba.
+From left to right the OriginalFilename for each sample is
+RKInstaller.exe, RKInstaller.exe, and POInstaller.exe. Note the similar naming scheme to the four samples
+that are in Neo4j (see Section 3.4). Interestingly, although these samples are not identical (we know this because
+their hashes are different), the middle and right hand side samples look visually indistinguishable from each other,
+whereas the left hand side sample is distinguishable from the other two. Overall, the structure of these PEs seem
+to be very similar which further indicates that these were likely developed by the same group or person.
+Figure 14 shows visualizations of three malware examples which contain a common PE resource artifact which
+was identified by its sha256 checksum. These three examples are different from the Neo4j nodes (specifically their
+hashes are not the same). These three distinct PE samples are also accessible on VirusTotal 17 18 19. This suggests
+that this particular indicator, while unique and not very common, is likely seen on other static analysis tools beyond
+these two datasets. These visualizations were generated using FileScan20 and PortEx21.
+B
+Large graph visualization
+Expanding out the network connections that exist in the Neo4j graph database is difficult to show visually because
+of the scale of the graphs in terms of edges and nodes. Here though in Figure 15 we show two large graph examples,
+which is possible using pygraphistry22 and networkx [51] in python3. The graphistry version which was used to
+generate these figures is 2.39.32, and the graph layout algorithm is ForceAtlas2Barnes. The primary observation
+that is important from Figure 15 is that just a couple of degrees out from an indicator can already result in a very
+large graph, even though the entire graph database is very sparse. One of the reasons for this is that different
+types of indicators, names, and vulnerability IDs will naturally be mentioned at different rates in the gathered
+internet text. For example, popular malware names will be very common in cybersecurity text, whereas hashes
+will generally not be mentioned very frequently. Therefore, it is important to filter down to the types of edges you
+want to follow in the database when you are searching for specific indicators - for example by searching only for
+17https://www.virustotal.com/gui/file/34aa24656d5527a5ff1f7eb4ce4e782085618ded3766730c81f8f16a15d7e0ce
+18https://www.virustotal.com/gui/file/4f0d5a81b8a5bc3f998a0ac7a37db5bad49e1a22173251ed916363375360d5a4
+19https://www.virustotal.com/gui/file/8b53201f1914764f384c6ec5a7a5c5ab2924afaf382d2bbe79f68e43e5dfa3ba
+20https://www.filescan.io
+21https://github.com/struppigel/PortEx
+22https://github.com/graphistry/pygraphistry
+16
+
+BytePlot (left)
+■OxFF
+0x00
+visible ASCll
+invisible ASCll
+non-AscIl
+Entropy (middle)
+0.2 (repetition)
+0.5(code)
+0.8 (packed)
+PE Structure (right)
+MSDoS Header
+COFF File Header
+Optional Header
+Section Table
+text
+.rdata
+rsrc
+Resources
+Imports
+Debug Info
+Entry Point
+OverlayBytePlot (left)
+■OxFF
+0x00
+visible ASCll
+invisible ASCll
+non-AscIl
+Entropy (middle)
+0.2 (repetition)
+0.5(code)
+ 0.8 (packed)
+PE Structure (riaht)
+MSDoS Header
+CoFF File Header
+Optional Header
+Section Table
+text
+.rdata
+rsrc
+Resources
+Imports
+Debug Info
+Entry Point
+OverlayBytePlot (left)
+■OxFF
+0x00
+visible ASCll
+invisible ASCll
+non-AscIl
+Entropy (middle)
+0.2 (repetition)
+0.5(code)
+ 0.8 (packed)
+PE Structure (riaht)
+MSDoS Header
+CoFF File Header
+Optional Header
+Section Table
+text
+.rdata
+rsrc
+Resources
+Imports
+Debug Info
+Entry Point
+OverlayIP and domain edges if you are searching for server infrastructure connections. It is also useful in these cases to
+use the CTC metadata and language detection metadata in order to search specifically for document nodes which
+are, with high confidence, English text discussing cybersecurity in order to get more relevant text. When there
+is limited information available on an indicator or document, searching through all available network connections
+(such as in the malware hash cases shown in Sections 3.1 and 3.4) can also be useful.
+Figure 15: Large graph visualizations which are expansions of the Qakbot IP address graph from Figure 7 in Section
+3.2. Degree 3 connections (left) has 7,862 nodes and 9,244 edges, degree 4 connections (right) has 46,631 nodes and
+112,935 edges. Note that the node and edge coloring is selected by the graphistry software and serves to delineate
+different aspects of the graph, but does not follow the same coloring scheme used in the Neo4j browser figures.
+These extremely large graph renderings show what the large scale behavior of this subgraph of the database looks
+like. Even though the database is overall quite sparse, there are clear clusters of the graph which behave similar to
+each other and there are also clearly very highly connected clusters.
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diff --git a/LNFLT4oBgHgl3EQfLy8m/content/tmp_files/load_file.txt b/LNFLT4oBgHgl3EQfLy8m/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf,len=967
+page_content='Cybersecurity Threat Hunting and Vulnerability Analysis Using a  Neo4j Graph Database of Open Source Intelligence  Elijah Pelofske∗1, Lorie M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Liebrock1, and Vincent Urias2  1New Mexico Cybersecurity Center of Excellence, New Mexico Tech  2Sandia National Laboratory  Abstract  Open source intelligence is a powerful tool for cybersecurity analysts to gather information both for analysis  of discovered vulnerabilities and for detecting novel cybersecurity threats and exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' However the scale of  information that is relevant for information security on the internet is always increasing, and is intractable for  analysts to parse comprehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Therefore methods of condensing the available open source intelligence, and  automatically developing connections between disparate sources of information, is incredibly valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In this  research, we present a system which constructs a Neo4j graph database formed by shared connections between  open source intelligence text including blogs, cybersecurity bulletins, news sites, antivirus scans, social media  posts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', Reddit and Twitter), and threat reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' These connections are comprised of possible indicators  of compromise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', IP addresses, domains, hashes, email addresses, phone numbers), information on known  exploits and techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', CVEs and MITRE ATT&CK Technique ID’s), and potential sources of information  on cybersecurity exploits such as twitter usernames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The construction of the database of potential IoCs is  detailed, including the addition of machine learning and metadata which can be used for filtering of the data  for a specific domain (for example a specific natural language) when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Examples of utilizing the graph  database for querying connections between known malicious IoCs and open source intelligence documents,  including threat reports, are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' We show that this type of relationship querying can allow for more effective  use of open source intelligence for threat hunting, malware family clustering, and vulnerability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  1  Introduction  Open source intelligence offers an extraordinary amount of information that a cybersecurity analyst can use for  threat detection, mitigation, and analysis [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  However, open source intelligence contains a large amount of  noise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', irrelevant information) and most importantly the scale of the data is too large to be useful in its  raw form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  To this end, automating the process of finding indicators of compromise and relevant relationships  between the indicators of compromise, has become increasingly important [5, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The central idea utilized in this  article is forming a network of associations between open source intelligence documents and potential indicators of  compromise (IoCs) that exist in the open source intelligence text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The term potential IoC is important because it  specifies that the text is potentially useful and quite unstructured, but there is a pattern match in the data that  does fit a particular form (for example, an IP address, or a CVE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Graphs are a natural way to express these types  of higher order connections and are used in a variety of cybersecurity contexts [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' These types of networks,  when they are intended for providing semantic meaning between heterogeneous data types are also referred to as  knowledge graphs [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In order to create a graph representation of a large amount of open source intelligence,  we utilize Neo4j which provides a visual interface to search the graph, allows a number of users to interact with the  data, and also provides an efficient query time for the database to interact with other analysis systems or to simply  query and display the raw OSint text documents that are connected to a relevant exploit or IoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Neo4j graph  databases have been used in other domains for the purpose of storing and querying data structures with complex  networks [14–17] including social network analysis [18] and typhoon disaster knowledge [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Utilizing graphs in  order to better evaluate potentially relevant connections that exist in a large dataset is a subject of considerable  interest in the literature [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  In this article we outline a methodology which consists of collecting and aggregating open source intelligence,  in conjunction with antivirus scan results, and presenting the information contained in this data in the form of  a network, or a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Specifically, edges represent a connection between a potential IoC and a document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The  ∗Email: elijahpelofske@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='12013v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content="CR]  27 Jan 2023  Neo4J Graph Database  of potential cybersecurity  IOC's  Arbitrarely  structured  text files  OSint web crawler  (aka Spider)  structured data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content="  Includes metadata  such as time stamps  File antivirus  scan  structured  data  Execute CTC to  determine if the natural  language is likely  discussing  cybersecurity or not  Execute Tram to  determine possible  Mitre Attack Technique  ID's  Execute rcATT to  determine possible Mitre  Attack Techniques or  Tactics  Extract metadata  and clean natural  language text when  required." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Deduplicate against  all previously  processed  documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content="  regex or search for  potential IOC's in the  natural language  Package documents  and metadata with  edges to their  potential IOC's in the  form of a Neo4J  transaction and add  to the database  Determine likely natural  languages being used  Executed  in parallel  Figure 1: IoC Neo4j graph database construction workflow diagram." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  challenge is that most open source intelligence text from the web is unstructured therefore analyzing the text for  relevant pattern matches is a means to extract the potential IoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Here a document is simply a collection of  natural language or data - for example a single tweet from Twitter could be a document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Furthermore, metadata  is included with the node document when constructing the database - several machine learning algorithms identify  whether the text is discussing cybersecurity and specifically what type of exploit techniques are being discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  This type of metadata is important because it can can serve as a signal for the quality of the given node document  meaning that a piece of open source intelligence text gathered from social media will have a higher confidence of  being relevant if the machine learning algorithm tells us it is likely discussing a cybersecurity topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In Section 2  we detail the construction of this type of cybersecurity intelligence graph, instantiated as a Neo4j graph database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  In Section 3 we give specific examples of where this type of open source intelligence graph database can provide  useful information and analysis in cybersecurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' We conclude in Section 4 with a discussion of what this type of  approach to open source intelligence analysis can be used for and future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  2  Methods  In this section we outline the pipeline used to construct the Neo4j database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1 the open source  intelligence data sources are defined, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2 the database construction is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1  Open source intelligence data sources  The open source intelligence documents that are fed into the system broadly fall into three categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' the three  green starting blocks in Figure 1 show these categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The first type of data is entirely unstructured text -  typically this type of data is simply a threat report or a list of indicators or vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This option allows any  collection of text sourced from any internet discussion to be analyzed for potential IoCs and to be added into the  Neo4j database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Figure 1 was created in Lucidchart1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The second type of data comes from a system of web crawlers (the details of this system are outlined in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The text found by the web crawlers is typically similarly unstructured and can originate from a variety of  sources such as Reddit, Twitter, blogs, cybersecurity bulletins, and news sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' However, the web crawler data also  contains metadata including links that the crawler followed to this current site, time and date information, and  1www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='lucidchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com  2  cybersecurity keywords that were found when parsing the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Therefore, this data is slightly more structured and  is therefore parsed differently from arbitrary text;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' specifically the metadata is kept separate from the text that will  be analyzed for potential IoCs, which is eventually added into the node information in Neo4j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The third type of data is structured antivirus scans of files that are potentially malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This data is entirely  machine readable and does not contain natural language information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Therefore, this data is also parsed differently  to the other two data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In particular, these antivirus scans include hashes of files and the names of those  files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Therefore, those specific fields (hashes and file names) that are present in the structured data are parsed for  creating edges in Neo4j, but no other natural language analysis or pattern matching for potential IoCs is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='Neo4j database construction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='Node type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='Count of unique nodes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='Count of edges  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='Document  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2128992  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='MD5 Hash  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='394826  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='944671  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='SHA1 Hash  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='323321  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='990817  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='SHA256 Hash  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='642535  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1941248  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='SHA512 Hash  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='18339  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='28112  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='Malware name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='365  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='117528  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='APT name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='457  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='165741  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='Email  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='85396  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='953757  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='CVE ID  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='174668  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1313206  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='Twitter username  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='174143  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='402899  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='Phone number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='23756  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='144397  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='IP address  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='119699  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='705386  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='domain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='214720  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3116508  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='File name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='351507  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1326480  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='MITRE ATT&CK Technique ID  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='445  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='21018  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='Table 1: Neo4j database statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The “Count of edges” column in the table is the number of edges between  the listed node type and Document nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' For each non-document node type, the edges between document nodes  and the unique nodes for that indicator are always labelled the same as the indicator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' for example CVE nodes are  connected to document nodes by edge labelled as CVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Those edge types are the counts displayed in the third  column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Therefore, there are no edges between document nodes which is why the edge count in that cell is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Figure 1 details the high level workflow that constructs the Neo4j database of potential IoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This entire system  is operating continuously by reading in new data from the independent web crawler infrastructure and then adding  relevant information into the Neo4j database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' First, the data is parsed depending on its source, as outlined in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This input data is referred to as a document because this collection of text is all originating from the  same source and is therefore logically linked, which could be helpful for identifying relationships when searching  the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' As an example, a single document could be a piece of text from social media, such as a Reddit  post or a tweet from Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Any metadata that is associated with the input text is parsed at this point, to be  added into the Neo4j database if a node representing this document is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Importantly, once the raw text  input has been extracted (this includes the JSON structured antivirus file scan data), a SHA256 checksum of the  data is created and this checksum is then checked against the checksums of all of the documents that are in the  Neo4j database;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' and if the document is a duplicate, it is not added to the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Next, several processes begin  executing in parallel, all of which are designed to extract useful meaning from the natural language input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' To  this end, the antivirus file scan data is not parsed using the machine learning algorithms or language detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  however the potential IoCs such as filenames and hashes are extracted and used to create edge relationships in the  database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The Cybersecurity Topic Classification (CTC) tool, which is compromised of multiple machine learning  algorithms trained to detect cybersecurity vs non-cybersecurity discussions from social media and developer forum  English text sources [24], is executed on the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The output of this algorithm is simply three states - either it is  likely cybersecurity related text, or not, or there was not enough data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', English words) to make a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The Reports Classification by Adversarial Tactics and Techniques (rcATT) machine learning python tool [25]  is also executed on the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This tool gives a list of likely MITRE ATT&CK tactics and techniques 2 that were  2https://attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='org/  3  mentioned in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The MITRE ATT&CK framework provides a consistent basis for tracking cybersecurity  techniques [26–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The rcATT tool was trained on threat report text and therefore the error rates on non-threat  report documents is expected to be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The tool is applied uniformly to all documents because it is not necessarily  known a-priori what the exact semantic content of the document is (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', whether it is a threat report, or a news  report, or entirely non-cybersecurity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' It is generally expected that based on the training data used to create the  machine learning models in the tool, cybersecurity content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', documents where CTC returned True) will have  higher accuracy results in regards to detecting discussions of specific MITRE ATT&CK tactics and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The Threat Report ATT&CK Mapper (TRAM) machine learning python tool3 is also executed on the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  TRAM, similar to rcATT, returns likely MITRE ATT&CK techniques that were mentioned in threat reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Therefore similarly to rcATT, this data will likely have high error rates for natural language text that is quite  different from threat reports, but could be more accurate for cybersecurity related text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Especially for the data coming from the web crawlers, the natural language text could be non-English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Having  some signal to indicate when this occurs in the Neo4j database could be useful (for example if one just wants to  query documents that are only English or Spanish text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The other reason that this signal is important is because  all of the natural language machine learning algorithms vectorize the input text using a very broadly defined English  dictionary - meaning that other languages are not used in these models, which means that their results will be  very inaccurate and should not be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Therefore, the python tool langdetect is also executed on the text and the  resulting language detection information is included in the node metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Lastly, pattern matches for all potential IoCs are performed on the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' With the exception of the structured  antivirus file scans, where the potential IoCs that can be extracted can be done automatically, all of the other  input text can be entirely unstructured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Therefore, simple pattern matching procedures are performed in order  to find potential IoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Hashes (md5, sha1, sha256, sha512) are found by searching for high entropy hexadecimal  text that fits within the required character length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' File names are found by matching tokenized words (NLTK  [32] was used for most of the tokenization procedures) that have a file ending that matches some standard file  ending (for example .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='py for python).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Advanced Persistent Threat (APT) group names and malware names are  all simply pattern matched against tokenized words in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Phone numbers, Email addresses, IP addresses,  twitter usernames, and domain names are all found by pattern match searching for the known standard formats  that they appear in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Some simple checks are used to rule out pattern matches which do not fit the expected format  of the data type and in the case of domain names, the top 1 million (Alexa top 1 million list) most searched domains  are removed in order to reduce noise in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Common Vulnerabilities and Exposures (CVEs) [33] and the  unique ID numbers of MITRE ATT&CK techniques [34] are also pattern matched for;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' both of these data types also  follow a standard format which can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Each of these pattern matches will correspond to an edge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=',  a connection) between the document node that contained this data and a node representing that unique pattern  match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This unique pattern match we broadly call a potential IoC, but it can also simply be a unique ID to track  a known vulnerability (for example a CVE), or it could be a potentially useful piece of information for connecting  two document nodes but not be a malicious IoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Once all of this pattern matching and data processing has been completed, a number of Neo4j transactions are  created in order to add the new data into the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' All of the processed documents have unique nodes created  which contain several different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The most important part of the node is the original raw text that  was in the node document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Next, web crawler metadata is in the nodes which originated from web crawling - here  the metadata includes the link, parent link, time stamps, keywords, and potentially other data such as checksums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Next, some document nodes are antivirus scans - in these cases there is a large amount of metadata about the  scan that was performed, but typically the data is not natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' All nodes also have language detection  metadata, if there was natural language text that could be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The natural language detection is useful  for filtering nodes which contain only a specific language of interest, however as expected most of the document  nodes are English text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The pattern matches that were found in the text that form the edges in the graph are also  included as a segment in the node data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Lastly, the machine learning results from CTC, rcATT, and TRAM are also  included as a separate data segment (if there is applicable English text that could be fed to these ML algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Each of these components of node data can be used to filter for specific nodes which have specific attributes that  are relevant for a specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Any documents which have no pattern matches are not added to the database  as they would simply be degree 0 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Next, for all unique pattern matches found in this set of documents,  check if the nodes already exist in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' If they do exist, do nothing, but if they do not exist, then create  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This is to avoid creating duplicate potential IoC or pattern match nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Lastly, edges are formed between  the document nodes and the nodes representing the pattern matches found in those documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Table 1 shows  the counts of nodes and edges in the database as of the writing of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Therefore, the constructed graph  3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/center-for-threat-informed-defense/tram/  4  database is always bipartite where one partition is the node documents (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', the sources of information) and the  other partition is the set of nodes representing different potential IoCs or vulnerability IDs that exist in the node  documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This structure allows users, or algorithms, to query for relationships based on potential IoCs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', a  path connecting two potential IoCs or the network of neighbors associated with a potential IoC) and then examine  the sources of the information within the node documents, including machine learning and language metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  3  IoC connections in the graph database  In this section some specific visual examples of the structure of the database indicates what is available to the user  when searching for IoCs or vulnerability IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The hard part of cybersecurity analysis is aggregating information,  in this case open source unstructured text, into a meaningful form that can provide a network of related documents  and potential indicators of compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1 we show the graph connectivity around an md5 hash of a  known malware file from the WannaCry ransomware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2 we show the graph connectivity surrounding  an IP address known to be a command and control server for the Qakbot trojan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3 we show the graph  connectivity around a CVE that is known to be exploited in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='4 shows a specific case where a sha256 hash of a contained resource was in several antivirus scans of  different portable executable malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This shared contained resource is reasonably unique, and indicates software  re-use during the development of otherwise seemingly unconnected malware samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='5 we show the distribution of CVE node degrees in the Noe4j database compared to their Common  Vulnerability Scoring System (CVSS) scores, and show that for reputable open source cybersecurity sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=',  threat reports), there is a weak-to-median linear relationship between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Lastly, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='6 the Neo4j  Graph Data Science implementation of PageRank is applied to the graph database, allowing a ranking of the most  influential CVEs across the entire database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Figure 2: Node and edge coloring legend for potential IoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  For the visual graph examples we query the database with the Neo4j browser using the Cypher query language4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The node and edge coloring’s encode the following information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Document nodes are large cyan nodes, SHA1 nodes  are magenta, SHA256 nodes are lime, SHA512 nodes are grey, MD5 nodes are purple, file name nodes are teal,  email address nodes are yellow, IP address nodes are orange, malware name nodes are brown, Twitter usernames  are maroon, APT name nodes are lavender, domain name nodes are green, phone number nodes are blue, CVE  nodes are red, and MITRE ATT&CK technique IDs are beige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The edge coloring matches the node coloring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' for  example an edge connecting a CVE to a document node where it was mentioned will also be colored red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Figure 2  shows the node and edge coloring legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In order to reduce visual clutter, if there are near duplicate documents  from the web crawlers which have the same connections to a group of nodes, we manually remove all but one of  the duplicate document nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Figure 3: Degree 1 connections associated with the md5 hash 84c82835a5d21bbcf75a61706d8ab549  4https://neo4j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/developer/cypher/  5  node_CVE  node_IP  node_phone_number  node_sha1  node_MitreAttack id  node_sha256  node_sha512  node_document  node_domain  node_twitter_username  node_email  node_file_name  node_APT_name  node_malware_name  node_md5Figure  4:  Expanded  connections  from  Figure  3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  degree  2  connections  out  from  the  md5  hash  84c82835a5d21bbcf75a61706d8ab549  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1  Visual analysis: md5 malware hash  In order to show a specific malware hash subgraph of the database, we will perform a query to the Neo4j database  for the md5 hash 84c82835a5d21bbcf75a61706d8ab549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The Cypher language query used to search for this  single md5 node is:  MATCH p=(find:node md5 {name:  ‘84c82835a5d21bbcf75a61706d8ab549’}) RETURN p  In order to illustrate the utility of the database, we have selected this hash because it is known to be a hash of  a piece of malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The query results show that there are three unique document nodes that reference this specific  hash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Figure 3 shows this connectivity graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The content of these three document nodes contains useful context  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The lower left hand node document is a cybersecurity blog-style website that is detailing a network  and security analysis tool, which used the example of this md5 hash for detecting malware using antivirus software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The lower right hand node document is a manalyzer report on this md5 hash5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The top node document is a 4chan  thread for which a user linked to this manalyzer report within a long discussion thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Next, we can query the neighbors of those three document nodes - the connectivity graph for this result is  shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Notably, there is a SHA1 and a SHA256 hash which were both in two of the node documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  These these two hashes and the md5 hash are all checksums of the same file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Next, we expand the neighbor relationships for these two SHA1 and SHA256 hashes that were both contained  in the two threat report nodes in order to see if there are relevant connections we can investigate further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This  neighborhood expansion is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' We see that there are two node documents that are linked to the  SHA256 hash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' These two nodes are threat reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' One of these threat reports is also linked to the earlier manalyzer  report node by a common filename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' However, this filename is simply the command prompt executable (cmd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='exe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  These two nodes represent slightly different versions of a threat report titled The Lazarus Constellation, authored  by Avisa Partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  This specific malware hash is associated with the WannaCry ransomware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  WannaCry is  ransomware [35–39] that propagated across the world in 2017 targeting computers running Windows OS primarily  using an exploit known as EternalBlue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Figure 6 shows the neighborhood of connections that the The Lazarus  Constellation threat reports contain, which includes a large number of hashes, CVEs, domains, and filenames that  5https://manalyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='org/report/84c82835a5d21bbcf75a61706d8ab549  6  Figure 5: Expanding the neighbors for two of the hash node connections from Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  are all associated with this APT group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This malware hash example shows how the graph database can retrieve  relevant information for an IoC - here we were able to find two other hashes of the same file, a place where the  hash was mentioned on a 4chan board, and finally threat reports which give the larger context of why these hashes  are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2  Visual analysis: Qakbot IP address  Next we will consider the connectivity graph around a known malicious IP address 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='139, which is the  IP address for a command and control server for the malware called Qakbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The connections showing where  this IP address was mentioned in the database are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' There were three document nodes which  mentioned this IP address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Two of these documents were from Github repositories where known IoCs on Qakbot  were published, and the third document is from a pastebin page which also posted Qakbot IoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Interestingly,  there are several IP addresses that are common to all, or a subset, of these documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Note that the malware  name (brown colored degree three node near the center of the graph) was the name Qakbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This graph shows  the relevant context around this IP address - namely that it is a command and control server for Qakbot and the  connectivity graph shows many additional IoCs that should be monitored in association with Qakbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Importantly,  these graph connections take advantage of multiple (separated) data sources - where if we were to have read only a  single one of these documents we would not have the aggregated data which includes many more IP addresses and  hash IoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' It is clear from this graph that this specific IP address is not a lone command and cotnrol server for  Qakbot, rather it is one of many IP addresses and domains that are connected to this malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Figure 15 in the  Appendix B shows significantly larger graph renderings for the degree 3 and 4 connections in the database from  this IP address node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3  Visual analysis: known exploited CVE  Here we show the connectivity graph for all neighboring nodes up to degree 2 away from a CVE that is known to be  exploited in the wild6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' CVE-2014-4404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' CVE-2014-4404 is a remote code execution exploit on Apple OS X, earlier  6https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='cisa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='gov/known-exploited-vulnerabilities-catalog  7  Figure 6: Expanding the two threat report node connections that were connected to the two shared hash nodes in  Figure 5  iOS versions and Apple TV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The neighborhood graph is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The neighboring nodes connected to this  specific node are showing the halo of data that is connected to this CVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In particular, these neighboring nodes show  filenames, other CVEs, twitter usernames, domain names, hashes etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', that were mentioned along with this CVE  in various documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Two of these nodes were threat reports which described the state of various cyberattacks  in 2021 - therefore they broadly discussed a number of different exploits, APT groups, twitter usernames, and  malware families with CVE-2014-4404 being one of the more exploited Apple software CVEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' As one would expect,  the CTC tool labelled each of these text documents that are in this halo as being cybersecurity related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This ML  metadata can be used to filter for a subgraph of the graph database which contains only cybersecurity related  (English) natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='4  Malware hash files: code and resource re-use  The static analysis antivirus [40] scan hash file data nodes that are formed in Neo4j include hashes of extracted  files, functions, or strings contained within the malware (as well as the malware itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' What these additional  hashes allow within the graph database is detection of shared code components, for example shared header files,  strings, or functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Of course some of these may not be incredibly useful indicators - for example they may be  standard Portable Executable manifest files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' However, if there are unique code artifacts in some subset of antivirus  file scans which are known to be malicious, then those could serve as indicators of shared code re-use from a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  These could even serve as malware signatures for antivirus products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The graph database is good for investigating  these shared resource hashes because it is easy to find where a set of antivirus scans have a shared resource hash  node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Here we provide a specific example - that is a SHA256 hash which we can search for in the Neo4j graph database  with the Cypher syntax of:  MATCH p=(find:node sha256 {name:  ’84f7c54dc015637a28f06867607c2e0b  dd225d10debb1390ff212d91cd2d042b’}) RETURN p  Using FileScan7, which contains references to this hash, we can find that this is a hash of the following English  7https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='filescan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='io  8  Figure 7: Degree 2 halo of connected nodes to the Qakbot 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='139 IP address, where the node representing  the IP of interesting is the orange degree 3 node near the bottom document node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  ASCII text: BundleInstall BundleInstall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Note that this representation is not necessarily capturing the full  hexadecimal data present in the data segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The terms bundle install suggest that the source code language is  Ruby, and this text artifact could be a result of packaging Ruby source code into a portable executable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Within the current Neo4j database, this SHA256 hash node has a degree of 5 (meaning that is referenced in 5  node documents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Two of these nodes are actually effectively duplicates - they are scans of the same file, meaning  that they have the same file name and associated extracted hashes (such as the hashes of contained resources in  the PE), but were scanned at different times and therefore have slightly different data so the direct de-duplication  did not remove one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Therefore as with the other Neo4j figures in order to reduce clutter, one of these  duplicates is removed in the displayed figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The graph renderings of these nodes and their degree 1 neighborhood  connections are shown in Figures 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  For the bottom graph in Figure 9 we can examine what some of the important information is that the nodes  have, such as the malware filename and the hash of the file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' These four nodes are known malware samples (this  statement is based on the high proportion of antivirus scans indicating malware) and additional information on  them is available on VirusTotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Top document node in Figure 9: the filename is rkinstaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='exe and the full sha256 hash is given in the  virustotal link8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Right document node in Figure 9: no associated filename and the full sha256 hash is given in the virustotal  link9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  8https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='virustotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/gui/file/5577ce9aa4e4ec2735247c5769f0e84db599825f2d95159b0102f3b30e80b6bb/details  9https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='virustotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/gui/file/06f11f4a555a4891c93f13f82dc06e8bcedda2a71c8a5e6aa5c18da871f41238/details  9  Figure 8: All nodes and edges connected to the node for CVE-2014-4404 up to two degrees away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The degree 11  red node approximately in the center of the graph is the CVE-2014-4404 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Bottom document node in Figure 9: the filename is rkinstaller364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='exe and the full sha256 hash is given in  the virustotal link10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Left document node in Figure 9: the filename is poinstaller257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='exe and the full sha256 hash is given in the  virustotal link11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The notable observation from this data is that this shared portable executable resource hash is a reasonably  unique artifact (meaning that it is not a commonly re-used ASCII text segment in PE development) and was found  across a small subset of malware samples, some of which also have other shared characteristics such as similar (or  identical) file names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This suggests, with reasonable confidence, that the development of these pieces of malware  are linked in a meaningful way, for example the same developer could have created these portable executables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This  demonstrates where the graph database construction of OSint allows a user to link together pieces of information  in order to group together seemingly unconnected documents and other potential indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Portable executable visualizations for three other pieces of known malware which contain this specific PE  resource sha256 hash are shown in Figure 14 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='5  CVE degree and CVSS score  Because this database is constructed in a largely unsupervised manner - i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', pattern matches are automatically  generated and new data is added to the database without human review - a natural question that arises is whether  the graph structure of the data represents the real world properties of the vulnerabilities or potential IoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' An easy  example of this that we can numerically compute is the degree of CVE nodes in the database (which corresponds to  how many times that CVE was mentioned in the text from the different data sources) and compare that against the  CVSS scores of CVEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' CVSS scores are intended to approximately represent the overall severity of the vulnerability  [41–44] where a CVSS score of 0 is the lowest severity and 10 is the maximum severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' There are two CVSS score  versions that we will compare - CVSS version 2 and version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Version 3 is the newest CVSS scoring method, which  is intended to be a more accurate rating scale for modern cybersecurity threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The relevant question is whether there exists a relationship of increasing degree of the CVE nodes in the graph  database with respect to CVE CVSS score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Intuitively, if the severity of a CVE corresponds to how frequently that  CVE is mentioned in social media, news, and threat reports, then higher CVSS scores will correspond to higher  10https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='virustotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/gui/file/f8d11b1e3e027355a11163049b530de4fd67183abd08a691d5d18744653ef575/details  11https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='virustotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/gui/file/f3efcfc7121f2348deb6f3b5ffde60878d978c25281e67defdc288feaef8b38c/details  10  Figure 9: Graph rendering of the neighboring nodes connected to the sha256 node hash of interest, which is the  degree four green-yellow node in the center of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The neighboring connections of these document nodes  have also been displayed in order to determine if there are any other connections of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Each of the document  nodes are antivirus scans of portable executables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The degree 1 expanded neighbors show that there is actually  another sha256 hash node that is shared by three out of the four document nodes, and there are no other edges  connecting the associated hashes and file names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' That sha256 hash is a standard manifest for creating PEs and is  common to a large number of PE samples (both benign-ware and malware) and is therefore not unique enough to  attribute a meaningful connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  degree CVE nodes in the Neo4j graph database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The CVSS scores are retrieved from the NIST NVD dataset 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  To this end, we compute the Pearson correlation coefficient between CVE node degrees and their CVSS score using  scipy in python [45–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Some CVEs do not have a version 2 or a version 3 score, and therefore are not able to  plotted in this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Figure 11 plots all CVE CVSS scores against CVE node degrees in the Neo4j graph database, which shows there  is not a positive or linear correlation between the CVE node degrees and CVSS scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This is notable because it  shows that across all of the cybersecurity mentioned natural text that was gathered, there is not a strong correlation  between the rate of CVE mentions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' CVE popularity) and CVSS scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' However, it could be the case that  more focused cybersecurity documents have a higher CVE CVSS score and Neo4j node degree correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' It could  also be the case there is not a strong signal of correlation for CVE nodes which are not commonly mentioned in  news, threat reports, and cybersecurity bulletins - and therefore only considering nodes which have at least a degree  of 2 could remove some noise in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Lastly, due to popularity of new CVEs, necessarily the web scraping  will have a temporal bias for the times during which the spiders were operating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Figure 12 shows the correlation plot for CVE node degrees vs CVSS score, but the Neo4j node degrees are  computed using a restricted set of document sources namely reputable blogs, information sites, and threat reports  specifically fireeye13, all threat reports, proofpoint14, exploitdb15, the Hackernews16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Additionally, CVE  nodes with degree 1 are not included in this computation since low degree nodes do not necessarily provide a  strong signal in regards to how referenced that CVE is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Lastly, the data in Figure 12 are restricted to the years  during which the OSint crawler system was operational [23] because there is an inherent temporal bias in regards  to what web pages are mentioned and scraped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Because only a subset of the CVEs had CVSS v3 scores, there  12https://nvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='gov  13https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='trellix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/en-us/about/newsroom/stories/threat-labs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='html  14https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='proofpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/us/blog/threat-insight  15https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='exploit-db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/  16https://thehackernews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/  11  Figure 10: This graph is a continuation of Figure 9 where further mentions of the filenames and hashes from the  antivirus scans are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' One of the sha256 nodes in the bottom document node is also contained in many  other node documents, however it is a hash of a standard DLL manifest file and therefore its neighbors are not  included in this figure to reduce visual clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The only other node that was referenced outside of this small  connected portion of the graph was the filename in the top document node, which was rkinstaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='exe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This  filename was also mentioned in a site that listed a large number of filenames known to be malware - that node  document along with the halo of its connected potential IoCs (most of which are other filenames) are shown in the  upper left hand portion of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  0  2  4  6  8  10  CVSS v2 score  0  200  400  600  800  CVE degree in Neo4j  Pearson correlation coefficient = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='032  2  4  6  8  10  CVSS v3 score  0  1000  2000  3000  4000  5000  CVE degree in Neo4j  Pearson correlation coefficient = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='007  Figure 11: CVSS score (x-axis) version 2 (left) and version 3 (right) vs CVE node degree in the Neo4j indicator  database (y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Right hand figure contains 97489 datapoints, and the left hand figure contains 150940 datapoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The pearson correlation coefficient for each dataset is shown in the plot titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The outlier CVE node in the right  hand plot, which has a degree of 4926, is CVE-2021-44228 (also known as Log4j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  were fewer data points available for those plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In Figure 12 we observe that there is a weak to median linear  positive correlation between CVSS score and Neo4j degree for the documents that are cybersecurity domain focused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Interestingly, there is a slightly higher Pearson correlation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='196) for CVSS version 2 compared to version 3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  This shows that taking into account the source of the data, temporal bias, and not considering degree 1 nodes  shows that there is a CVSS and CVE node degree correlation, as opposed to these correlations shown in the entire  dataset in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' There are two potential reasons why the CVSS score - node degree correlations are only  12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' :2  4  6  8  10  CVSS v2 score  0  20  40  60  80  CVE degree in Neo4j  Pearson correlation coefficient = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='189  2  3  4  5  6  7  8  9  10  CVSS v3 score  0  20  40  60  80  CVE degree in Neo4j  Pearson correlation coefficient = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='211  Figure 12: CVSS score (x-axis) version 2 (left) and version 3 (right) vs CVE node degree in the Neo4j indicator  database (y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The datapoints plotted is based on the degree of the CVE nodes connected to a subset of the  document nodes which are more reputable for the domain of cybersecurity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', threat reports), and CVE nodes  with degree 1 are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Additionally, only CVEs which were released in the time period of the web  scraping upon which the graph database is built, are plotted in order to remove temporal bias which have existed  in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The pearson correlation coefficient is shown in the plot titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Right hand figure contains 1898  datapoints, and the left hand figure contains 1666 datapoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  weakly linear in Figure 12:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The inherent bias present in the documents from the web crawlers - i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', a CVE could be more commonly  discussed on social media or news sites because of a reason other than its significance for cybersecurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This  reason seems to be the most prevalent due to the presence of social media and news content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In essence, the  frequency of mentions of a CVE are more related to what catches attention than severity of the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The CVSS score does not perfectly reflect the real world severity of a given CVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  2  3  4  5  6  7  8  9  10  CVSS v2 score  10  20  30  40  50  CVE node PageRank score in Neo4j  Pearson correlation coefficient = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='149  2  3  4  5  6  7  8  9  10  CVSS v3 score  0  100  200  300  400  500  600  700  CVE node PageRank score in Neo4j  Pearson correlation coefficient = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='102  Figure 13: CVSS score (x-axis) version 2 (left) and version 3 (right) vs CVE node PageRank score in the Neo4j  indicator database (y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Right hand figure contains 715 datapoints, and the left hand figure contains 567  datapoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The pearson correlation coefficient is shown in the plot titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='6  CVE Page Rank  Neo4j has a library called Graph Data Science (GDS) which contains various graph algorithms that can be executed  on a graph that is stored in Neo4j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' These graphs algorithms could be useful for identifying patterns and clusters  in this specific potential IoC graph database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' As a simple example of how the structure of the graph yields node  rankings based only on the reference to potential IOCs and vulnerability IDs, Table 2 details the top 11 highest  13  CVE ID  CVE node  Page Rank  score  in  Neo4j  Vulnerability name and description  In  CISA  known  exploited  vulner-  ability  catalog  CVSS  score  v2  CVSS  score  v3  CVE-2021-44228  758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1  Apache Log4j2 Remote Code Execution  Vulnerability  Yes  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='0  CVE-2021-45046  113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='06  It was found that the fix to address CVE-  2021-44228 in Apache Log4j 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='0 was in-  complete in certain non-default configura-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  No  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='0  CVE-2021-34527  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='21  “PrintNightmare” - Microsoft Windows  Print Spooler Remote Code Execution Vul-  nerability  Yes  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8  CVE-2017-11882  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='64  Microsoft Office memory corruption vul-  nerability  Yes  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8  CVE-2012-0158  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='71  Microsoft  MSCOMCTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='OCX  Remote  Code Execution Vulnerability  Yes  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3  N/A  CVE-2014-0160  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='13  OpenSSL Information Disclosure Vulnera-  bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “heartbleed”  Yes  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='5  CVE-2021-34481  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='47  Windows Print Spooler Elevation of Privi-  lege Vulnerability  No  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='6  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8  CVE-2021-45105  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='92  Apache Log4j2 versions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='0-alpha1 through  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='0 (excluding 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1) did not  protect from uncontrolled recursion from  self-referential lookups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  No  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='9  CVE-2021-1675  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='43  Microsoft Windows Print Spooler Remote  Code Execution Vulnerability  Yes  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8  CVE-2021-40444  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='45  Microsoft MSHTML Remote Code Execu-  tion Vulnerability  Yes  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8  CVE-2017-0199  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='32  Microsoft Office/WordPad Remote Code  Execution  Vulnerability  with  Windows  API  Yes  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8  Table 2: The top 11 most referenced CVEs ranked by their PageRank score in the Neo4j database in descending  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The PageRank computation was performed on all edges, regardless of type, in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' PageRank  scores are rounded to two decimal places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Note that the CISA known exploits catalog is continuously updated -  this information is correct at the time this paper is written, but may change in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The PageRank score  ranking is the most important part of the table, but the numerical page rank scores do approximately correspond  to the degree of popularity and relevance of the CVE id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' For example, it is clear that in the current dataset  CVE-2021-44228 is significantly more referenced than every other CVE in the graph database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The intention of  including the CVSS scores and whether the vulnerability is currently in the catalog is to give additional context  on the severity of the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  ranked CVEs in the (undirected) graph database according to the PageRank algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' PageRank [49, 50] is an  algorithm, originally designed for search engine ranking, which can be applied to any network data structure to  determine which nodes are the most influential and referenced (here a reference is simply an edge in the network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  All parameters were set to default for the PageRank computation with the exception of maxIterations which was  set to 300 and dampingFactor which was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='75 instead of the typical 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The reasoning for selecting a  smaller damping factor than what is typically used in search engines is that in this specific graph we are interested  in potentially longer range influences on the relevance of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The PageRank scores in Table 2 clearly show that the most referenced CVE nodes in the graph database are high  impact and well known CVEs including the heartbleed, Log4j, and PrintNightmare vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The only two  CVEs which are not currently in the CISA known exploited catalog are follow up CVEs to the Log4j vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  14  All of these CVEs not only are high impact and typically rank high on CVSS scores, but they are also notable  because of how widespread their discussion was throughout the various social media channels, in large part because  of the wide user base of the software these vulnerabilities exploit, that this graph database is constructed from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Another natural question that arises from the PageRank scores, is how those scores relate to the CVE CVSS  score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Table 2 shows that there does seem to be at least some correlation, where the high PageRank score CVEs  have high CVSS scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Similar to the degree correlation plots in Figures 11 and 12, in Figure 13 CVE node  PageRank scores are plotted against CVSS scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Because the majority of the OSint crawler data is temporally  biased towards the more recent years of data gathering, the datapoints plotted in Figure 13 are restricted to the  years during which the OSint crawler system was operational [23], the same as in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Additionally, in order  to filter only for nodes which have a robust distribution of neighbors in the graph, only points with a PageRank  score of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='5 or over is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Figure 13 shows that there is a low to medium linear correlation for the relevance,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' the PageRank score, against the CVE CVSS scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Interestingly, the CVSS v3 scores are more correlated with  the PageRank scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This could indicate that the CVSS v3 scores more accurately represent the severity of the  CVE’s compare to CVSS v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  4  Discussion and Conclusion  There are primarily two technical challenges that still need to be improved in this data gathering and analysis  pipeline:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' There are many instances of near duplicate content from the web crawling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' For example, the content  of a web page could be slightly altered from day to day (for example even if the web page content includes  the current date and time);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' and if the web crawlers end up at that same web page multiple times direct  de-duplication will not remove the content for being a near duplicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Another example of near duplicate  content that occurs often is that the web-crawling will catch a social media conversation (for example a Reddit  thread) while it is occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' We want the crawlers to catch this type of conversation because they could save  content that is removed at some point in the future, but it also leads to potentially a large number of near  duplicates of the same social media thread as it evolves over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The challenge is in quantifying how close  a document is to being a duplicate in order to remove it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' in particular computing a distance metric pairwise  between all documents can be very computationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' A reasonable solution could be to not remove  near duplicates, but instead to take the union of near duplicate documents, and to compute similarity metrics  among close clusters of documents (for example determined by their relationship in the Neo4j potential IoC  graph) for finding near duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Reduce the amount of noise in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  It is difficult to know a-priori which content is relevant,  and therefore in this work we tend to gather more information rather than remove potentially not useful  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This allows the database to be able to catch interesting edge cases and atypical cybersecurity  content, but it comes at the cost of increased noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' However, there are some consistent sources of noise  which are almost always not relevant which can be manually filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' For example localhost IP addresses  or standard file names from popular programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Another example of noise in the data set is software version  numbers being identified as IP addresses due to the similarity in their format in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' It is not clear how  to reduce the noise in the data set uniformly across all document types of potential IoCs, however, machine  learning algorithms which more closely identify relevant information in a piece of text could be used to better  filter the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  There are additional potential indicators and vulnerability tracking ids that could be extracted from natural  language text in future work, such as CWE’s (Common Weakness Enumeration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Aggregating and condensing open source intelligence into a human readable and easily searchable form is an  important task given the scale of the data that is available in the form of social media, news, blogs and threat  reports in the cybersecurity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Here we present one possible way to address this problem by parsing and  transforming the open source data into a graph structure where each document can be associated with potential  cybersecurity indicators of compromise, other infrastructure, CVEs, or MITRE ATT&CK Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Querying  this database based on the indicators then allows analysts to find open source intelligence documents connected  to that indicator and review their content;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' this graph database thus reduces the overhead required in searching a  massive amount of open source documents to a succinct cloud of relevant documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  15  5  Acknowledgements  Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and  Engineering Solutions of Sandia, LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', a wholly owned subsidiary of Honeywell International, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=', for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  The New  Mexico Cybersecurity Center of Excellence (NMCCoE) is a statewide Research and Public Service Project sup-  ported center for economic development, education, and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The authors would like to thank both SNL and  NMCCoE for funding and computing system access and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  A  Portable Executable malware visualization  Figure  14:  Visualization  of  three  different  PE32  malware  samples  byteplot,  entropy,  and  struc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Each  of  these  contain  at  least  one  common  resource  file  (detailed  in  Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='4),  which  was  sufficiently  unique  to  link  these  pieces  of  malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  From  left  to  right  the  sha256  check-  sums  of  these  samples  are  34aa24656d5527a5ff1f7eb4ce4e782085618ded3766730c81f8f16a15d7e0ce,  4f0d5a81b8a5bc3f998a0ac7a37db5bad49e1a22173251ed916363375360d5a4,  8b53201f1914764f384c6ec5a7a  5c5ab2924afaf382d2bbe79f68e43e5dfa3ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  From left to right the OriginalFilename for each sample is  RKInstaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='exe, RKInstaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='exe, and POInstaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='exe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Note the similar naming scheme to the four samples  that are in Neo4j (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Interestingly, although these samples are not identical (we know this because  their hashes are different), the middle and right hand side samples look visually indistinguishable from each other,  whereas the left hand side sample is distinguishable from the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Overall, the structure of these PEs seem  to be very similar which further indicates that these were likely developed by the same group or person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Figure 14 shows visualizations of three malware examples which contain a common PE resource artifact which  was identified by its sha256 checksum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' These three examples are different from the Neo4j nodes (specifically their  hashes are not the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' These three distinct PE samples are also accessible on VirusTotal 17 18 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' This suggests  that this particular indicator, while unique and not very common, is likely seen on other static analysis tools beyond  these two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' These visualizations were generated using FileScan20 and PortEx21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  B  Large graph visualization  Expanding out the network connections that exist in the Neo4j graph database is difficult to show visually because  of the scale of the graphs in terms of edges and nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Here though in Figure 15 we show two large graph examples,  which is possible using pygraphistry22 and networkx [51] in python3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The graphistry version which was used to  generate these figures is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='32, and the graph layout algorithm is ForceAtlas2Barnes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' The primary observation  that is important from Figure 15 is that just a couple of degrees out from an indicator can already result in a very  large graph, even though the entire graph database is very sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' One of the reasons for this is that different  types of indicators, names, and vulnerability IDs will naturally be mentioned at different rates in the gathered  internet text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' For example, popular malware names will be very common in cybersecurity text, whereas hashes  will generally not be mentioned very frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Therefore, it is important to filter down to the types of edges you  want to follow in the database when you are searching for specific indicators - for example by searching only for  17https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='virustotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/gui/file/34aa24656d5527a5ff1f7eb4ce4e782085618ded3766730c81f8f16a15d7e0ce  18https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='virustotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/gui/file/4f0d5a81b8a5bc3f998a0ac7a37db5bad49e1a22173251ed916363375360d5a4  19https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='virustotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/gui/file/8b53201f1914764f384c6ec5a7a5c5ab2924afaf382d2bbe79f68e43e5dfa3ba  20https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='filescan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='io  21https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/struppigel/PortEx  22https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='com/graphistry/pygraphistry  16  BytePlot (left)  ■OxFF  0x00  visible ASCll  invisible ASCll  non-AscIl  Entropy (middle)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2 (repetition)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='5(code)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8 (packed)  PE Structure (right)  MSDoS Header  COFF File Header  Optional Header  Section Table  text  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='rdata  rsrc  Resources  Imports  Debug Info  Entry Point  OverlayBytePlot (left)  ■OxFF  0x00  visible ASCll  invisible ASCll  non-AscIl  Entropy (middle)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2 (repetition)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='5(code)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8 (packed)  PE Structure (riaht)  MSDoS Header  CoFF File Header  Optional Header  Section Table  text  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='rdata  rsrc  Resources  Imports  Debug Info  Entry Point  OverlayBytePlot (left)  ■OxFF  0x00  visible ASCll  invisible ASCll  non-AscIl  Entropy (middle)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2 (repetition)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='5(code)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8 (packed)  PE Structure (riaht)  MSDoS Header  CoFF File Header  Optional Header  Section Table  text  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='rdata  rsrc  Resources  Imports  Debug Info  Entry Point  OverlayIP and domain edges if you are searching for server infrastructure connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' It is also useful in these cases to  use the CTC metadata and language detection metadata in order to search specifically for document nodes which  are, with high confidence, English text discussing cybersecurity in order to get more relevant text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' When there  is limited information available on an indicator or document, searching through all available network connections  (such as in the malware hash cases shown in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='4) can also be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  Figure 15: Large graph visualizations which are expansions of the Qakbot IP address graph from Figure 7 in Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Degree 3 connections (left) has 7,862 nodes and 9,244 edges, degree 4 connections (right) has 46,631 nodes and  112,935 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Note that the node and edge coloring is selected by the graphistry software and serves to delineate  different aspects of the graph, but does not follow the same coloring scheme used in the Neo4j browser figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  These extremely large graph renderings show what the large scale behavior of this subgraph of the database looks  like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Even though the database is overall quite sparse, there are clear clusters of the graph which behave similar to  each other and there are also clearly very highly connected clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  References  [1]  Michael Glassman and Min Ju Kang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “Intelligence in the internet age: The emergence and evolution of Open  Source Intelligence (OSINT)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: Computers in Human Behavior 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1016/  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='chb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Systematic literature review to investigate the application of open  source intelligence (osint) with artificial intelligence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: Journal of Applied Security Research 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='3 (2021),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1080/19361610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1761737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  [3]  Robert David Steele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “Open source intelligence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: Handbook of Intelligence Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Routledge, 2007,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' In: 2020 IEEE International Conference on Intelligence and Security Informatics (ISI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1109/ISI49825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='9280512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  [5]  Peng Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “A System for Efficiently Hunting for Cyber Threats in Computer Systems Using Threat  Intelligence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: 2021 IEEE 37th International Conference on Data Engineering (ICDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1109/ICDE51399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='00309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  17  [6]  Nidhi Rastogi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “MALOnt: An Ontology for Malware Threat Intelligence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: Deployable Machine  Learning for Security Defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' by Gang Wang, Arridhana Ciptadi, and Ali Ahmadzadeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Cham: Springer  International Publishing, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' 28–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' isbn: 978-3-030-59621-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  [7]  Onur Catakoglu, Marco Balduzzi, and Davide Balzarotti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “Automatic extraction of indicators of compro-  mise for web applications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: Proceedings of the 25th International Conference on World Wide Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' 2016,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1145/2872427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='  [8]  Yuta Kazato, Yoshihide Nakagawa, and Yuichi Nakatani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “Improving maliciousness estimation of indicator  of compromise using graph convolutional networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: 2020 IEEE 17th Annual Consumer Communications  & Networking Conference (CCNC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1109/CCNC46108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='9045113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  [9]  Ryan Christian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “An Ontology-Driven Knowledge Graph for Android Malware”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: Proceedings of  the 2021 ACM SIGSAC Conference on Computer and Communications Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' CCS ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Virtual Event,  Republic of Korea: Association for Computing Machinery, 2021, 2435–2437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' isbn: 9781450384544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='  [10]  Benjamin Bowman and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Howie Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “Towards Next-Generation Cybersecurity with Graph AI”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In:  SIGOPS Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='1 (2021), 61–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' issn: 0163-5980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1145/3469379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' url: https:  //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='3469386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  [11]  Yongfu Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “The Analysis Method of Security Vulnerability Based on the Knowledge Graph”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In:  2020 the 10th International Conference on Communication and Network Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' ICCNS 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Tokyo, Japan:  Association for Computing Machinery, 2021, 135–145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' isbn: 9781450389037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='  url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='  [12]  Shuqin Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “Threat Analysis of IoT Security Knowledge Graph Based on Confidence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: Emerging  Technologies for Education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' by Weijia Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Cham: Springer International Publishing, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' 254–  264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' isbn: 978-3-030-92836-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  [13]  Sharmishtha Dutta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' “Knowledge Graph for Malware Threat Intelligence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' In: (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' url: http://rgdoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='13140/RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='27340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='95367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' In: Proceedings of the Southern  Association for Information Systems Conference, Atlanta, GA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Graph Databases: Neo4j Analysis.” In: ICEIS  (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Graph databases: their power and limitations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: Ifip International Conference on Com-  puter Information Systems and Industrial Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Research on architecture and query performance based on distributed  graph database Neo4j”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: 2013 3rd International Conference on Consumer Electronics, Communications  and Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Using Neo4j graph database in social network analysis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: Studia Informatica 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='  In: 2020 Chinese Control And Decision Conference (CCDC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' In: Handbook of  Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' In: First  International Workshop on Graph Data Management Experiences and Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Designing a Modular and Distributed Web Crawler  Focused on Unstructured Cybersecurity Intelligence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “A Robust Cybersecurity Topic Classification Tool”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  In: International Journal of Network Security & Its Applications, V14, N1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='02473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Retrieving ATT&CK tactics and techniques in cyber threat reports”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' MA  thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' University of Twente, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='1109/RWS50334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='9241271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Mitre ATT&CK”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Learning the associations of mitre ATT&CK  adversarial techniques”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='1109/CNS48642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' eprint: 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='01654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Linking CVE’s to MITRE ATT&CK Techniques”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  In: The 16th International Conference on Availability, Reliability and Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' Investigating co-occurrences of MITRE ATT&CK Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='06495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='06495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' An investigation of security controls and MITRE ATT&CK  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' Natural language processing with Python: analyzing text with  the natural language toolkit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='1109/IRI54793.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' In: Technical report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “The static analysis of WannaCry ransomware”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='8323680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='  In: 2020 International Conference on Cyber-Enabled Distributed Computing and Knowledge Discovery (Cy-  berC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='1109/CyberC49757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Analyzing WannaCry Ransomware Considering the  Weapons and Exploits”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' In: 2019 21st International Conference on Advanced Communication Technology  (ICACT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' “Automated Behavioral Analysis of Malware: A Case Study of Wan-  naCry Ransomware”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='1109/ICMLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content='23919/  ICACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+page_content=' Exploring network structure, dynamics, and function using  NetworkX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' Los Alamos National Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
+page_content=' (LANL), Los Alamos, NM (United States), 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFLT4oBgHgl3EQfLy8m/content/2301.12013v1.pdf'}
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+On Bayes risk of the posterior mean in linear inverse problems
+Alen Alexanderian*
+Abstract
+We recall some basics regarding the concept of Bayes risk in the context of finite-
+dimensional ill-posed linear inverse problem with Gaussian prior and noise models. In
+particular, we rederive the following basic result: in the present Gaussian linear setting,
+the Bayes risk of the posterior mean, relative to the sum of squares loss function, equals
+the trace of the posterior covariance operator.
+1
+Introduction
+We recall some basics from theory of point estimation regarding the concept of the risk for
+an estimator. Then, taking a Bayesian point of view, we discuss the notion of Bayes risk. We
+focus on ill-posed linear inverse problems with Gaussian prior and additive Gaussian noise
+models. In this case, we show that the Bayes risk, relative to the sum of squares loss func-
+tion, equals the trace of the posterior covariance operator; see Proposition 4.1. The results
+discussed herein are well-known. The purpose of this note is to provide an accessible and
+self-contained summary. For further details on the concepts of risk and Bayes risk of an
+estimator discussed below, see e.g., [7, 3]. Herein, we consider only inverse problems with
+finite-dimensional parameters. However, the result on Bayes risk (Proposition 4.1) can be
+extended to Bayesian linear inverse problems in infinite-dimensional Hilbert spaces [2]. In
+fact, the style of presentation in this note, while limited to a finite-dimensional setup, is
+motivated by the more general infinite-dimensional Hilbert space setting.
+In a statistical inverse problem, Bayes risk can be used as a means of assessing the sta-
+tistical quality of the estimated parameter. This is important, for example, in the context
+of optimal experimental design (OED). In an OED problem, one seeks measurements that
+optimize the quality of the estimated parameters. Bayes risk is one from among several
+possibilities for an optimal experimental design criterion; see e.g, [5, 9, 1].
+2
+Preliminaries
+In this section, we recall some basics from probability and statistics that are needed in this
+note. Throughout, RN is always equipped with the Euclidean inner product, denoted by
+⟨·, ·⟩, and the Euclidean norm ∥ · ∥ = ⟨·, ·⟩1/2. If we want to emphasize the dimension N of
+the space, when using the Euclidean inner product, we use the notation ⟨·, ·⟩N.
+2.1
+Background concepts from probability
+We denote the Borel sigma-algebra on RN by B(RN). Let us recall the definition of the
+mean and covariance operator of a probability measure µ on (RN, B(RN)) that has finite
+*North Carolina State University, Raleigh, NC, USA. E-mail: alexanderian@ncsu.edu
+Last revised: January 12, 2023.
+arXiv:2301.04270v1  [math.ST]  11 Jan 2023
+
+Bayes risk in linear inverse problem
+first and second moments. Let {e1, . . . , eN} be the standard basis in RN. The mean c and
+covariance operator Q of µ are defined as follows:
+ci = ⟨ei, c⟩ =
+�
+RN ⟨ei, x⟩ µ(dx)
+and
+Qij =
+�
+RN ⟨ei, x − c⟩⟨ej, x − c⟩ µ(dx),
+with i, j ∈ {1, . . . , N}.1 Using the definition of mean and covariance operator, it is straight-
+forward to show
+�
+RN ∥x∥2 dx = tr(Q) + ∥c∥2.
+Let µ be a probability measure on (RN, B(RN)). The law of a random variable Z : (RN, B(RN), µ) →
+(RM, B(RM)) is the probability measure LZ defined by
+LZ(E) = µ(Z−1(E))
+E ∈ B(RM).
+See [11] for more details. We recall the following well-known result: a Borel measurable
+function φ : RM → R is integrable with respect to LZ precisely when φ ◦ Z is integrable
+with respect to µ. In addition, we have the change of variable formula
+�
+RN φ(Z(x))µ(dx) =
+�
+RM φ(z)LZ(dz).
+This result holds in more general measure theoretic settings; see [4, Section 3.6]. We will
+also need the following well-known result regarding Gaussian measures [6]. We include a
+proof of this in the present finite-dimensional setting, for completeness.
+Proposition 2.1. Let µ = N(c, Q) be a Gaussian measure on (RN, B(RN)). Let A ∈ RM×N
+and s ∈ RM. Define the random variable Z(x) = Ax + s on (RN, B(RN), µ). Then, Z is a
+Gaussian random variable with law LZ = N
+�
+Ac + s, AQAT�
+.
+Proof. See the Appendix A.
+Let us also recall the following related result: let X be a N-dimensional Gaussian random
+vector with law N(c, Q) and let A ∈ RM×N and s ∈ RM. Then, Z = AX + s is also a
+Gaussian and
+Z ∼ N
+�
+Ac + s, AQAT�
+.
+(2.1)
+See e.g., [8, p. 121] for a direct proof of this for Gaussian random vectors.
+2.2
+Frequentist risk
+In what follows, H denotes the N-dimensional Euclidean space, m ∈ H is an inference
+parameter, and �
+m(y) is an estimator (point-estimate) for m obtained using a realization of
+data y. We let Loss [m, �
+m(y)] denote a loss function. An example (and that’s the only one
+we consider) is the sum of squares loss function:
+Loss [m, �
+m(y)] = ∥m − �
+m(y)∥2 ,
+where ∥ · ∥ is the Euclidean norm. The data y takes values in Y = RD. Herein, we let
+πlike(y|m) be a likelihood function.
+From a frequentist point of view, there is nothing random about the parameter vector m.
+This parameter is an underlying truth we seek to estimate. On the other hand, the estima-
+tor �
+m is a random variable—it is a function of the random variable y. (The distribution of
+1The notions of mean and covariance operator of measures extend naturally to the case of probability measures
+on infinite-dimensional Hilbert spaces [6].
+2
+
+Bayes risk in linear inverse problem
+y is specified by a likelihood model.) We denote the (frequentist) risk of the estimator �
+m
+by Risk(�
+m; m) and define it as follows,
+Risk(�
+m; m) = Ey|m {Loss [m, �
+m(·)]} =
+�
+Y
+Loss [m, �
+m(y)] πlike(y|m) dy.
+A common choice for the loss function is the sum of squares loss function, where we use
+Risk(�
+m; m) = Ey|m
+�
+∥m − �
+m(y)∥2�
+=
+�
+Y
+∥m − �
+m(y)∥2 πlike(y|m) dy.
+The above risk is often referred to as the mean-squared-error and is denoted by MSE; i.e.,
+MSE(�
+m; m) = Ey|m
+�
+∥m − �
+m(y)∥2�
+. An elementary calculation shows,
+MSE(�
+m; m) = ∥m − E {�
+m}∥2 + E
+�
+∥�
+m − E {�
+m}∥2�
+,
+(2.2)
+where for brevity we have used E {·} in place of Ey|m {·}. Note that the first term in (2.2)
+quantifies the magnitude of estimation bias, ∥m − E {�
+m}∥2 = ∥Bias(�
+m)∥2, and the second
+one describes the variability of the estimator around its mean.
+3
+The MSE for the solution of an ill-posed linear inverse problem
+Consider a linear forward operator F ∈ RD×N and the following data model
+y = Fm + η,
+where η is a centered Gaussian with covariance Γnoise. This implies, y|m ∼ N(Fm, Γnoise).
+Given a realization of y, we can obtain a point-estimate for the parameter m by computing
+�
+m(y) = arg min
+m∈H
+1
+2(Fm − y)TΓ−1
+noise(Fm − y) + β
+2 (m − m0)TR(m − m0).
+(3.1)
+Here R ∈ RN×N is a (symmetric positive definite) regularization operator, needed due
+to ill-posed nature of the inverse problemm, β is a regularization parameter, and m0 is a
+reference parameter value. In many cases m0 = 0 is used. The solution to the regularized
+least-squares problem is
+�
+m(y) = H−1(FTΓ−1
+noisey + βRm0),
+(3.2)
+with
+H = Hmisfit + βR,
+where
+Hmisfit = FTΓ−1
+noiseF.
+Note that H is the Hessian of the objective function in (3.1), and Hmisfit is the Hessian of
+the data-misfit part in the objective function.
+The following result describes the MSE(�
+m; m) in the present setting:
+Proposition 3.1. Let �
+m in (3.2) be the estimator of m. Then,
+MSE(�
+m; m) = β2 ��H−1R(m − m0)
+��2 + tr(H−2Hmisfit).
+(3.3)
+Proof. First we note that
+Bias(�
+m) = Ey|m {�
+m} − m = Ey|m
+�
+H−1FTΓ−1
+noisey
+�
++ βH−1Rm0 − m
+= H−1FTΓ−1
+noiseFm − m + βH−1Rm0
+= H−1(FTΓ−1
+noiseF − H)m + βH−1Rm0
+= −βH−1R(m − m0).
+3
+
+Bayes risk in linear inverse problem
+Thus, the first term in (2.2) is given by β2 ��H−1R(m − m0)
+��2.
+Next, we compute the
+second term in (2.2). To this end, let us define Z(y) = �
+m(y) − E {�
+m(·)}, and note that Z
+has a Gaussian law LZ with mean zero and covariance operator,
+(H−1FTΓ−1
+noise)Γnoise(H−1FTΓ−1
+noise)T = H−1FTΓ−1
+noiseFH−1 = H−1HmisfitH−1.
+Therefore,
+E{∥Z(y)∥2} =
+�
+H
+∥z∥2 Lz(dz) = tr(H−1HmisfitH−1) = tr(H−2Hmisfit).
+4
+Bayes risk and the case of the Bayesian linear inverse problem
+As seen in the previous section, the MSE of the least-squares estimate �
+m has a contribution
+from the estimation bias; see, e.g., (3.3), where the first term in the sum is the size of the
+estimation bias. We cannot compute the bias term because it depends on the unknown
+parameter m we are trying estimate. As seen shortly, this issue can be dealt with naturally
+in a Bayesian framework, where we view m as a random variable.
+We equip H with the corresponding Borel sigma algebra and consider the measurable
+space (H, B(H)). In the Bayesian framework, our prior knowledge regarding the parame-
+ter m is encoded in a probability measure µpr on (H, B(H)). We refer to µpr as the prior
+measure. This setup, in particular, enables computing the expected value of the bias term
+and leads to the notion of Bayes risk. Namely, we define the Bayes risk as the expected
+value of the frequentist risk with respect to the prior law of m. In particular, using the
+MSE as the risk measure, we define the Bayes risk of the estimator �
+m by
+Risk(�
+m) = Eµpr {MSE(�
+m; m)} =
+�
+H
+�
+Y
+∥m − �
+m(y)∥2 πlike(y|m) dy µpr(dm).
+We consider a Gaussian prior measure µpr = N(mpr, Γpr).
+Next, we cast the linear inverse problem (3.1) in a Bayesian framework. The following
+is based on standard results on Gaussian linear inverse problems [10]. In the Bayesian
+formulation, we relabel components of (3.1) and (3.2) as follows:
+• m0 is replaced with mpr;
+• βR is replaced with Γ−1
+pr ;
+• H−1 is replaced with the covariance operator Γpost; and
+• �
+m(y) becomes the posterior mean my
+post.
+The expression (3.3) for the MSE, cast in the present Bayesian framework, is given by
+MSE(my
+post; m) =
+��ΓpostΓ−1
+pr (m − mpr)
+��2 + tr(Γ2
+postHmisfit).
+The following result, which is the main point of this note, concludes our discussion.
+Proposition 4.1. We have Risk(my
+post) = tr(Γpost).
+Proof. Consider the random variable Z(m) = ΓpostΓ−1
+pr (m−mpr), defined on (H, B(H), µpr),
+and note that LZ = N
+�
+0, ΓpostΓ−1
+pr Γpost
+�
+. Thus,
+�
+H
+��ΓpostΓ−1
+pr (m − mpr)
+��2 µpr(dm) =
+�
+H
+∥z∥2 Lz(dz) = tr(ΓpostΓ−1
+pr Γpost) = tr(Γ2
+postΓ−1
+pr ).
+Therefore,
+Risk(my
+post) =
+�
+H
+MSE(mpost) µpr(dm) =
+�
+H
+��ΓpostΓ−1
+pr (m − mpr)
+��2 µpr(dm) + tr(Γ2
+postHmisfit)
+= tr(Γ2
+postΓ−1
+pr ) + tr(Γ2
+postHmisfit) = tr(Γpost).
+4
+
+Bayes risk in linear inverse problem
+References
+[1] Alen Alexanderian. Optimal experimental design for infinite-dimensional Bayesian inverse prob-
+lems governed by PDEs: A review. Inverse Problems, 37(4):043001, 2021.
+[2] Alen Alexanderian, Philip J. Gloor, and Omar Ghattas. On Bayesian A-and D-optimal experimen-
+tal designs in infinite dimensions. Bayesian Analysis, 11(3):671–695, 2016.
+[3] James O Berger. Statistical decision theory and Bayesian analysis. Springer Science & Business
+Media, 2013.
+[4] V. I. Bogachev. Measure theory. Vol. I. Springer-Verlag, Berlin, 2007.
+[5] Kathryn Chaloner and Isabella Verdinelli. Bayesian experimental design: A review. Statist. Sci.,
+10(3):273–304, 1995.
+[6] Giuseppe Da Prato. An introduction to infinite-dimensional analysis. Springer Science & Busi-
+ness Media, 2006.
+[7] Jayanta K Ghosh, Mohan Delampady, and Tapas Samanta. An introduction to Bayesian analysis:
+theory and methods, volume 725. Springer, 2006.
+[8] Allan Gut. An intermediate course in probability. Springer Texts in Statistics. Springer, New
+York, second edition, 2009.
+[9] Eldad Haber, Lior Horesh, and Luis Tenorio. Numerical methods for experimental design of
+large-scale linear ill-posed inverse problems. Inverse Problems, 24(5):055012, 2008.
+[10] Albert Tarantola. Inverse problem theory and methods for model parameter estimation. SIAM,
+2005.
+[11] David Williams. Probability with martingales. Cambridge Mathematical Textbooks. Cambridge
+University Press, Cambridge, 1991.
+5
+
+Bayes risk in linear inverse problem
+A
+Proof of Propositon 2.1
+Before getting into the proof, we recall the notion of the the Fourier transform of a mea-
+sure. Let µ be a measure on (RN, B(RN)). The Fourier transform of µ is given by
+�µ(h) =
+�
+RN exp (i⟨h, x⟩) µ(dx),
+h ∈ RN.
+The Fourier transform uniquely characterizes a measure; see e.g., [6, Proposition 1.7]. It
+is known that a Gaussian measure µ = N(c, Q) on (RN, B(RN)) has Fourier transform
+�µ(h) = exp
+�
+i⟨h, c⟩ − 1
+2⟨Qh, h⟩
+�
+,
+h ∈ RN.
+We are now ready to prove Proposition 2.1.
+Proof of Proposition 2.1. We prove the result by showing that
+�
+LZ(h) = exp
+�
+⟨h, Ac + s⟩M + 1
+2⟨AQATh, h⟩M
+�
+,
+for all h ∈ RM.
+Let h ∈ RM. We have
+�
+LZ(h) =
+�
+RM exp
+�
+i⟨h, z⟩M
+�
+LZ(dz)
+=
+�
+RN exp
+�
+i⟨h, Z(x)⟩M
+�
+µ(dx)
+=
+�
+RN exp
+�
+i⟨h, Ax + s⟩M
+�
+µ(dx)
+= exp
+�
+i⟨h, s⟩M
+� �
+RN exp
+�
+i⟨h, Ax⟩M
+�
+µ(dx)
+= exp
+�
+i⟨h, s⟩M
+� �
+RN exp
+�
+i⟨ATh, x⟩N
+�
+µ(dx)
+= exp
+�
+i⟨h, s⟩M
+�
+exp
+�
+i⟨ATh, c⟩N − 1
+2⟨QATh, ATh⟩N
+�
+= exp
+�
+i⟨h, s⟩M
+�
+exp
+�
+i⟨h, Ac⟩M − 1
+2⟨AQATh, h⟩M
+�
+= exp
+�
+i⟨h, Ac + s⟩M − 1
+2⟨AQATh, h⟩M
+�
+.
+6
+
diff --git a/N9E3T4oBgHgl3EQfBwmo/content/tmp_files/load_file.txt b/N9E3T4oBgHgl3EQfBwmo/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d4956917127c792eb72ea9dd522d88f766d689f9
--- /dev/null
+++ b/N9E3T4oBgHgl3EQfBwmo/content/tmp_files/load_file.txt
@@ -0,0 +1,216 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf,len=215
+page_content='On Bayes risk of the posterior mean in linear inverse problems  Alen Alexanderian*  Abstract  We recall some basics regarding the concept of Bayes risk in the context of finite-  dimensional ill-posed linear inverse problem with Gaussian prior and noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' In  particular, we rederive the following basic result: in the present Gaussian linear setting,  the Bayes risk of the posterior mean, relative to the sum of squares loss function, equals  the trace of the posterior covariance operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  1  Introduction  We recall some basics from theory of point estimation regarding the concept of the risk for  an estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Then, taking a Bayesian point of view, we discuss the notion of Bayes risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' We  focus on ill-posed linear inverse problems with Gaussian prior and additive Gaussian noise  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' In this case, we show that the Bayes risk, relative to the sum of squares loss func-  tion, equals the trace of the posterior covariance operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' The results  discussed herein are well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' The purpose of this note is to provide an accessible and  self-contained summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' For further details on the concepts of risk and Bayes risk of an  estimator discussed below, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=', [7, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Herein, we consider only inverse problems with  finite-dimensional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' However, the result on Bayes risk (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1) can be  extended to Bayesian linear inverse problems in infinite-dimensional Hilbert spaces [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' In  fact, the style of presentation in this note, while limited to a finite-dimensional setup, is  motivated by the more general infinite-dimensional Hilbert space setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  In a statistical inverse problem, Bayes risk can be used as a means of assessing the sta-  tistical quality of the estimated parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' This is important, for example, in the context  of optimal experimental design (OED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' In an OED problem, one seeks measurements that  optimize the quality of the estimated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Bayes risk is one from among several  possibilities for an optimal experimental design criterion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='g, [5, 9, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  2  Preliminaries  In this section, we recall some basics from probability and statistics that are needed in this  note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Throughout, RN is always equipped with the Euclidean inner product, denoted by  ⟨·, ·⟩, and the Euclidean norm ∥ · ∥ = ⟨·, ·⟩1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' If we want to emphasize the dimension N of  the space, when using the Euclidean inner product, we use the notation ⟨·, ·⟩N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1  Background concepts from probability  We denote the Borel sigma-algebra on RN by B(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Let us recall the definition of the  mean and covariance operator of a probability measure µ on (RN, B(RN)) that has finite  North Carolina State University, Raleigh, NC, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' E-mail: alexanderian@ncsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='edu  Last revised: January 12, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='04270v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='ST]  11 Jan 2023  Bayes risk in linear inverse problem  first and second moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Let {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' , eN} be the standard basis in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' The mean c and  covariance operator Q of µ are defined as follows:  ci = ⟨ei, c⟩ =  �  RN ⟨ei, x⟩ µ(dx)  and  Qij =  �  RN ⟨ei, x − c⟩⟨ej, x − c⟩ µ(dx),  with i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1 Using the definition of mean and covariance operator, it is straight-  forward to show  �  RN ∥x∥2 dx = tr(Q) + ∥c∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Let µ be a probability measure on (RN, B(RN)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' The law of a random variable Z : (RN, B(RN), µ) →  (RM, B(RM)) is the probability measure LZ defined by  LZ(E) = µ(Z−1(E))  E ∈ B(RM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  See [11] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' We recall the following well-known result: a Borel measurable  function φ : RM → R is integrable with respect to LZ precisely when φ ◦ Z is integrable  with respect to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' In addition, we have the change of variable formula  �  RN φ(Z(x))µ(dx) =  �  RM φ(z)LZ(dz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  This result holds in more general measure theoretic settings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' see [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' We will  also need the following well-known result regarding Gaussian measures [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' We include a  proof of this in the present finite-dimensional setting, for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Let µ = N(c, Q) be a Gaussian measure on (RN, B(RN)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Let A ∈ RM×N  and s ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Define the random variable Z(x) = Ax + s on (RN, B(RN), µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Then, Z is a  Gaussian random variable with law LZ = N  �  Ac + s, AQAT�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' See the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Let us also recall the following related result: let X be a N-dimensional Gaussian random  vector with law N(c, Q) and let A ∈ RM×N and s ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Then, Z = AX + s is also a  Gaussian and  Z ∼ N  �  Ac + s, AQAT�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1)  See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=', [8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' 121] for a direct proof of this for Gaussian random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='2  Frequentist risk  In what follows, H denotes the N-dimensional Euclidean space, m ∈ H is an inference  parameter, and �  m(y) is an estimator (point-estimate) for m obtained using a realization of  data y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' We let Loss [m, �  m(y)] denote a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' An example (and that’s the only one  we consider) is the sum of squares loss function:  Loss [m, �  m(y)] = ∥m − �  m(y)∥2 ,  where ∥ · ∥ is the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' The data y takes values in Y = RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Herein, we let  πlike(y|m) be a likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  From a frequentist point of view, there is nothing random about the parameter vector m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  This parameter is an underlying truth we seek to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' On the other hand, the estima-  tor �  m is a random variable—it is a function of the random variable y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' (The distribution of  1The notions of mean and covariance operator of measures extend naturally to the case of probability measures  on infinite-dimensional Hilbert spaces [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  2  Bayes risk in linear inverse problem  y is specified by a likelihood model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=') We denote the (frequentist) risk of the estimator �  m  by Risk(�  m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' m) and define it as follows,  Risk(�  m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' m) = Ey|m {Loss [m, �  m(·)]} =  �  Y  Loss [m, �  m(y)] πlike(y|m) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  A common choice for the loss function is the sum of squares loss function, where we use  Risk(�  m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' m) = Ey|m  �  ∥m − �  m(y)∥2�  =  �  Y  ∥m − �  m(y)∥2 πlike(y|m) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  The above risk is often referred to as the mean-squared-error and is denoted by MSE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=',  MSE(�  m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' m) = Ey|m  �  ∥m − �  m(y)∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' An elementary calculation shows,  MSE(�  m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' m) = ∥m − E {�  m}∥2 + E  �  ∥�  m − E {�  m}∥2�  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='2)  where for brevity we have used E {·} in place of Ey|m {·}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Note that the first term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='2)  quantifies the magnitude of estimation bias, ∥m − E {�  m}∥2 = ∥Bias(�  m)∥2, and the second  one describes the variability of the estimator around its mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  3  The MSE for the solution of an ill-posed linear inverse problem  Consider a linear forward operator F ∈ RD×N and the following data model  y = Fm + η,  where η is a centered Gaussian with covariance Γnoise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' This implies, y|m ∼ N(Fm, Γnoise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Given a realization of y, we can obtain a point-estimate for the parameter m by computing  �  m(y) = arg min  m∈H  1  2(Fm − y)TΓ−1  noise(Fm − y) + β  2 (m − m0)TR(m − m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1)  Here R ∈ RN×N is a (symmetric positive definite) regularization operator, needed due  to ill-posed nature of the inverse problemm, β is a regularization parameter, and m0 is a  reference parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' In many cases m0 = 0 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' The solution to the regularized  least-squares problem is  �  m(y) = H−1(FTΓ−1  noisey + βRm0),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='2)  with  H = Hmisfit + βR,  where  Hmisfit = FTΓ−1  noiseF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Note that H is the Hessian of the objective function in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1), and Hmisfit is the Hessian of  the data-misfit part in the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  The following result describes the MSE(�  m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' m) in the present setting:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Let �  m in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='2) be the estimator of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Then,  MSE(�  m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' m) = β2 ��H−1R(m − m0)  ��2 + tr(H−2Hmisfit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' First we note that  Bias(�  m) = Ey|m {�  m} − m = Ey|m  �  H−1FTΓ−1  noisey  �  + βH−1Rm0 − m  = H−1FTΓ−1  noiseFm − m + βH−1Rm0  = H−1(FTΓ−1  noiseF − H)m + βH−1Rm0  = −βH−1R(m − m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  3  Bayes risk in linear inverse problem  Thus, the first term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='2) is given by β2 ��H−1R(m − m0)  ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Next, we compute the  second term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' To this end, let us define Z(y) = �  m(y) − E {�  m(·)}, and note that Z  has a Gaussian law LZ with mean zero and covariance operator,  (H−1FTΓ−1  noise)Γnoise(H−1FTΓ−1  noise)T = H−1FTΓ−1  noiseFH−1 = H−1HmisfitH−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Therefore,  E{∥Z(y)∥2} =  �  H  ∥z∥2 Lz(dz) = tr(H−1HmisfitH−1) = tr(H−2Hmisfit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  4  Bayes risk and the case of the Bayesian linear inverse problem  As seen in the previous section, the MSE of the least-squares estimate �  m has a contribution  from the estimation bias;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=', (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='3), where the first term in the sum is the size of the  estimation bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' We cannot compute the bias term because it depends on the unknown  parameter m we are trying estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' As seen shortly, this issue can be dealt with naturally  in a Bayesian framework, where we view m as a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  We equip H with the corresponding Borel sigma algebra and consider the measurable  space (H, B(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' In the Bayesian framework, our prior knowledge regarding the parame-  ter m is encoded in a probability measure µpr on (H, B(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' We refer to µpr as the prior  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' This setup, in particular, enables computing the expected value of the bias term  and leads to the notion of Bayes risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Namely, we define the Bayes risk as the expected  value of the frequentist risk with respect to the prior law of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' In particular, using the  MSE as the risk measure, we define the Bayes risk of the estimator �  m by  Risk(�  m) = Eµpr {MSE(�  m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' m)} =  �  H  �  Y  ∥m − �  m(y)∥2 πlike(y|m) dy µpr(dm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  We consider a Gaussian prior measure µpr = N(mpr, Γpr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Next, we cast the linear inverse problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1) in a Bayesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' The following  is based on standard results on Gaussian linear inverse problems [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' In the Bayesian  formulation, we relabel components of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='2) as follows:  m0 is replaced with mpr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  βR is replaced with Γ−1  pr ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  H−1 is replaced with the covariance operator Γpost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' and  �  m(y) becomes the posterior mean my  post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  The expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='3) for the MSE, cast in the present Bayesian framework, is given by  MSE(my  post;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' m) =  ��ΓpostΓ−1  pr (m − mpr)  ��2 + tr(Γ2  postHmisfit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  The following result, which is the main point of this note, concludes our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' We have Risk(my  post) = tr(Γpost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Consider the random variable Z(m) = ΓpostΓ−1  pr (m−mpr), defined on (H, B(H), µpr),  and note that LZ = N  �  0, ΓpostΓ−1  pr Γpost  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Thus,  �  H  ��ΓpostΓ−1  pr (m − mpr)  ��2 µpr(dm) =  �  H  ∥z∥2 Lz(dz) = tr(ΓpostΓ−1  pr Γpost) = tr(Γ2  postΓ−1  pr ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Therefore,  Risk(my  post) =  �  H  MSE(mpost) µpr(dm) =  �  H  ��ΓpostΓ−1  pr (m − mpr)  ��2 µpr(dm) + tr(Γ2  postHmisfit)  = tr(Γ2  postΓ−1  pr ) + tr(Γ2  postHmisfit) = tr(Γpost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  4  Bayes risk in linear inverse problem  References  [1] Alen Alexanderian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Optimal experimental design for infinite-dimensional Bayesian inverse prob-  lems governed by PDEs: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Inverse Problems, 37(4):043001, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  [2] Alen Alexanderian, Philip J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Gloor, and Omar Ghattas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' On Bayesian A-and D-optimal experimen-  tal designs in infinite dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Bayesian Analysis, 11(3):671–695, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  [3] James O Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Statistical decision theory and Bayesian analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Springer Science & Business  Media, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Bogachev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Springer-Verlag, Berlin, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  [5] Kathryn Chaloner and Isabella Verdinelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Bayesian experimental design: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=',  10(3):273–304, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  [6] Giuseppe Da Prato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' An introduction to infinite-dimensional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Springer Science & Busi-  ness Media, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  [7] Jayanta K Ghosh, Mohan Delampady, and Tapas Samanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' An introduction to Bayesian analysis:  theory and methods, volume 725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  [8] Allan Gut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' An intermediate course in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Springer Texts in Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Springer, New  York, second edition, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  [9] Eldad Haber, Lior Horesh, and Luis Tenorio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Numerical methods for experimental design of  large-scale linear ill-posed inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Inverse Problems, 24(5):055012, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  [10] Albert Tarantola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Inverse problem theory and methods for model parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' SIAM,  2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  [11] David Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Probability with martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Cambridge Mathematical Textbooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Cambridge  University Press, Cambridge, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  5  Bayes risk in linear inverse problem  A  Proof of Propositon 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1  Before getting into the proof, we recall the notion of the the Fourier transform of a mea-  sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' Let µ be a measure on (RN, B(RN)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' The Fourier transform of µ is given by  �µ(h) =  �  RN exp (i⟨h, x⟩) µ(dx),  h ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  The Fourier transform uniquely characterizes a measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=', [6, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' It  is known that a Gaussian measure µ = N(c, Q) on (RN, B(RN)) has Fourier transform  �µ(h) = exp  �  i⟨h, c⟩ − 1  2⟨Qh, h⟩  �  ,  h ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  We are now ready to prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' We prove the result by showing that  �  LZ(h) = exp  �  ⟨h, Ac + s⟩M + 1  2⟨AQATh, h⟩M  �  ,  for all h ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  Let h ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content=' We have  �  LZ(h) =  �  RM exp  �  i⟨h, z⟩M  �  LZ(dz)  =  �  RN exp  �  i⟨h, Z(x)⟩M  �  µ(dx)  =  �  RN exp  �  i⟨h, Ax + s⟩M  �  µ(dx)  = exp  �  i⟨h, s⟩M  � �  RN exp  �  i⟨h, Ax⟩M  �  µ(dx)  = exp  �  i⟨h, s⟩M  � �  RN exp  �  i⟨ATh, x⟩N  �  µ(dx)  = exp  �  i⟨h, s⟩M  �  exp  �  i⟨ATh, c⟩N − 1  2⟨QATh, ATh⟩N  �  = exp  �  i⟨h, s⟩M  �  exp  �  i⟨h, Ac⟩M − 1  2⟨AQATh, h⟩M  �  = exp  �  i⟨h, Ac + s⟩M − 1  2⟨AQATh, h⟩M  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
+page_content='  6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfBwmo/content/2301.04270v1.pdf'}
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+Reinforcement learning and decision making via single-photon quantum walks
+Fulvio Flamini,1, ∗ Marius Krumm,1, ∗ Lukas J. Fiderer,1 Thomas Müller,2 and Hans J. Briegel1
+1Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria
+2Department of Philosophy, University of Konstanz, Universitätsstraße 10, 78464 Konstanz, Germany
+Variational quantum algorithms represent a promising approach to quantum machine learning where classi-
+cal neural networks are replaced by parametrized quantum circuits. Here, we present a variational approach to
+quantize projective simulation (PS), a reinforcement learning model aimed at interpretable artificial intelligence.
+Decision making in PS is modeled as a random walk on a graph describing the agent’s memory. To implement
+the quantized model, we consider quantum walks of single photons in a lattice of tunable Mach-Zehnder in-
+terferometers. We propose variational algorithms tailored to reinforcement learning tasks, and we show, using
+an example from transfer learning, that the quantized PS learning model can outperform its classical counter-
+part. Finally, we discuss the role of quantum interference for training and decision making, paving the way for
+realizations of interpretable quantum learning agents.
+INTRODUCTION
+Machine learning methods are revolutionizing science and
+technology, with applications ranging from drug discovery
+[1] to the study of quantum phases of matter [2].
+Within
+this toolbox, reinforcement learning (RL) has demonstrated
+the capacity to solve tasks better than the human state of the
+art [3]. Currently, the leading method in machine learning
+is given by artificial neural networks, with performance be-
+yond theoretical expectations. The main near-term approach
+to quantum machine learning replaces these artificial neural
+networks with variational quantum circuits, typically consist-
+ing of fixed two-qubit gates and parametrized Pauli rotations.
+Popular platforms include, e.g., superconducting circuits [4]
+and ion traps [5]. Another promising platform is offered by
+photonic circuits, consisting of a fixed, regular mesh of tun-
+able Mach-Zehnder interferometers [6, 7].
+In this paper, we propose a variational approach to quantum
+RL based on linear-optical interferometers and on a classical
+learning model called projective simulation (PS) [8, 9]. In PS,
+the agent’s decision making process is modeled as a random
+walk of an excitation in an episodic memory. Following an
+approach already investigated by some of the authors [10], we
+replace this classical random walk with a single-photon quan-
+tum walk in an optical interferometer. On the one hand, we
+are able to preserve the core features of classical PS, so that
+the main ideas developed in the field can be transferred to the
+quantum domain. On the other hand, this approach can be
+tailored to end applications, with actual constraints and needs
+in mind. These two points link to the main motivation of this
+work, that is, the development of a learning agent that operates
+in distributed quantum optical networks [11], and that can in-
+teract with other agents by exchanging quantum information
+encoded in light.
+The article is structured as follows: In Sec. I, we review
+the standard PS model that provides the background of our
+work. In Sec. II, we present our approach to quantizing PS
+and, based on that, to designing a photonic implementation.
+∗ These authors contributed equally to this work.
+marius.krumm@uibk.ac.at
+In Sec. III, we introduce two variational training algorithms
+tailored to the proposed framework, whose applicability goes
+beyond the scope of this work. In Sec. IV, we provide nu-
+merical evidence that such quantum PS agents can generalize
+better than their classical counterparts. Most of our analyses
+can be directly tested with a package we make available on-
+line [12] (see App. A), and can be further extended in various
+directions (using the same package; see Apps. B-D).
+I.
+PROJECTIVE SIMULATION
+In this section, we review the basics of RL and PS. RL is
+a paradigm of machine learning that involves an agent inter-
+acting with an environment (see Fig. 1). The agent receives
+from the environment an input s ∈ S, called state or percept,
+and decides what action a ∈ A to take next. RL typically
+considers Markov decision problems, for which the agent’s
+policy can be described by a conditional probability distribu-
+tion π(a|s). The agent receives immediate feedback on its
+actions through a reward R; its goal is to find the policy that
+maximizes the long-term expected reward.
+In this work we focus on PS, a physically motivated RL
+model that has been applied to the study of collective behav-
+ior [13], robotics [14], automated experiment design [15] and
+quantum error correction [16]. PS relies on a memory struc-
+ture called episodic and compositional memory (ECM), de-
+scribed by a directed graph (see Fig. 1). The vertices of this
+graph, called clips, are associated with memories of the agent
+such as remembered percepts, actions or intermediate, more
+complex steps in the decision making process. Importantly,
+this means that a clip carries semantic information [17]. When
+the environment delivers a percept s, the corresponding clip is
+excited, and a random walk of a single excitation is initiated
+in its clip network (see Fig. 2), until the excitation hits an ac-
+tion clip a. Each directed edge from clip ci to clip cj is asso-
+ciated with an unnormalized transition probability hij called
+h-value. The transition probability p(cj|ci) is computed by
+normalizing over all clips that can be reached from ci:
+pij ≡ p(cj|ci) =
+hij
+�
+j′ hij′ .
+(1)
+arXiv:2301.13669v1  [quant-ph]  31 Jan 2023
+
+2
+a
+PERCEPT
+REWARD
+ACTION
+PS agent
+𝑐2
+𝑐𝑘
+1
+2
+k
+Ƹ𝑐1
+†
+1
+2
+k
+𝑈𝑘
+Environment
+b
+c
+𝑠
+𝑎
+ℎ
+(𝑠)
+(𝑅)
+(𝑎)
+𝑐4
+𝑐5
+ℎ2𝑘
+Ƹ𝑐2
+†
+Ƹ𝑐𝑘
+†
+Ƹ𝑐1
+†
+Ƹ𝑐2
+†
+Ƹ𝑐𝑘
+†
+𝑐3
+ℎ1𝑘
+𝑐1
+FIG. 1:
+Projective Simulation (PS). This work uses PS as a frame-
+work for reinforcement learning and its quantization. a) When a PS agent
+receives a percept s from the environment, it initiates a random walk in the
+graph that describes its memory, until it hits a vertex representing an action
+a. The performed action may be rewarded, leading to an update of the pa-
+rameters (h-values) that control its decision making process. b) In classical
+PS, the vertices of the graph represent clips (ck). Directed edges are asso-
+ciated with transition probabilities that depend on the h-values. c) In the
+quantized PS, clips are represented by mode operators ˆc†
+k describing a single
+excitation, while the connectivity is governed by mode-mixing transforma-
+tions Uk. Each Uk acts on the mode operators associated with the parent
+clips of ck (here c1, c2), and its complex elements are the transition ampli-
+tudes that replace the classical probabilities of transitioning to ck. All panels:
+all edges that contributed to the decision making process are in green.
+Learning occurs by tuning the h-values, which are initially set
+to 1. To this end, each edge with h-value hij is additionally
+equipped with a so-called glow factor gij, which contributes
+to the standard PS update rule [18]:
+h(t+1)
+ij
+= 1 + (1 − γ)(h(t)
+ij − 1) + g(t)
+ij R.
+(2)
+Here, the role of the glow factor gij is to remember transitions
+(ci → cj) taken in past interactions with the environment, so
+that they are reinforced when a reward R is received. Further,
+γ ∈ [0, 1] is a forgetting factor: its main purpose is to gradu-
+ally dampen previously learned transition probabilities, to let
+the agent adapt to changing environments. γ is also used as
+a soft cut-off to prevent the h-values to overflow. Each gij is
+updated as:
+g(t+1)
+ij
+=
+�
+1
+if ci → cj was taken at t,
+(1 − η) g(t)
+ij
+otherwise,
+(3)
+where g(0)
+ij
+= 0 and η ∈ [0, 1] is a discount factor for the
+actions taken further in the past. We refer to the literature on
+PS for a more detailed description and additional mechanisms,
+such as association [19] and reflection [20, 21].
+II.
+QUANTUM PROJECTIVE SIMULATION
+Since decision making in PS is generated via a random walk
+of an excitation, it is possible to quantize it by giving learning
+agents access to a quantum walk in a quantum memory. In the
+literature, several variants of this idea have been investigated.
+In Ref. [20], an algorithm based on discrete quantum walks
+was presented which provides a quadratic speedup. However,
+this algorithm relies on sophisticated oracles that likely will
+not be available in the near future. Furthermore, since the al-
+gorithm is described as a quantum circuit, it is not clear how it
+can be interpreted as a random walk of a physical excitation.
+On the other hand, in Ref. [8], it was proposed to quantize
+PS by considering a quantum walk of an excitation in a quan-
+tum many-body system. Since Hamiltonians are hermitian,
+in such systems the probability amplitude of a transition is as
+large as the amplitude of the backwards transition. To break
+this symmetry, the authors proposed to use dissipative effects
+(‘quantum jumps’), at the cost of coherence.
+In this section, we propose a quantum extension of PS that
+allows for an optical implementation. Our guiding principle
+is to design a PS agent natively as a quantum walk of a physi-
+cal excitation. In particular, our proposal does not rely on the
+availability of quantum oracles. Our scheme uses an optical
+interferometer that induces a natural directionality of transi-
+tions, without the need for dissipation. We will first describe
+how to implement the main elements of PS in the quantum do-
+main in general and then, in Sec. II B, on an optical platform.
+A.
+From classical to quantum projective simulation
+In order to quantize PS and facilitate an optical implemen-
+tation, we need to quantize its main elements: the episodic
+memory, realized by the clip network, and the decision-
+making process, described by the propagation of an excita-
+tion through the clip network. A physical realization of a clip
+network could be a system of harmonic oscillators (modes)
+with non-trivial many-body interactions between sets of oscil-
+lators [8]. In case of an optical implementation, clips ci ∈ C
+(including percept clips, action clips and intermediate clips)
+then correspond to modes in an optical interferometric net-
+work [10] (see Sec. II B). For the quantization, we associate
+with each clip ci a pair of creation and annihilation operators
+of the corresponding (optical) mode, {ˆc†
+i, ˆci}. An excitation
+of memory clip ci at time τ0 is then described by the quantum
+state ˆc†
+i(τ0) |vac⟩, where |vac⟩ denotes the vacuum state, de-
+scribing the inactive state of the agent’s memory. Here we use
+the Heisenberg picture where the operators carry the agent’s
+internal time dependence. Note that we index RL steps with
+t, while physical time within a step is denoted by τk.
+A decision-making process is initiated by the excitation of
+some clip ci at time τ0.
+The subsequent evolution of the
+agent’s memory will be described by a unitary of the form
+ˆc†
+j(τn) = U†(τn, τ0)ˆc†
+j(τ0)U(τn, τ0) =
+|C|
+�
+i=1
+U ECM
+ji
+ˆc†
+i(τ0) ,
+(4)
+which relates the excitation operators of the clip network at
+different times. The Heisenberg time evolution of the ECM
+U(τn, τ0) is thus realized by the interferometric scattering ma-
+trix U ECM connecting the modes, which, in turn, is composed
+
+3
+റ𝜃1
+𝑠1…|𝑆|
+Classical ECM
+Quantum ECM
+a
+𝑐4
+𝑎1
+𝑠2
+𝑐5
+𝑎2
+𝑐3
+𝑐2
+𝑐1
+b
+c
+𝑐1…|𝐼|
+𝑎1…|𝐴|
+𝑠1
+𝑠2
+𝑐1
+𝑐2
+𝑎1
+𝑎2
+𝑐3
+𝑐4
+𝑐5
+𝑠1
+𝑠2
+𝑐1
+𝑐2
+𝑎1
+𝑎2
+𝑐3
+𝑐4
+𝑐5
+𝑈1
+𝑈2
+𝑈𝑛
+𝑈1
+𝑈3
+𝑈4
+𝑈5
+𝑈6
+𝑈7
+ℎ15
+ℎ12
+ℎ11
+…
+𝑠1…|𝑆|
+𝑐1…|𝐼|
+റ𝜃2
+റ𝜃𝑛
+𝑈2
+𝑎1…|𝐴|
+𝑠1
+FIG. 2: Decision making in classical and quantum projective simulation (PS). a) In PS, decision making occurs via a classical random walk over a
+network of clips {c1, . . . , c|C|} that describes the agent’s episodic and compositional memory (ECM) [17]. When the agent receives a percept s (blue), an
+excitation triggers and performs a random walk over the network of intermediate clips {c1, . . . , c|I|} (green) until it hits one of the action clips {a1, . . . , a|A|}
+(red), which is immediately coupled out. b) In the proposed quantized PS, these random walks become quantum walks of single excitations through layers
+k = (1, . . . , n) implementing the transformations Uk, which mix sets of clips in a certain order (see App. H). Here, clip-to-clip transition probabilities,
+classically described by the h-values, are replaced by transition amplitudes described by the matrix elements of Uk. c) The evolution can be seen as hopping
+between eigenstates (of the modes) in the Hilbert space (empty disks: in/out modes; filled disks: output modes). This panel reproduces the quantized evolution
+in the classical ECM in panel (a). Clips are now visited in superposition (green).
+of a sequence of n intermediate transformations U k,
+U ECM(⃗θ) =
+n
+�
+k=1
+U k(⃗θ) .
+(5)
+The U k are updated during the learning process via the set of
+parameters ⃗θ (see Fig. 2). The precise role of the U k(⃗θ) will
+be discussed in more detail in the following. For convenience
+and to facilitate an optical implementation, in this work we
+will consider ECMs that are described by a directed acyclic
+graph (DAG; see App. H).
+Percept encoding — When the agent receives a classical
+percept at time τ0, this leads to the excitation of a percept
+clip s. The quantum state of the memory is then described by
+c†
+s(τ0) |vac⟩. Subsequently, a quantum walk through the clip
+network starts [8].
+Quantum walk in the ECM — To connect the random walk
+in the ECM to the quantum walk considered in this work, we
+refer to the examples in Figs. 1 and 2. As shown in panel
+Fig. 1b, a clip effectively acts as a node that routes incoming
+edges from sets of clips to other clips in the next step of the
+random walk. In the quantum ECM sketched in Fig. 1c and
+Fig. 2b-c, this mechanism translates into the application of a
+transformation U k, which mixes creation operators of incom-
+ing clips with ˆc†
+k. A general clip-to-clip transition ci → cj is
+then modeled by transformations U k that mix subsets of cre-
+ation operators, so that the stochastic nature of PS manifests
+itself at the level of quantum amplitudes.
+A crucial departure from classical PS is that clips are now
+visited in parallel (in superposition, see Fig. 1c), and all re-
+gions of the ECM that are reachable from a percept via a se-
+quence of U k contribute to the decision making process. As
+a consequence, the probability to find an excitation in action
+clip a at time τn (described by the quantum state c†
+a(τn) |vac⟩)
+when starting with an excitation of percept clip s at time τ0
+(described by c†
+s(τ0) |vac⟩) is given by the modulus square of
+the corresponding transition amplitude
+psa = |⟨a|s⟩|2 ≡ |⟨a; τn|s; τ0⟩|2 =
+���U ECM
+as
+(⃗θ)
+���
+2
+,
+(6)
+where |ci; τk⟩ = ˆc†
+i(τk) |vac⟩ are the time-dependent eigen-
+states of the number operators ˆc†
+i(τk)ˆci(τk) corresponding to
+an excitation being present in clip ci at time τk. Eq. (6) entails
+a quantum interference of all paths. While in classical PS the
+excitation has a well-defined trajectory in the ECM, now the
+excitation can be delocalized over all the allowed modes (i.e.,
+the quantum amplitude is in general non-zero for more than
+one mode at the same time). This aspect bears an interesting
+potential for the learning model, which has already been
+discussed in the literature [8, 22] and that is further discussed
+in Secs. III-IV and in the Appendix.
+Action decoding — The decision-making process ends
+when U n is applied and the output states are measured. When
+the excitation is detected in an output state associated with an
+action, this action is coupled out. In all other cases, the exci-
+tation can be (i) routed back to other regions of the reachable
+ECM; (ii) discarded, exciting again the same percept. Solu-
+tion (ii) can be interpreted as the agent’s capacity to not take
+action during this decision-making process (see also the no-
+tion of suspension of judgment [23]).
+Practically, the agent takes action by post-selecting over
+the outcomes that correspond to actions. In general, one can
+associate multiple output states with a single action.
+This
+makes it possible to avoid post-selection in settings with an
+action space smaller than the percept space, or to implement
+quantum channels that are more general than unitaries.
+Training the quantum PS agent — To train the agent, we
+consider a variational approach based on a loss function L that
+is inspired by classical PS and aims to reproduce its learning
+mechanism. L depends on the probabilities psa(⃗θ) to take ac-
+tion a given a percept s (that is, the policy) and the variational
+parameters ⃗θ. These probabilities can be estimated either nu-
+merically (by simulating the quantum walk) or experimentally
+
+4
+(via multiple measurements with fixed input). In Sec. III A
+we introduce and motivate our choice for L (see Eq. 8), along
+with several modifications and training methods.
+B.
+Quantum optical projective simulation
+In this section, we describe our approach to implement
+quantum PS on a photonic platform. Table I summarizes the
+main connections with classical PS.
+Related works — Photonic circuits provide a promising
+platform in both classical and quantum domains [6, 7], with
+results of increasing complexity being continuously reported
+[24, 25], in particular for (un)supervised learning [26–32] and
+RL [10, 22] based on path encoding.
+In this work, we focus on RL and aim to design a quantum
+PS agent that operates with a native quantum walk of physical
+excitations.
+Hence, we will replace the single excitation
+with a single photon propagating in a linear-optical circuit.
+This approach, while retracing some of the intuitions in Ref.
+[10] where the psa were reproduced by optical binary trees,
+promises the following advantages: (1) it requires only a
+single (possibly larger) circuit for all percepts, instead of one
+binary tree per percept and layer, with no explicit need for fast
+active switching; (2) it allows to take advantage of quantum
+coherence and interference in the decision making process;
+(3) it allows for more advanced training algorithms, which are
+applicable beyond the scope of this work (see Sec. III); (4)
+it suggests concrete directions for further developments (see
+Apps. B-D), such as a framework for learning with quantum
+data. In the following, we outline how the quantized version
+of PS can be realized on a photonic circuit.
+Photonic architecture — For the implementation of PS on
+quantum hardware, we want to identify quantum extensions of
+the Markov chain process that govern the evolution of a clas-
+sical excitation. These extensions, usually referred to as quan-
+tum channels (completely positive trace-preserving maps),
+would have to reproduce the dynamics in classical PS as a
+limiting case. A direct implementation of classical PS would
+require the {U k} to reproduce the stochastic maps that form
+the random walk in the ECM graph described in Sec. I. Specif-
+ically, the real-valued matrices P associated with these trans-
+formations need to (i) reproduce the probabilities pij induced
+by the h-values, and (ii) fulfill the connectivity in the ECM
+(P can be seen as a weighted adjacency matrix). Such a class
+of transformations can in principle be realized on a photonic
+platform by recalling that any real-valued matrix P can be de-
+composed as P = V1ΣV †
+2 via singular value decomposition,
+where Σ is a diagonal matrix with non-negative real entries
+(i.e., optical attenuators) and V1,2 are unitary transformations
+[26]. Hence, the architecture we use for the variational circuit
+is the canonical square one [33], since it can implement any
+unitary transformation by suitably tuning its phase shifters.
+Indeed, we recall that any such U k can be implemented by
+a homogeneous mesh of Mach-Zehnder interferometers, each
+comprising two 50:50 beam splitters and two phase shifters
+𝜃2
+𝑐1…|𝐼|
+𝑎1…|𝐴|
+50:50
+50:50
+1
+2
+3
+4
+1
+2
+3
+4
+𝜃1
+𝜃2
+π
+0
+𝑠1…|𝑆|
+𝑐1…|𝐼|
+𝑎1…|𝐴|
+𝑠1…|𝑆|
+𝜃1
+a
+d
+b
+θ
+→
+Tunable phase
+Tunable MZI
+c
+Iden�ty
+FIG. 3: Optical implementation of PS. To enable PS on photonic cir-
+cuits, we use as variational ansatz the square architecture [33]. a) The update
+of the agent’s memory translates into an update of the phase shifters ⃗θ. b)
+Any unitary U can be implemented with a regular mesh of Mach-Zehnder
+interferometers (MZI), consisting of two 50:50 beam splitters (BS, blue) and
+two phase shifters (θ1, θ2) [33, 34]. c) When θ = π all light entering a MZI
+is reflected (bar state), while when θ = 0 it is transmitted (cross state). d)
+Square, universal architectures can be carved inside larger ones by tuning the
+phases around them. In this panel, a universal 4 × 4 circuit (in/out modes are
+labeled) is carved in a larger mesh. Tilted elements represent tunable phase
+shifters (those inside the carved structure are in fuchsia). BSs with arbitrary
+transmissivities (orange) are implemented using tunable MZIs. Light can be
+confined to the pink waveguides by setting the green optical elements.
+(see Fig. 3 and App. F) [33, 34]. The variational parameters ⃗θ
+correspond to the phase settings that control U k.
+In the fully coherent case, without additional attenuators,
+one can set Σ = I in the singular value decomposition,
+meaning that the U k become unitary transformations.
+In
+this case, due to unitarity constraints and since each U k
+acts as an adjacency matrix, additional edges between clips
+are created that are not present in the original ECM. This
+limitation represents an additional departure from classical
+PS, which can open up interesting opportunities in the context
+of learning, for instance leveraging the constraints to improve
+generalization.
+Percept encoding — We associate each percept with a
+photon input in a spatial mode of the circuit. The size of the
+Hilbert space grows linearly in the number of percepts.
+ECM structure — We associate each clip with one optical
+mode of the architecture. The unnormalized transition prob-
+abilities hij that describe the transitions ci → cj are now re-
+placed by transition amplitudes between optical modes, de-
+scribed by a complex amplitude U k
+ji.
+Our chosen architecture is especially useful for reasons that
+go beyond its universality, noise resilience and compactness
+
+5
+Element
+Classical PS
+Quantum PS (→ Optical implementation)
+Decision making
+Random walk of one excitation
+Quantum walk of a single excitation (photon)
+Agent’s memory
+Directed graph of clips
+Variational (optical) circuit
+Clips
+Vertices of the graph
+(Optical) mode operators
+Clip-to-clip transitions
+h-values
+Quantum scattering amplitudes
+Input/Output encoding
+Percept/Action clips
+Input/Output modes
+Learning mechanism
+Update of h-values via Eq. (2)
+Update of variational parameters (phase shifters) via Eq. (8)
+TABLE I: Links between the main elements of projective simulation (PS) and its proposed optical quantization.
+[33], since it also promises to support key features of PS:
+• Clip creation/deletion: new clips (i.e., optical modes)
+can be added during the learning process, while clips
+that are rarely traversed can be removed.
+• Edge creation/deletion: the random walk can be manip-
+ulated by creating or deleting links between clips.
+• Clip composition: fictitious clips can also be created, by
+random variation or merging of existing clips.
+We observe that this mesh allows one to carve substructures
+that are in turn universal for unitary transformations, by
+adjusting the surrounding phases (without the need to modify
+the hardware; see Fig. 3d).
+This means that the Hilbert
+space describing the ECM can be dynamically enlarged
+to accommodate new clips, and the U k
+ji can be controlled
+by intervening on the regions of the circuit that implement U k.
+Action decoding — In PS, the decision-making process
+ends when an action clip is hit. In the proposed implemen-
+tation, this corresponds to detecting photons from one of the
+output modes that correspond to an action clip. Whenever a
+photon is detected in a mode that does not correspond to an ac-
+tion, that event is discarded. In the future, these photons could
+be routed back into the circuit, or associated with mechanisms
+of suspension of belief [23].
+III.
+TRAINING THE ARCHITECTURE
+In this section, we discuss two algorithms to train the quan-
+tum optical PS agent. To this end, we first recall that the stan-
+dard update rule in PS has no straightforward quantum ana-
+logue: while only one path is taken by the classical excitation
+and rewarded by the update rule (technically, the h-values as-
+sociated with its edges), all paths from percept to action are
+visited by the quantum excitation, and the very notion of path
+loses its meaning. Hence, also motivated by the advances in
+quantum machine learning, here we take a different, varia-
+tional approach to update the parameters ⃗θ.
+A.
+Variational approach to mimic classical PS
+The first algorithm to train the quantum PS agent is based
+on a variational optimization of a loss function L inspired by
+classical standard PS.
+Since a quantum circuit does not have access to the inter-
+mediate states of the computation, the update rule we propose
+adapts that of a classical 2-layer ECM. This does not mean
+that the quantum agent is necessarily 2-layered (in fact, the
+architecture supports multi-layer ECMs, as shown in Fig. 2);
+rather, it means that we update the observable transition prob-
+abilities psa(⃗θ). In turn, also the intermediate transition am-
+plitudes in the circuit are affected. However, the update rule of
+PS requires the use of unnormalized h-values instead of (nor-
+malized) probabilities. Thus, let us first introduce the classical
+normalization factor of the h-values hs, i.e. hs := �
+a hsa,
+from which we get hsa = psahs and the classical 2-layer PS
+update rule
+p(t+1)
+sa
+(⃗θ)h(t+1)
+s
+= 1 + (1 − γ)
+�
+p(t)
+sa h(t)
+s
+− 1
+�
++ g(t)
+sa R. (7)
+A loss function L that implements Eq. (7) is
+L(⃗θ(t+1),⃗h(t+1)
+s
+) =
+�
+s
+D
+�
+p(t+1)
+sa
+(⃗θ)h(t+1)
+s
+− 1, (1 − γ)
+�
+p(t)
+sa h(t)
+s
+− 1
+�
++ g(t)
+sa R
+�
+(8)
+Here, ⃗θ(t+1) and ⃗h(t+1)
+s
+are variational parameters, while D
+is a suitable distance measure such as the Kullback-Leibler
+divergence (after rearranging the arguments of D), R is the
+reward, and gsa is the glow factor described in Eq. (3). The
+expression in Eq. (3) is still valid in the proposed quantum
+extension, since the update rule underlying the loss function
+is taken from a classical two-layer PS, and both input (s) and
+output (a) are classical. However, we emphasize again that
+the quantum ECM itself is multi-layered.
+Note that only the first argument of D in Eq. (8) contains
+tunable parameters, while the right entry contains the target
+values calculated from the previous state of the agent, as a
+
+6
+reference, similar to deep-Q learning. Similarly, one can use
+experience replay [35] to collect rewards and glows for a batch
+of percept-action pairs, and add up or merge the loss functions
+in appropriate ways. That is because continuous online learn-
+ing (i.e. updating after each action) has the disadvantage that
+the unitarity constraints will also affect the transition proba-
+bilities of undetected percept-action pairs, without taking their
+rewards into account.
+In the following, we discuss suitable heuristic simplifica-
+tions of this loss function, which make it more amenable to an
+actual implementation. First, by approximating h(t)
+s
+≈ hs and
+introducing an effective reward rs :=
+R
+hs in the exact update
+rule in Eq. (7), we obtain
+p(t+1)
+sa
+(⃗θ) − 1
+hs
+≈ (1 − γ)
+�
+p(t)
+sa − 1
+hs
+�
++ g(t)
+sa rs.
+(9)
+Without a reward (i.e. rs = 0), the forgetting mechanism
+after the approximation seeks to reset p(t)
+sa to
+1
+hs . However,
+in information theory, the no-information distribution is the
+uniform distribution. This motivates us to replace
+1
+hs with a
+uniform distribution
+1
+|A|,
+p(t+1)
+sa
+(⃗θ) =
+1
+|A| + (1 − γ)
+�
+p(t)
+sa − 1
+|A|
+�
++ g(t)
+sa rs.
+(10)
+So far, we did not make any choices concerning the source of
+the rewards R. For our simplifications, we now take the point
+of view that the reward R is designed by a programmer to
+achieve an intended agent behavior. Then, a further simplifi-
+cation can be made by directly modelling the effective reward
+r, rather than R. Therefore, we replace rs with r. With this
+change in mind, Eq. (10) now looks like an analogue of the
+exact update rule in Eq. (7), but for probabilities instead of
+h-values. To ensure that the agent updates get smaller over
+time, it is recommended to use an “annealing schedule” for
+the reward, i.e. to put an additional time dependence into the
+reward of the form r := f(t) · r0. Here, f(t) is a function that
+monotonically decreases from 1 to 0. This allows the training
+algorithm to make finer and finer updates to the probabilities.
+In order to stay as close as possible to classical PS, the up-
+date rule would have to be applied to all percept-action pairs.
+This would require implementing the forgetting mechanism
+(γ > 0) for unvisited percept-action pairs, or enforcing the
+transition probability of unvisited pairs to remain unperturbed
+(γ = 0), leading to a complicated loss function. To avoid
+this overhead, we include only the actually observed percept-
+action pairs in the loss function, while the transition probabil-
+ities of unobserved pairs are allowed to change according to
+the unitarity constraints. Combining all of the above simplifi-
+cations, we obtain the loss function
+L(⃗θ) =
+�
+(s,a) | g(t)
+sa >0
+D
+�
+p(t+1)
+sa
+(⃗θ) ,
+(11)
+C
+� 1
+|A| + (1 − γ)
+�
+p(t)
+sa − 1
+|A|
+�
++ g(t)
+sa r
+� �
+where C is a (smooth) cutoff function to enforce that proba-
+bilities are in [0, 1], and only (s, a)-pairs with non-zero glow
+are considered.
+B.
+Variational approach based on causal diamonds
+In the previous section, we implicitly assume that at each
+step one tunes all phase shifters (i.e., parameters ⃗θ) in the cir-
+cuit. Indeed, this is possible and can be beneficial to train the
+agent faster and with a better performance. At the same time,
+updating all phases at each step might not be necessary for the
+following reason.
+In a square architecture with |C| input/output modes, the
+number of phase shifters is quadratic in |C|. However, for a
+fixed transition probability psa(⃗θ), one can see that only some
+phases play a role: Those at the intersection of the future and
+past light cones of s and a, respectively (see Fig. 4a). In the
+literature on relativity, this intersection is called causal dia-
+mond. A number of considerations follow:
+i. To update psa(⃗θ), one could simply tune the phase
+shifters inside the causal diamond.
+ii. For light to reach a from s, what matters is that light
+does not leak out of their causal diamond. Hence, one
+can also just tune the O(|C|) phases that lie on the sur-
+face of the causal diamond.
+iii. Since leakage only occurs on the surface of the past
+light cone, it is sufficient to focus on its intersection
+with the causal diamond (see Fig. 4). We call these
+phase shifters leaking nodes.
+Overall, in terms of reconfiguration time, stability and
+power consumption, the above observations allow for progres-
+sively simpler experimental requirements. The price to pay is
+that the surfaces of different causal diamonds (for different
+(s, a) pairs) intersect and, for this reason and due to unitarity
+constraints, updating one transition probability also affects the
+other transitions. Also, the fewer phase shifters are tuned, the
+smaller the set of values that solve the optimization problem
+when multiple (s, a) pairs are rewarded. The desired trade-off
+can be found with the experimental conditions in mind.
+We consider two main strategies to tune the sets of relevant
+phases described by points (i-iii) above: (a) phases are in-
+dependently updated by gradient descent, and (b) phases are
+updated sequentially (from left to right, each phase in a layer
+is set to a new value before the phases in the next layers are
+adjusted). The latter strategy is possible because we assume
+that light propagates only forward, and it leads to a faster and
+smoother training.
+A quantitative comparison between the
+two strategies, and a study of the most suitable one with multi-
+ple rewarded percept/action pairs, can be done with our shared
+package, keeping experimental requirements in mind [12]. An
+illustrative analysis for strategy (b) is shown in Fig. 4c. Addi-
+tional information can be found in App. E 2.
+Importantly, we point out that this approach is not restricted
+to the regular mesh considered in this work. Rather, similar
+considerations also apply to circuits with arbitrary mode con-
+nectivity and even higher-level structures in the ECM [12].
+
+7
+a
+c
+b
+Steps
+psa
+0
+250
+500
+0
+0.25
+0.5
+0.75
+1
+FIG. 4: Update via causal diamonds. To reinforce a transition probability psa in a 2-layer PS, it is not necessary to update all phase shifters at each step.
+a) Only the phases inside the causal diamond of (s, a) influence psa (colored nodes indicate the surface of the causal diamond). Since light can only leave the
+causal diamond from the purple nodes (leaking nodes), it is sufficient to tune their phases. b) The approach works also with multiple transitions, by iteratively
+tuning different sets of leaking nodes (here shown with five different colors, including gray for the overlaps, and representing pairs of adjacent phase shifters
+and beam splitters as a single node), even if the updates compete with each other (some leaking nodes belong to multiple causal diamonds). c) The agent learns
+to maximize psa for the four (s, a) pairs in panel (b), by tuning only the leaking nodes. In some cases, it may be necessary to tune more nodes to maximize all
+probabilities. Each percept/action is encoded in a pair of adjacent input/output modes.
+C.
+Direct updating via a Gram-Schmidt process
+The previous sections consider a variational approach to the
+optimization problem, where phases are gradually adjusted
+to produce a new transformation that behaves better under a
+given figure of merit. We now consider a complementary ap-
+proach, where we first update the unitary evolution U of the
+ECM, and then use a well-known decomposition [33] to re-
+trieve the ⃗θ that produce the desired U. In this case, one di-
+rectly adjusts the matrix elements of U to implement an up-
+date rule such as Eqs. (7) or (11): the new matrix is no longer
+unitary; however, a Gram-Schmidt orthonormalization (GSO)
+process yields a new U ′ that approximates the intended U ′
+as.
+We add a few remarks on this approach (see also App. E 1).
+First, if U ′
+as is close to the old value Uas, one step of GSO
+only causes a small update of the other entries of U. In this
+sense, this approach allows for a smooth, flexible and con-
+trolled update of U (see Fig. 5). Also, both the GSO and the
+unitary decomposition are efficient and relatively fast subrou-
+tines, which could even be faster than collecting statistics for
+other variational approaches. In other words, the overhead
+is outsourced to the classical CPU. The limitation of this ap-
+proach is that it assumes perfect knowledge of U and a precise
+control of the experimental settings. The impact of imperfec-
+tions can be estimated using the shared package [12].
+IV.
+QUANTUM VERSUS CLASSICAL PROJECTIVE
+SIMULATION IN A TRANSFER-LEARNING SCENARIO
+In this section, we apply the proposed quantum PS to an
+actual learning problem.
+The PS update rule of Eqs.
+(7)
+and (8) has already been used in various applications [13–
+16]. Here, we consider a scenario which pronounces the dif-
+ferences between classical and quantum PS. We consider a
+transfer-learning scenario [36]: a machine learning setting
+where a model trained for one task can solve another task.
+The scenario we choose is that of Eva et al. [37] (see Fig. 6).
+0
+25
+50
+75
+0
+0.5
+1
+p  ∼
+Steps
+sa
+p
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+FIG. 5: Update via a Gram-Schmidt process. We can directly update
+the unitary transformation U that describes the ECM using a Gram-Schmidt
+orthonormalization. Here, a learning curve (purple) is shown for a single
+rewarded (s, a) pair in a 2-layer PS described by a 10 × 10 Haar-random U.
+All other transition probabilities (other colors), from s and to a, decrease to
+0. Curves correspond to a single agent, without average. Inset: (|Uij|2)ij
+after training (arrows indicate (s, a)).
+In this scenario, one considers a list of particles character-
+ized by discretized observables, and asks whether two observ-
+ables have two particular values. To answer these questions,
+or tasks, in our adapted version of this thought experiment the
+agent attempts to build a meaningful representation of the par-
+ticles in two stages (corresponding to the middle layer and a
+task layer, see Fig. 6a): In stage 1, the agent learns a represen-
+tation from the percept layer to the middle layer, where each
+node corresponds to one observable and one value of that ob-
+servable. While Eva et al. intended that this representation
+is learned implicitly, in our case we directly train the middle
+layer to achieve this representation. In stage 2, the connec-
+tions from percept to middle layer are kept fixed. The task
+layer has two nodes, which correspond to yes/no-answers. For
+each question (task), we train a separate set of connections
+from the middle layer to the corresponding task layer.
+
+8
+a
+c
+?
+?
+?
+1,2,3
+1,2,3
+1,2,3
+Percept 𝑠
+Values
+Tasks
+?
+?
+Values
+1 … 3
+Not used
+Yes
+No
+𝑠
+𝑈
+Tasks
+1 … 3
+1 … 3
+𝑠
+𝑣
+𝑒
+Value
+(27)
+b
+(9)
+d
+Aq
+c
+...
+...
+Experiments
+1
+27
+0.85
+0.9
+0.95
+1
+Steps
+Aq
+A
+FIG. 6: A transfer-learning scenario for PS agents. a) Structure of the
+ideal, classical ECM considered in the transfer-learning scenario in Sec. IV.
+Panel adapted from Ref. [37]. b) Percepts s consist of the values of three ob-
+servables which characterize particles. The experiment consists of two stages.
+(i) In the first stage, the PS agent is set up to learn a middle layer of nodes,
+in which each node corresponds to one observable and one value of that ob-
+servable. (ii) In the second stage, this middle layer is kept fixed. The task
+layer is exchangeable and it represents one yes/no question about pairs of ob-
+servables. Since the excitation of the classical ECM is localized, in the middle
+layer the classical agent can only use the value of one observable. Interference
+allows the quantum PS agent to combine knowledge of several observables at
+once. c) Layout of the quantum PS agent. To fulfill the unitarity constraints,
+the connection from percept to middle layer is an interferometer tree with 9
+outputs (one tree per percept). The connection from the middle layer to the
+task layer is a 9 × 9 square architecture, shared by all percepts but different
+for each task layer. Two output modes correspond to the yes/no answers, the
+others are discarded. d) Prediction accuracy AQ of the quantum PS agent
+for each of the 27 experiments (yes/no question). Inset: Learning curve for
+a single experiment (dark orange). A classical agent can achieve at most an
+accuracy AC = 8/9 (horizontal line, for both plots), always lower than AQ.
+The contrast between classical and quantum learning is
+rooted in the way the decision-making process handles the
+structured representation of the PS agents: Since the exci-
+tation of the classical PS agent is localized, it can only use
+knowledge about one observable and, therefore, it cannot
+reliably answer questions involving two observables.
+The
+quantum PS agent, however, can use interference to combine
+knowledge about two observables.
+To be more specific, Eva et al. [37] consider three observ-
+ables, each of which can take three values. We label these
+observables and their values 0, 1, 2. Each percept can be writ-
+ten as (v0, v1, v2) with vj ∈ {0, 1, 2}, and vj is the value of
+observable Oj. In total, there are 33 = 27 percepts. In ad-
+dition to the percept layer, there is a middle layer and a final
+layer (see Fig. 6). The purpose of the middle layer is to explic-
+itly represent the value of each observable. More specifically,
+this layer contains 32 = 9 nodes, each corresponding to one
+observable-value pair. For each percept (v0, v1, v2), the idea is
+that the agent learns to visit the middle clips (Oj, vj) with uni-
+form probability 1
+3, while not visiting the other middle clips
+(Oj, v) corresponding to wrong values v of the observables.
+For
+the
+proposed
+quantum
+PS,
+we
+represent
+the
+correct
+middle-layer
+state
+of
+percept
+(v0, v1, v2)
+as
+1
+√
+3 (|v0⟩ + |3 + v1⟩ + |6 + v2⟩).
+For example,
+(2, 1, 2)
+is represented by
+1
+√
+3 (|2⟩ + |4⟩ + |8⟩). Since these states can
+be non-orthogonal for different percepts, for each percept we
+use a separate binary tree (see Fig. 6b) with 9 output leaves
+(27 such trees in total), instead of the unitaries U k shown in
+Figs. 1 and 2.
+To make a fair comparison to a classical PS agent, we con-
+sider the following training. Each of the 27 percepts is sam-
+pled with uniform probability. Then the middle layer is mea-
+sured to output one (observable, value) pair. Depending on
+whether the pair is predicted right or wrong, a fixed reward
+±r is given. We use the simplified loss function of Eq. (11)
+without forgetting-mechanism (γ = 0), and only reward the
+current percept-action pair (η = 1). To enforce that all relative
+complex phases are close to zero, we add the ℓ1 norm of these
+phases to the loss function. Since we use separate binary trees
+for each percept, and the goal of the tree is just to prepare one
+simple state, the middle layer can be trained to achieve the
+intended representation essentially perfectly.
+After the middle layer is trained, we fix its weights, mean-
+ing that we will not adjust the binary trees for anything that
+follows. Now, we introduce the final layer, which we call the
+task layer. The task layer represents yes-no-questions about
+the values of observables. Our transfer learning scenario re-
+quires that the task layer be exchangeable, while we keep the
+middle layer fixed and independent of the task layer. The task
+layers we consider here pick two observables O, O′ and ask
+whether these observables have specific values vO and vO′.
+There are 27 =
+�3
+2
+�
+· 32 such task layers. The quantum PS
+agent uses a 9x9 square architecture for each task layer, as
+shown in Fig. 6c). We emphasize that the same task unitary
+is used for each percept. Only two output modes are used,
+which correspond to yes/no-answers.
+The 27 percepts are
+sampled uniformly. Fig. 6d) shows the achieved (weighted)
+accuracies for all two-observable task layers. Our agent con-
+sistently achieved over 97% (weighted) accuracy for all task
+layers. The analysis was implemented with PyTorch; details
+are provided in App. G.
+While the proposed quantum PS agent can achieve near per-
+fect transfer learning, one can use analytical reasoning to see
+that the classical PS agent cannot achieve near perfect accu-
+racy: Since the classical excitation is always localized in one
+node, in the middle layer the classical PS agent can at most
+know the value of one relevant observable. Therefore, the best
+thing the agent can do is to guess about the other observable.
+Since two out of three possible values of that other observable
+lead to a right no-answer, the best policy the classical PS agent
+can represent is to always answer no. Hence, the performance
+of the classical PS agent is upper bounded by 8
+9.
+
+9
+DISCUSSION
+In this work, we introduce a variational approach to quan-
+tum reinforcement learning that can be implemented with in-
+tegrated photonic quantum technologies.
+To this end, we
+started from preliminary results by some of the authors [10],
+based on the projective simulation (PS) learning model [8].
+This allowed us to establish a map between the key elements
+of classical PS and its counterparts in the quantum domain.
+At the core of this analogy lies the random walk of a physi-
+cal excitation, which becomes a quantum walk in the agent’s
+memory. We then discussed an approach to implement the de-
+cision making process and training of the quantum PS agent
+using photonic circuits.
+To this end, in Sec. III we introduced several variational
+methods to train the agent, all compatible with, or tailored to,
+a quantum optical implementation. The variational parame-
+ters are represented by the phases in the optical architecture,
+given by the canonical mesh of Mach-Zehnder interferome-
+ters. More specifically, we proposed a variational method that
+is general and can accommodate custom loss functions in-
+side the update rule. In particular, it allows to draw a close
+connection to the update rule of classical PS. We proposed a
+modification of the variational training that exploits features
+of the interferometer implementation to ease the experimental
+requirements (by reducing the number of parameters that need
+to be updated during the learning process), which makes train-
+ing the architecture more efficient and flexible. Furthermore,
+we proposed an alternative method for updating that consists
+of a deterministic and exact algorithm to update the overall
+unitary transformation that describes the quantum PS agent’s
+memory. All algorithms can also be combined to refine the
+update or to address more complex structures in the memory.
+As a specific numerical experiment that demonstrates im-
+portant differences between classical and quantum PS, we
+considered a transfer learning scenario. We show how quan-
+tum interference allows the quantum PS agent to combine in-
+formation from several clips, while the classical PS agent only
+has access to the information of one clip at a time.
+This work also sets the stage for a number of follow-up
+investigations, which we outline in the Appendix with addi-
+tional numerical analyses. In particular, Apps. B and C dis-
+cuss the potential offered by multi-photon (i.e., multi-percept)
+and multi-frequency quantum information processing, respec-
+tively. Finally, App. D takes a first step towards a notion of
+interpretability in quantum artificial intelligence, an issue that
+has attracted more and more attention in the last few years and
+that is especially relevant within PS. All results can be tested
+with a package we release online [12].
+ACKNOWLEDGMENTS
+This project has received funding from the European
+Union’s Horizon 2020 research and innovation program un-
+der the Marie Skłodowska-Curie grant agreement, Grants No.
+801110 and No. 885567. It reflects only the author’s view,
+the EU Agency is not responsible for any use that may be
+made of the information it contains. ESQ has received fund-
+ing from the Austrian Federal Ministry of Education, Science
+and Research (BMBWF). This work was supported in part by
+the Austrian Science Fund (FWF) through the SFB BeyondC
+F7102 and the Volkswagen Foundation (Az:97721).
+This
+work was also co-funded by the European Research Coun-
+cil (ERC) under Project No. 101055129. Views and opinions
+expressed are however those of the authors only and do not
+necessarily reflect those of the European Union or the Euro-
+pean Research Council. Neither the European Union nor the
+granting authority can be held responsible for them.
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+11
+a
+b
+c
+C1
+C4
+C2
+C3
+C5
+C6
+C7
+C8
+C9
+C1
+C4
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+FIG. 7: Simulating a quantum ECM. a) Directed acyclic graph describ-
+ing the ECM shown in Fig. 2, where each vertex corresponds to a clip ck (here
+including percepts and actions). b) Vertices, labeled as in panel (a), are rear-
+ranged in a layered structure and recolored depending on the order in which
+the corresponding unitary acts on the excitation (see App. H for a discussion
+on the ordering of the unitaries). Some unitaries that appear consecutively
+in the ordering can be seen to commute trivially and are thus grouped in the
+same layer with the same color. c) Subgraphs corresponding to percepts c1
+and c2. Vertices are colored as in panel (b).
+APPENDIX
+Appendix A: Package overview
+The above considerations can be tested and explored with a
+Mathematica package we make available online [12, 38].
+Below we describe its main functionalities.
+•
+Function 1 allows to study the structure of the quantum
+ECM proposed in this work, starting from the directed acyclic
+graph (DAG) that describes the connectivity of the ECM. It
+can be used to test the learning model and to develop exten-
+sions. This function operates at an abstract level and, specif-
+ically, does not presume an optical implementation. Func-
+tions 2-4 below can interact with its output by manipulating
+the transformations associated with each vertex of the DAG.
+Simple examples and details can be found in Fig. 7.
+Function 1 — InitializeQuantumECM
+Input: The directed acyclic graph (DAG) G describing an ECM.
+Output: An equivalent G with explicit layers; the vertices of G that cor-
+respond to percepts and actions; all subgraphs of G that are reachable
+from each percept (where quantum walks take place); a description of
+the clip-to-clip connectivity in G; extra information.
+•
+Functions 2-4 have been adapted from a package from
+Ref. [39]. They make it possible to manipulate unitary trans-
+formations on a linear-optical circuit, with a focus on the
+canonical decompositions. These routines serve to instanti-
+ate the abstract representation of a quantum ECM, and al-
+0
+20
+0
+0.5
+1
+psa
+(α)
+α
+1.1
+1.2
+1.3
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+Steps
+FIG. 8: Combining causal diamonds and Gram-Schmidt process. The
+Gram-Schmidt orthonormalization can be applied to the Uk associated with
+an arbitrary path in the ECM. In this example, we reinforce the transition
+probability psa by updating the Uk along the longest path (1,4,5,20) that
+connects a percept (s: green, k = 1) to an action (a: red, k = 20) for
+the ECM shown in the inset. Here, we assume that at each step the same
+(s, a) gets rewarded, for three different factors α (see Sec. E 1). Inset: The
+method based on causal diamonds can be extended to the ECM graph. While
+in Sec. III B the analyses zoomed in the structure of each Uk, one can also
+zoom out and reinforce psa by tuning the leaking nodes (“leaking Uk”) of (s,
+a) along the quantum walk (black vertices, k ∈ {5, 6, 14}). Other percepts
+(actions) are colored in light green (light red).
+low to study arbitrary circuits as graphs. They handle the ⃗θ
+parametrization in two formats, one for canonical architec-
+tures and one for arbitrary connectivities.
+Function 2 — UnitaryDecomposition
+Input: A unitary U and the choice of an algorithm for its decomposition.
+Output: For canonical architectures [33, 34], it decomposes U into phases
+and transmissivities. If beam splitters are implemented as Mach-Zehnder
+interferometers, it returns the appropriate phase settings.
+Function 3 — GetUfromParameters
+Input: Parameters in the format handled by Function 2, or in a layer-wise
+format that allows arbitrary connectivity.
+Output: Unitary generated by any of three methods with the input parame-
+ters (see Function 2). Optional noise and/or losses can be set.
+Function 4 — GetGraph
+Input: Information on the mode connectivity in the circuit, in the format
+handled by Functions 2 and 3.
+Output: Graph representation of the input circuit.
+•
+Functions 5-6 enable the update based on the causal
+diamond for a 2-layer PS (see Sec. III B and Fig. 4) and multi-
+layer PS (see Fig. 8). They make use of the graph representa-
+tions given by Functions 1 and 4. Function 7 implements the
+update based on the Gram-Schmidt process (see Sec. III C).
+Function 5 — FindLeakingNodes(AnyDAG)
+Input: Graph G describing an optical circuit / a quantum ECM (see Func-
+tion 4); a percept/action pair (s, a).
+Output: Vertices of G in the causal diamond and leaking nodes of (s, a).
+
+12
+Function 6 — UpdateCausalDiamond
+Input: Layered parametrization of the circuit (partially pre-computed for a
+faster evaluation); a percept/action pair (s, a); phase shift δθ.
+Output: New layered parametrization and the corresponding unitary.
+Function 7 — UpdateGramSchmidt
+Input: A unitary U; a percept/action pair (s, a); a rescaling factor α.
+Output: Updated U after one Gram-Schmidt process around Uas.
+Appendix B: Multi-photon quantum walks
+In this work, we replaced the excitation at the core of the
+decision making process of PS with a single-photon excitation
+in a quantum walk. In this picture, a photon is always injected
+into the mode that corresponds to a percept, and the new ac-
+tion is given by the output mode the photon is measured in. An
+interesting extension of this framework considers the simulta-
+neous injection of multi-photon states. A quantum PS agent
+equipped with this feature could be able to process multiple
+percepts at the same time, or a percept with a more complex
+structure. Furthermore, information encoded in n indistin-
+guishable photons injected in n different input modes would
+interfere at the level of quantum amplitudes, potentially open-
+ing up advantages with respect to the classical scheme, where
+such a feature is currently and independently under investiga-
+tion. Foreseen potential advantages in this sense are: (i) the
+advantages inherited by a multi-excitation classical PS; (ii) the
+exponentially larger (in n) percept/action (Hilbert) space; (iii)
+quantum computational advantages related to the hardness of
+simulating the dynamics [40].
+Appendix C: Quantum walks and frequency encoding
+An interesting possibility offered by an optical implemen-
+tation is to use additional degrees of freedom of photons, be-
+sides the spatial modes, to encode information and/or to train
+multiple agents in parallel. In particular, frequency appears
+to be a convenient candidate since the operation of integrated
+photonic circuits is wavelength-dependent. In fact, both beam
+splitters’ and phase shifters’ transformations depend on the
+wavelength as follows [41]:
+U bs
+λ ∝
+�
+cos π
+4λ − sin π
+4λ
+sin π
+4λ
+cos π
+4λ
+�
+,
+U ps
+λ (P) ∝
+�1
+0
+0 ei P
+λ
+�
+(C1)
+where P is the (suitably rescaled) power setting that controls
+the phase shifter. With these simplified expressions, we can
+already get an idea about the average impact of frequency en-
+coding on unitary transformations U of increasing size. Re-
+sults for this analysis are shown in Fig. 9a. We see that, as
+expected since the architecture consists of hundreds of opti-
+cal elements, a tiny variation in wavelength λ leads to signifi-
+cantly different effective Us. This means that photons injected
+with these deviations would undergo different Us, possibly
+implementing quantum walks in different quantum ECMs.
+a
+b
+p
+Size of U
+Steps
+sa
+(λ)
+F
+δ
+0.005
+0.01
+0
+250
+500
+750
+0
+0.5
+1
+0.015
+20
+30
+40
+50
+60
+70
+0.8
+0.85
+0.9
+0.95
+1
+FIG. 9: Wavelength-dependent quantum walks in the agent’s mem-
+ory. The wavelength dependence of beam splitters and phase shifters turn a
+limitation of optical technologies into a potential advantage (see Sec. C). a)
+Fidelity of random effective unitary transformations U of increasing size, as
+perceived by photons with different separation in wavelength λ (see legend),
+averaged over 100 pairs of random unitaries. Each U is numerically generated
+by uniformly sampling power settings in the phase shifters (see Eq. C1) of a
+square architecture where beam splitters are implemented as tunable Mach-
+Zehnder interferometers. Standard deviations are smaller than the point size.
+b) Four learning curves for a 2-layer PS agent based on a 10×10 square archi-
+tecture (four rewarded percept/action pairs; each percept/action is encoded in
+a pair of adjacent input/output optical modes). The figure of merit optimized
+by the variational approach based on causal diamonds (see Sec. III B) is the
+geometric mean of the transition probability p(λ)
+sa for three values of λ.
+Finally, we investigate whether a variational approach can
+work also while additional information is encoded in the fre-
+quency degree of freedom. To do so, we simulate quantum
+learning agents that attempt to solve the same RL task (same
+rewarded percept/action pairs) using different λs. Inciden-
+tally, this analysis is similar to, but more sophisticated than,
+the one presented in Fig. 4d. Results are shown in Fig. 9b,
+using the approach based on the causal diamond in a 2-layer
+PS (see Sec. III B). We see that the agent learns to map each
+of five percepts to the corresponding rewarded action, at the
+same time for three different λs. Importantly, all phase set-
+tings are the same for all λ, that is, the phase configuration
+found by the algorithm operates well (as shown) when input
+with different λ.
+The above considerations suggest that there is a large poten-
+tial in the use of additional degrees of freedom that influence
+the quantum walk. Foreseen potential advantages of this as-
+pect are: (i) extra room for more complex RL mechanism; (ii)
+parallelism in the agent’s decision making process.
+
+13
+Appendix D: Towards interpretable quantum artificial
+intelligence
+While neural networks show excellent performance, their
+black-box-like nature can be problematic for some purposes.
+For example, in delicate decisions that directly impact hu-
+mans, one may wish to have a justification before following
+the advice of an enigmatic machine. Similar concerns apply
+in science where, rather than developing an oracle, one might
+aim to acquire a deeper understanding of a given phenomenon.
+These issues have recently sparked interest in interpretable
+and explainable artificial intelligence [42]. PS offers one vi-
+able framework towards this goal. Differently from neural
+networks [17], in classical PS an interpretation can be found
+by studying the path taken from percept to action. However,
+this is no longer true in the quantum domain, where we lose
+the notion of observable path. Since in a quantum walk all al-
+lowed paths contribute to the decision making process, a quan-
+tum PS agent loses interpretability in the classical sense (there
+is no longer a unique path that can be retraced).
+A notion of interpretability that is applicable to quantum
+agents is currently being explored. Here we make a first step
+in this direction by observing that a notion of partial trace can
+still be recovered, if one studies the likelihood that a photon
+(or more photons, as in App. B) would be measured in any
+given mode of the circuit (see Fig. 10). This information can
+be retrieved by classically simulating the single-photon quan-
+tum walk, since all phase settings are known. This possibility
+still holds for multi-photon quantum walks despite the com-
+putational complexity of the task, which becomes practically
+unfeasible only from n ∼ 30 [43].
+Appendix E: Variational algorithms
+In this section, we provide additional information on the
+two variational algorithms presented in Sec. III.
+The following algorithms naturally take into account the
+unitarity constraint inherent to the architecture. We point out
+that this constraint is not necessarily a limitation and, rather,
+it could be advantageous to improve the agent’s performance.
+In fact, since a machine learning ansatz with more freedom is
+also more prone to overfitting, it is reasonable to believe that
+a specialized ansatz with restrictions and symmetries might
+generalize better. A reduced number of degrees of freedom
+might also help during the training stage.
+1.
+Update via Gram-Schmidt process
+Here we outline the update via a Gram-Schmidt process.
+The algorithm consists of four steps (i-iv):
+i. Compute U (t+1)
+a0s0
+from the current U (t)
+as according to a
+desired update rule. Here, s0 and a0 are the observed
+percept and action of the current round, respectively.
+In the examples in Sec. III C, we considered a constant
+rescaling factor α, i.e. U (t+1)
+a0s0
+= α U (t)
+a0s0, which acts
+like a learning rate.
+ii. Normalize the row U (t+1)
+a0,• .
+iii. Use GSO on the other rows to obtain a new unitary, in
+the following way: For each row a ̸= a0, define
+˜U (t+1)
+as
+:= U (t)
+as −
+�
+˜s∈S
+�
+˜a∈Aa
+U (t+1)
+˜a˜s
+U (t)
+a˜s U (t+1)
+˜as
+,
+(E1)
+where the overbar denotes the complex conjugation,
+and Aa is the set of rows ˜a for which we have already
+defined U (t+1)
+˜a,•
+, i.e. the previous ˜a in the loop. Deter-
+mine U (t+1)
+a,•
+from ˜U (t+1)
+a,•
+via normalization.
+iv. Use the decomposition [33] of canonical square archi-
+tectures to find the phase settings that implement the
+new unitary. The decomposition is exact and requires a
+number of operations quadratic in the size of U.
+The main advantage of this approach is that it allows one
+to control any theoretical aspect of the learning process, from
+the update rule to the learning speed. One limitation is that
+it assumes perfect knowledge and control of the experimental
+settings. Appending to it one step of variational fine-tuning, as
+the one we describe in the following section, can help mitigate
+this issue and speed up the learning process.
+2.
+Update via causal diamonds
+Here we provide additional considerations on the update
+based on causal diamonds. The advantage of this approach
+is that it is efficient (only few phase shifters are adjusted at
+each step) and that it is variational (it takes into account the
+imperfect control over the experimental settings).
+We consider two main approaches (a,b) to update a given
+subset Θ = (θ1, . . . , θN) of N phase shifters at each training
+step, albeit more sophisticated strategies can be devised and
+tailored to the hardware. These phases can correspond to the
+set of leaking nodes described in Sec. III B, or to a larger set
+selected via other criteria.
+a. Gradient ascent — Each phase θ ∈ Θ is independently
+tuned to maximize a predefined figure of merit F.
+b. Sequential — Each phase θ ∈ Θ, in the order given by
+the light propagation, is tuned to independently maxi-
+mize a figure of merit F, and is immediately set to the
+new value before the next phase is probed.
+The latter approach (b) (see Fig. 4 and Fig. 8) is usually
+faster than (a), since each phase update starts from a progres-
+sively higher value of F, and the individual updates within a
+training step do not sabotage each other. However, an update
+based on gradient ascent as in (a) can be beneficial to escape
+local minima, which are more likely to cause problems in (b).
+Appendix F: Optical architecture
+In this section we review the basics behind the operation of
+the optical architecture discussed in the main text. In Sec. II B,
+
+14
+FIG. 10: Partial traceability in the decision making process of quantum PS agents. a-c) Three quantum walks, associated with as many 2-layer ECMs,
+are numerically simulated with a square optical architecture. Each panel shows the probability P to measure a photon in one of the output modes before training.
+Each column (label: Layers) displays the squared column (for a fixed percept) of the unitary matrix obtained by adding a new layer of optical elements to the
+square architecture, from left (input layer) to right (full circuit). The rightmost column of each plot corresponds to the output probabilities of a quantum walk
+over the full ECM. d-f) Same analysis repeated after training with the variational approach based on the causal diamond (see Sec. III B). Panels (d), (e) and (f)
+correspond to the evolutions in panels (a), (b) and (c) after training, respectively. We observe that a notion of partial trace emerges inside the circuit, which
+might be leveraged to study interpretability in quantum learning agents. Analogous results (not shown) are found for the numerical simulations in Fig. 8.
+we proposed to consider the square canonical layout [33] for
+several reasons, above all because it supports the implemen-
+tation of any unitary transformation U, with no modifications
+at the hardware level. This is possible because any m × m
+U can be decomposed as U = D �
+b Ub, i.e., the product of
+a diagonal matrix D (whose elements are complex and have
+modulo 1) and m(m − 1)/2 unitaries Ub implementing com-
+plex rotations in the plane spanned by two adjacent modes:
+Ub =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1 . . .
+0
+0
+. . . 0
+...
+...
+...
+...
+...
+0 . . . u11 u12 . . . 0
+0 . . . u21 u22 . . . 0
+...
+...
+...
+...
+...
+0 . . .
+0
+0
+. . . 1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(F1)
+where the uij (i, j ∈ {1, 2}) describe the transformation act-
+ing on the two modes.
+This 2 × 2 transformation is im-
+plemented by means of a pair of phase shifter (θ1) and im-
+balanced beam splitter, whose transformation U BS is imple-
+mented as a Mach-Zehnder interferometer with tunable phase
+shifter (θ2):
+U BS(θ2) = 1
+2
+�
+1 i
+i 1
+� �
+eiθ2 0
+0
+1
+� �
+1 i
+i 1
+�
+.
+(F2)
+Overall, the proposed ansatz can be implemented using
+well-established, integrated photonic components that only
+involve tunable phase shifters and fixed, balanced beam split-
+ters (directional couplers).
+Appendix G: Numerical details of the transfer learning scenario
+In this section, we report additional details on the numerical
+analysis described in Sec. IV, performed with Pytorch.
+Both classical and quantum PS agents have three layers:
+percept, middle and task layer. While in classical PS each
+percept s is represented by a vertex in the ECM graph, in
+this analysis the quantum PS agent associates with it a (linear-
+optical) binary tree [10], to overcome the limitation due to
+the unitarity constraints in the middle layer. All its phases
+are initialized to π
+4 to yield a balanced output distribution.
+The classical PS agent is expected to pick an observable uni-
+formly at random, and then send the excitation to the mid-
+dle clip with the right value. For the quantum PS agent, we
+wish that the percept (v0, v1, v2) is represented by the state
+1
+√
+3
+�
+|v0⟩ + |3 + v1⟩ + |6 + v2⟩
+�
+. To this end, we first train
+the binary trees, considering a RL scenario where percepts are
+sampled i.i.d. uniformly. We use the simplified loss function
+LPS of Eq. (11), with the Kullback-Leibler divergence applied
+to the probability distribution {psa, 1 − psa}. Furthermore,
+we use ReLU functions to cut off the target values in Eq. (11)
+such that they are in [0, 1]:
+p(t+1)
+sa
+= 1.0 − ReLU
+�
+1.0 − ReLU(p(t)
+sa + r)
+�
+(G1)
+As a reward, we assign r := ±0.1 depending on whether or
+not the agent predicts the right value. To ensure that each ob-
+servable is picked uniformly, we implement a curiosity mech-
+anism by including the Shannon neg-entropy of the observ-
+
+15
+ables in the loss function:
+LShannon = log(3) +
+2
+�
+j=0
+psOj log psOj
+(G2)
+where psOj is the probability of the j-th observable Oj, given
+a percept s. To enforce that the relative phases among the
+modes of the middle layer vanish, we also add to the loss
+function the ℓ1 norm of the complex phases, i.e. Lphase :=
+�8
+m=0 |Φsm|. Here, Φsm is the phase in the middle-layer
+mode m when percept s is considered. The full loss function
+for the middle layer is then:
+Lfull = 1.0 LPS + 10.0 LShannon + 1.0 Lphase
+(G3)
+We use experience replay, and add up the loss functions of a
+batch of 600 sampled percepts before running 10 optimizer
+steps (Adam optimizer [44]) and wiping the batch. The learn-
+ing rate is set to 0.01, decreased to 0.001 once the accuracy
+of the last batch was at least 0.95. Training stops when the
+accuracy of the middle layer exceeds 0.99 for 10 consecu-
+tive rounds and, at the same time, the ℓ1-norm of the phases
+weighted by psm is smaller than 0.1.
+Once the middle layer is trained, we fix its phase shifters so
+that the trees do not change anymore. Now, we introduce the
+task layer, which is exchangeable and independent of the mid-
+dle layer. While Eva et al. only considered single-observable
+experiments, our task layers pick two observables (O0, O1)
+and ask whether they have specific values vO0 and vO1.
+The quantum PS agent uses one 9x9 interferometer for each
+task layer. Again, we use experience replay, with a batch size
+of 500 (much larger than the number of percepts) and train the
+agent for 40000 rounds. Since our tasks ask for specific val-
+ues of two observables simultaneously, only 1
+9 of the percepts
+will correspond to a yes-answer. Hence, in addition to the raw
+accuracy, we use a weighted accuracy where yes-answers are
+given an additional weight 8 (otherwise the agent could just
+always answer no, which would yield a seemingly good ac-
+curacy of 8
+9). The loss function is the one in Eq. (8), with a
+reward equal to ±0.1 for a right or wrong answer, and an addi-
+tional factor 8 if the right answer is yes, the Kullback-Leibler
+divergence for {psa, 1 − psa}, ReLU functions to cut off the
+target values and neither glow nor forgetting. We normalize
+the probability of the yes- and no-mode of the task layer to
+one, i.e. we post-select on getting one of these two modes.
+For the learning rate and number of optimizer steps we use
+the same settings as before.
+Overall, the quantum PS agent consistently achieves over
+0.97 accuracy in all two-observable tasks (see Fig. 6).
+Appendix H: Classical-quantum equivalence of ECM graphs
+In this section we formalize the connection between the di-
+rected graph of a classical ECM and the quantum walks that
+derive from it. We refer to this set of quantum walks (from
+input to output) as unitary routing. We begin by introducing
+some basic elements from graph theory.
+Definition 1 (Topological ordering). A topological ordering
+oG of a directed graph G = (V, E) is a map, o : V →
+{1, . . . , |V |}, which defines a total ordering of vertices of G
+such that for every directed edge (vj, vk) from vertex vj to ver-
+tex vk, vj comes before vk in the ordering, i.e., o(vj) < o(vk).
+Let OG be the set of all topological orderings of G.
+An example of topological ordering is presented in Fig. 11.
+Directed acyclic graphs (DAGs) are directed graphs that do
+not contain loops. They can be defined as follows [45]:
+Definition 2 (Directed acyclic graph). A directed graph is
+acyclic if and only if it admits a topological ordering.
+Thus, the graph in Fig. 11 is an example of a DAG. It is worth
+noticing that the topological ordering of a DAG such as in
+Fig. 11 is not unique, because swapping consecutive vertices
+(in the topological ordering) which are not connected by an
+edge yields another valid topological ordering.
+Below we recall two additional, useful definitions:
+Definition 3 (Ordered DAG). An ordered DAG ⃗G is a
+DAG (V, E) with V
+= {1, . . . , |V |} such that o : V
+→
+{1, . . . , |V |}, o(j) = j is a topological ordering of (V, E).
+For any DAG G = (V, E) with topological ordering o we
+denote the induced ordered DAG by
+⃗G(o) =
+�
+{o(vi) | vi ∈ V }i,
+(H1)
+{(o(vj), o(vk)) | (vj, vk) ∈ E}
+�
+.
+Let ocDAG be the set of all ordered and connected DAGs.
+𝑐4
+𝑎1
+𝑠2
+𝑐5
+𝑎2
+𝑐3
+𝑐2
+𝑐1
+𝑠1
+𝑠1
+𝑠2
+𝑐1
+𝑐2
+𝑐3
+𝑐4
+𝑐5
+𝑎1
+𝑎2
+𝑐4
+𝑎1
+𝑠2
+𝑐5
+𝑎2
+𝑐2
+a
+b
+𝑠1
+𝑐3
+𝑈𝑐5𝑠1
+5
+𝑈𝑐2𝑠1
+2
+𝑈𝑐1𝑠1
+1
+𝑐1
+FIG. 11: a) Directed acyclic graph shown in Fig. 2 (top) and one of its
+topological orderings (bottom). Notice that, for the sake of simplicity, all ver-
+tices had already been labeled with subscripts in ascending order. b) Fulfilling
+unitarity constraints in the mode-mixing transformations Uk in Eq. (H3) may
+generate edges (blue) that are not present in the classical ECM.
+
+16
+Object
+Intuition
+Routing
+Sequence of parametrized unitary transformations,
+constructed from a graph G.
+Route
+A specific sequence of the routing, where all param-
+eters have been fixed.
+Router
+The map from Go to its routing
+TABLE II: Summary of the main quantities introduced in App. H.
+Definition 4 (Parents and children). Let G = (V, E) be a
+graph. For any vertex vi ∈ V , let PG(vi) be the set of parents
+of vi, i.e., PG(vi) = {vj|vj ∈ V, ∃(vj, vi) ∈ E}. For any
+vertex vi ∈ VG, let CG(vi) be the set of children of vi, i.e.,
+CG(vi) = {vj|vj ∈ V, ∃(vi, vj) ∈ E}.
+Equipped with the above definitions, we proceed by defin-
+ing an ECM graph. Here we focus on ECMs described by
+connected DAGs.
+Definition 5 (ECM graph). An ECM graph is a connected
+DAG G = (V, E) whose vertices are called either percepts,
+S = {vj|vj ∈ V, PG(vj) = ∅}, actions A = {vj|vj ∈
+V, CG(vj) = ∅}, or intermediate clips C = V \(S ∪ A).
+The quantum version of a classical ECM graph is defined
+via a unitary routing. We refer to Table II for an overview of
+the terminology introduced in this section.
+Definition 6 (Unitary route). Let G = (V, E) be an ECM
+graph and o a topological ordering of G. A unitary route of
+the induced ordered DAG ⃗G(o) is a sequence of |V | × |V |
+unitaries, ordered in ascending order of their superscripts,
+route( ⃗G(o)) =
+�
+U o(vi)�
+vi∈C∪A ,
+(H2)
+whose matrix elements are given by
+U i
+jk =
+�
+ui
+jk
+if j, k ∈
+�
+{i} ∪ P ⃗G(o)(i)
+�
+δjk
+else,
+(H3)
+where the ui
+jk ∈ C are only subject to unitarity constraints.
+For example, with reference to the example in Fig. 11
+and the topological ordering (o(s1), ..., o(a2)) = (1, ..., 9),
+PG(c2) = (s1, s2) and P ⃗G(o)(4) = (1, 2), hence
+U 4 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+u4
+11 u4
+12 0 u4
+14
+0
+· · · 0
+u4
+21 u4
+22 0 u4
+24
+0
+0
+1
+0
+u4
+41 u4
+42 0 u4
+44
+0
+0
+0
+1
+...
+...
+... 0
+0
+0
+· · ·
+0
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(H4)
+Definition 7 (Unitary routing). Let routing(⃗G(o)) be the
+set of all unitary routes of ⃗G(o), generated by the freedom in
+choosing the coefficients ui
+jk within the constraints of unitar-
+ity. Let also ocROUTING = {routing(⃗G(o))} ⃗G(o)∈ocDAG.
+Definition 8 (Unitary router). We define a unitary router as
+the function
+router : ocDAG → ocROUTING,
+(H5)
+⃗G(o) �→ routing(⃗G(o)).
+Given a unitary route route( ⃗G(o)), the unitary matrix de-
+scribing the whole quantum ECM is then given by
+U =
+�
+U i∈route( ⃗G(o))
+U i,
+(H6)
+where the product is ordered from left to right in descending
+order of the superscripts. Note that here the superscript i refers
+to the position of the vertices in the ordering o, while in the
+main text the superscript refers to an unspecified ordering, and
+that in general the U i do not commute.
+If an ECM graph G admits more than one topological or-
+dering, G can have different unitary routings. Nevertheless,
+there is the following one-to-one quantum-classical corre-
+spondence:
+Lemma 9 (Quantum-classical correspondence). The func-
+tion router is bijective.
+Proof.
+To show bijection we will first show injectivity,
+that is, we must show that router(⃗G(o)) = router( ⃗H(o′)) ⇒
+⃗G(o) = ⃗H(o′). According to the definition of routing, a ⃗G(o)
+has routing(⃗G(o)) only if (i) it has vertices V = {1, . . . , n},
+where n is the number of rows (or columns) of each U i in any
+of the routes in routing(⃗G(o)), and (ii) it has directed edges
+E = {(j, k) | ∃ route( ⃗G(o)) ∈ routing(⃗G(o)) and ∃ U k ∈
+route( ⃗G(o)) such that U k
+jk ̸= 0}. This completely determines
+⃗G(o) as ⃗G(o) = (V, E). Thus router(⃗G(o)) = router( ⃗H(o′))
+can only be true if ⃗G(o) = ⃗H(o′).
+Surjectivity is trivial because the codomain of router is de-
+fined as its range, which completes the proof that router is
+bijective. □
+While a one-to-one correspondence exists on the level of
+ordered DAGs ⃗G(o) and unitary routings, there is an interest-
+ing freedom on the level of ECM graphs and/or unitary routes.
+In the latter case, a unitary route is not sufficient to uniquely
+identify a ⃗G(o) because if ui
+jk = ui
+kj = 0 for some j ̸= k, the
+corresponding directed edge (j, k) in ⃗G(o) cannot be recov-
+ered just from the unitary route. Instead, one would recover
+a ⃗G(o)′ without that edge. Classically, this corresponds to the
+case of a vanishing h-value, hjk = 0. In the former case, it is
+worth noticing that ECM graphs are defined without ordering.
+However, ordering matters for unitary routes since in general
+the U is do not commute. This represents an interesting depar-
+ture from classical ECMs: different orderings can give rise to
+different unitary routings.
+
diff --git a/ONFRT4oBgHgl3EQf4jgk/content/tmp_files/load_file.txt b/ONFRT4oBgHgl3EQf4jgk/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9ca82540c2a1eac89a94da5177e5e351f5253f0a
--- /dev/null
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@@ -0,0 +1,1394 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf,len=1393
+page_content='Reinforcement learning and decision making via single-photon quantum walks  Fulvio Flamini,1, ∗ Marius Krumm,1, ∗ Lukas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Fiderer,1 Thomas Müller,2 and Hans J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Briegel1  1Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria  2Department of Philosophy, University of Konstanz, Universitätsstraße 10, 78464 Konstanz, Germany  Variational quantum algorithms represent a promising approach to quantum machine learning where classi-  cal neural networks are replaced by parametrized quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here, we present a variational approach to  quantize projective simulation (PS), a reinforcement learning model aimed at interpretable artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Decision making in PS is modeled as a random walk on a graph describing the agent’s memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To implement  the quantized model, we consider quantum walks of single photons in a lattice of tunable Mach-Zehnder in-  terferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We propose variational algorithms tailored to reinforcement learning tasks, and we show, using  an example from transfer learning, that the quantized PS learning model can outperform its classical counter-  part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Finally, we discuss the role of quantum interference for training and decision making, paving the way for  realizations of interpretable quantum learning agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  INTRODUCTION  Machine learning methods are revolutionizing science and  technology, with applications ranging from drug discovery  [1] to the study of quantum phases of matter [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Within  this toolbox, reinforcement learning (RL) has demonstrated  the capacity to solve tasks better than the human state of the  art [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Currently, the leading method in machine learning  is given by artificial neural networks, with performance be-  yond theoretical expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The main near-term approach  to quantum machine learning replaces these artificial neural  networks with variational quantum circuits, typically consist-  ing of fixed two-qubit gates and parametrized Pauli rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Popular platforms include, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=', superconducting circuits [4]  and ion traps [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Another promising platform is offered by  photonic circuits, consisting of a fixed, regular mesh of tun-  able Mach-Zehnder interferometers [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In this paper, we propose a variational approach to quantum  RL based on linear-optical interferometers and on a classical  learning model called projective simulation (PS) [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In PS,  the agent’s decision making process is modeled as a random  walk of an excitation in an episodic memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Following an  approach already investigated by some of the authors [10], we  replace this classical random walk with a single-photon quan-  tum walk in an optical interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' On the one hand, we  are able to preserve the core features of classical PS, so that  the main ideas developed in the field can be transferred to the  quantum domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' On the other hand, this approach can be  tailored to end applications, with actual constraints and needs  in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' These two points link to the main motivation of this  work, that is, the development of a learning agent that operates  in distributed quantum optical networks [11], and that can in-  teract with other agents by exchanging quantum information  encoded in light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The article is structured as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' I, we review  the standard PS model that provides the background of our  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' II, we present our approach to quantizing PS  and, based on that, to designing a photonic implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  ∗ These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  marius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='krumm@uibk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='at  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III, we introduce two variational training algorithms  tailored to the proposed framework, whose applicability goes  beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' IV, we provide nu-  merical evidence that such quantum PS agents can generalize  better than their classical counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Most of our analyses  can be directly tested with a package we make available on-  line [12] (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A), and can be further extended in various  directions (using the same package;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' see Apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' B-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  PROJECTIVE SIMULATION  In this section, we review the basics of RL and PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' RL is  a paradigm of machine learning that involves an agent inter-  acting with an environment (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The agent receives  from the environment an input s ∈ S, called state or percept,  and decides what action a ∈ A to take next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' RL typically  considers Markov decision problems, for which the agent’s  policy can be described by a conditional probability distribu-  tion π(a|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The agent receives immediate feedback on its  actions through a reward R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' its goal is to find the policy that  maximizes the long-term expected reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In this work we focus on PS, a physically motivated RL  model that has been applied to the study of collective behav-  ior [13], robotics [14], automated experiment design [15] and  quantum error correction [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' PS relies on a memory struc-  ture called episodic and compositional memory (ECM), de-  scribed by a directed graph (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The vertices of this  graph, called clips, are associated with memories of the agent  such as remembered percepts, actions or intermediate, more  complex steps in the decision making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Importantly,  this means that a clip carries semantic information [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' When  the environment delivers a percept s, the corresponding clip is  excited, and a random walk of a single excitation is initiated  in its clip network (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 2), until the excitation hits an ac-  tion clip a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Each directed edge from clip ci to clip cj is asso-  ciated with an unnormalized transition probability hij called  h-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The transition probability p(cj|ci) is computed by  normalizing over all clips that can be reached from ci:  pij ≡ p(cj|ci) =  hij  �  j′ hij′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (1)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='13669v1  [quant-ph]  31 Jan 2023  2  a  PERCEPT  REWARD  ACTION  PS agent  𝑐2  𝑐𝑘  1  2  k  Ƹ𝑐1  †  1  2  k  𝑈𝑘  Environment  b  c  𝑠  𝑎  ℎ  (𝑠)  (𝑅)  (𝑎)  𝑐4  𝑐5  ℎ2𝑘  Ƹ𝑐2  †  Ƹ𝑐𝑘  †  Ƹ𝑐1  †  Ƹ𝑐2  †  Ƹ𝑐𝑘  †  𝑐3  ℎ1𝑘  𝑐1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 1:  Projective Simulation (PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This work uses PS as a frame-  work for reinforcement learning and its quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a) When a PS agent  receives a percept s from the environment, it initiates a random walk in the  graph that describes its memory, until it hits a vertex representing an action  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The performed action may be rewarded, leading to an update of the pa-  rameters (h-values) that control its decision making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' b) In classical  PS, the vertices of the graph represent clips (ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Directed edges are asso-  ciated with transition probabilities that depend on the h-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' c) In the  quantized PS, clips are represented by mode operators ˆc†  k describing a single  excitation, while the connectivity is governed by mode-mixing transforma-  tions Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Each Uk acts on the mode operators associated with the parent  clips of ck (here c1, c2), and its complex elements are the transition ampli-  tudes that replace the classical probabilities of transitioning to ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' All panels:  all edges that contributed to the decision making process are in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Learning occurs by tuning the h-values, which are initially set  to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To this end, each edge with h-value hij is additionally  equipped with a so-called glow factor gij, which contributes  to the standard PS update rule [18]:  h(t+1)  ij  = 1 + (1 − γ)(h(t)  ij − 1) + g(t)  ij R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (2)  Here, the role of the glow factor gij is to remember transitions  (ci → cj) taken in past interactions with the environment, so  that they are reinforced when a reward R is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Further,  γ ∈ [0, 1] is a forgetting factor: its main purpose is to gradu-  ally dampen previously learned transition probabilities, to let  the agent adapt to changing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' γ is also used as  a soft cut-off to prevent the h-values to overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Each gij is  updated as:  g(t+1)  ij  =  �  1  if ci → cj was taken at t,  (1 − η) g(t)  ij  otherwise,  (3)  where g(0)  ij  = 0 and η ∈ [0, 1] is a discount factor for the  actions taken further in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We refer to the literature on  PS for a more detailed description and additional mechanisms,  such as association [19] and reflection [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  QUANTUM PROJECTIVE SIMULATION  Since decision making in PS is generated via a random walk  of an excitation, it is possible to quantize it by giving learning  agents access to a quantum walk in a quantum memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In the  literature, several variants of this idea have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' [20], an algorithm based on discrete quantum walks  was presented which provides a quadratic speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' However,  this algorithm relies on sophisticated oracles that likely will  not be available in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Furthermore, since the al-  gorithm is described as a quantum circuit, it is not clear how it  can be interpreted as a random walk of a physical excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  On the other hand, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' [8], it was proposed to quantize  PS by considering a quantum walk of an excitation in a quan-  tum many-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Since Hamiltonians are hermitian,  in such systems the probability amplitude of a transition is as  large as the amplitude of the backwards transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To break  this symmetry, the authors proposed to use dissipative effects  (‘quantum jumps’), at the cost of coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In this section, we propose a quantum extension of PS that  allows for an optical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Our guiding principle  is to design a PS agent natively as a quantum walk of a physi-  cal excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In particular, our proposal does not rely on the  availability of quantum oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Our scheme uses an optical  interferometer that induces a natural directionality of transi-  tions, without the need for dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We will first describe  how to implement the main elements of PS in the quantum do-  main in general and then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' II B, on an optical platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  From classical to quantum projective simulation  In order to quantize PS and facilitate an optical implemen-  tation, we need to quantize its main elements: the episodic  memory, realized by the clip network, and the decision-  making process, described by the propagation of an excita-  tion through the clip network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A physical realization of a clip  network could be a system of harmonic oscillators (modes)  with non-trivial many-body interactions between sets of oscil-  lators [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In case of an optical implementation, clips ci ∈ C  (including percept clips, action clips and intermediate clips)  then correspond to modes in an optical interferometric net-  work [10] (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' II B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' For the quantization, we associate  with each clip ci a pair of creation and annihilation operators  of the corresponding (optical) mode, {ˆc†  i, ˆci}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' An excitation  of memory clip ci at time τ0 is then described by the quantum  state ˆc†  i(τ0) |vac⟩, where |vac⟩ denotes the vacuum state, de-  scribing the inactive state of the agent’s memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here we use  the Heisenberg picture where the operators carry the agent’s  internal time dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Note that we index RL steps with  t, while physical time within a step is denoted by τk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  A decision-making process is initiated by the excitation of  some clip ci at time τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The subsequent evolution of the  agent’s memory will be described by a unitary of the form  ˆc†  j(τn) = U†(τn, τ0)ˆc†  j(τ0)U(τn, τ0) =  |C|  �  i=1  U ECM  ji  ˆc†  i(τ0) ,  (4)  which relates the excitation operators of the clip network at  different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The Heisenberg time evolution of the ECM  U(τn, τ0) is thus realized by the interferometric scattering ma-  trix U ECM connecting the modes, which, in turn, is composed  3  റ𝜃1  𝑠1…|𝑆|  Classical ECM  Quantum ECM  a  𝑐4  𝑎1  𝑠2  𝑐5  𝑎2  𝑐3  𝑐2  𝑐1  b  c  𝑐1…|𝐼|  𝑎1…|𝐴|  𝑠1  𝑠2  𝑐1  𝑐2  𝑎1  𝑎2  𝑐3  𝑐4  𝑐5  𝑠1  𝑠2  𝑐1  𝑐2  𝑎1  𝑎2  𝑐3  𝑐4  𝑐5  𝑈1  𝑈2  𝑈𝑛  𝑈1  𝑈3  𝑈4  𝑈5  𝑈6  𝑈7  ℎ15  ℎ12  ℎ11  …  𝑠1…|𝑆|  𝑐1…|𝐼|  റ𝜃2  റ𝜃𝑛  𝑈2  𝑎1…|𝐴|  𝑠1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 2: Decision making in classical and quantum projective simulation (PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a) In PS, decision making occurs via a classical random walk over a  network of clips {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' , c|C|} that describes the agent’s episodic and compositional memory (ECM) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' When the agent receives a percept s (blue), an  excitation triggers and performs a random walk over the network of intermediate clips {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' , c|I|} (green) until it hits one of the action clips {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' , a|A|}  (red), which is immediately coupled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' b) In the proposed quantized PS, these random walks become quantum walks of single excitations through layers  k = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' , n) implementing the transformations Uk, which mix sets of clips in a certain order (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here, clip-to-clip transition probabilities,  classically described by the h-values, are replaced by transition amplitudes described by the matrix elements of Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' c) The evolution can be seen as hopping  between eigenstates (of the modes) in the Hilbert space (empty disks: in/out modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' filled disks: output modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This panel reproduces the quantized evolution  in the classical ECM in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Clips are now visited in superposition (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  of a sequence of n intermediate transformations U k,  U ECM(⃗θ) =  n  �  k=1  U k(⃗θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (5)  The U k are updated during the learning process via the set of  parameters ⃗θ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The precise role of the U k(⃗θ) will  be discussed in more detail in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' For convenience  and to facilitate an optical implementation, in this work we  will consider ECMs that are described by a directed acyclic  graph (DAG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Percept encoding — When the agent receives a classical  percept at time τ0, this leads to the excitation of a percept  clip s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The quantum state of the memory is then described by  c†  s(τ0) |vac⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Subsequently, a quantum walk through the clip  network starts [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Quantum walk in the ECM — To connect the random walk  in the ECM to the quantum walk considered in this work, we  refer to the examples in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' As shown in panel  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 1b, a clip effectively acts as a node that routes incoming  edges from sets of clips to other clips in the next step of the  random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In the quantum ECM sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 1c and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 2b-c, this mechanism translates into the application of a  transformation U k, which mixes creation operators of incom-  ing clips with ˆc†  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A general clip-to-clip transition ci → cj is  then modeled by transformations U k that mix subsets of cre-  ation operators, so that the stochastic nature of PS manifests  itself at the level of quantum amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  A crucial departure from classical PS is that clips are now  visited in parallel (in superposition, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 1c), and all re-  gions of the ECM that are reachable from a percept via a se-  quence of U k contribute to the decision making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' As  a consequence, the probability to find an excitation in action  clip a at time τn (described by the quantum state c†  a(τn) |vac⟩)  when starting with an excitation of percept clip s at time τ0  (described by c†  s(τ0) |vac⟩) is given by the modulus square of  the corresponding transition amplitude  psa = |⟨a|s⟩|2 ≡ |⟨a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' τn|s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' τ0⟩|2 =  ���U ECM  as  (⃗θ)  ���  2  ,  (6)  where |ci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' τk⟩ = ˆc†  i(τk) |vac⟩ are the time-dependent eigen-  states of the number operators ˆc†  i(τk)ˆci(τk) corresponding to  an excitation being present in clip ci at time τk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (6) entails  a quantum interference of all paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' While in classical PS the  excitation has a well-defined trajectory in the ECM, now the  excitation can be delocalized over all the allowed modes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=',  the quantum amplitude is in general non-zero for more than  one mode at the same time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This aspect bears an interesting  potential for the learning model, which has already been  discussed in the literature [8, 22] and that is further discussed  in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III-IV and in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Action decoding — The decision-making process ends  when U n is applied and the output states are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' When  the excitation is detected in an output state associated with an  action, this action is coupled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In all other cases, the exci-  tation can be (i) routed back to other regions of the reachable  ECM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (ii) discarded, exciting again the same percept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Solu-  tion (ii) can be interpreted as the agent’s capacity to not take  action during this decision-making process (see also the no-  tion of suspension of judgment [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Practically, the agent takes action by post-selecting over  the outcomes that correspond to actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In general, one can  associate multiple output states with a single action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  This  makes it possible to avoid post-selection in settings with an  action space smaller than the percept space, or to implement  quantum channels that are more general than unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Training the quantum PS agent — To train the agent, we  consider a variational approach based on a loss function L that  is inspired by classical PS and aims to reproduce its learning  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' L depends on the probabilities psa(⃗θ) to take ac-  tion a given a percept s (that is, the policy) and the variational  parameters ⃗θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' These probabilities can be estimated either nu-  merically (by simulating the quantum walk) or experimentally  4  (via multiple measurements with fixed input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III A  we introduce and motivate our choice for L (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 8), along  with several modifications and training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Quantum optical projective simulation  In this section, we describe our approach to implement  quantum PS on a photonic platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Table I summarizes the  main connections with classical PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Related works — Photonic circuits provide a promising  platform in both classical and quantum domains [6, 7], with  results of increasing complexity being continuously reported  [24, 25], in particular for (un)supervised learning [26–32] and  RL [10, 22] based on path encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In this work, we focus on RL and aim to design a quantum  PS agent that operates with a native quantum walk of physical  excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Hence, we will replace the single excitation  with a single photon propagating in a linear-optical circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  This approach, while retracing some of the intuitions in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  [10] where the psa were reproduced by optical binary trees,  promises the following advantages: (1) it requires only a  single (possibly larger) circuit for all percepts, instead of one  binary tree per percept and layer, with no explicit need for fast  active switching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (2) it allows to take advantage of quantum  coherence and interference in the decision making process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (3) it allows for more advanced training algorithms, which are  applicable beyond the scope of this work (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (4)  it suggests concrete directions for further developments (see  Apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' B-D), such as a framework for learning with quantum  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In the following, we outline how the quantized version  of PS can be realized on a photonic circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Photonic architecture — For the implementation of PS on  quantum hardware, we want to identify quantum extensions of  the Markov chain process that govern the evolution of a clas-  sical excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' These extensions, usually referred to as quan-  tum channels (completely positive trace-preserving maps),  would have to reproduce the dynamics in classical PS as a  limiting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A direct implementation of classical PS would  require the {U k} to reproduce the stochastic maps that form  the random walk in the ECM graph described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Specif-  ically, the real-valued matrices P associated with these trans-  formations need to (i) reproduce the probabilities pij induced  by the h-values, and (ii) fulfill the connectivity in the ECM  (P can be seen as a weighted adjacency matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Such a class  of transformations can in principle be realized on a photonic  platform by recalling that any real-valued matrix P can be de-  composed as P = V1ΣV †  2 via singular value decomposition,  where Σ is a diagonal matrix with non-negative real entries  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=', optical attenuators) and V1,2 are unitary transformations  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Hence, the architecture we use for the variational circuit  is the canonical square one [33], since it can implement any  unitary transformation by suitably tuning its phase shifters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Indeed, we recall that any such U k can be implemented by  a homogeneous mesh of Mach-Zehnder interferometers, each  comprising two 50:50 beam splitters and two phase shifters  𝜃2  𝑐1…|𝐼|  𝑎1…|𝐴|  50:50  50:50  1  2  3  4  1  2  3  4  𝜃1  𝜃2  π  0  𝑠1…|𝑆|  𝑐1…|𝐼|  𝑎1…|𝐴|  𝑠1…|𝑆|  𝜃1  a  d  b  θ  →  Tunable phase  Tunable MZI  c  Iden�ty  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 3: Optical implementation of PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To enable PS on photonic cir-  cuits, we use as variational ansatz the square architecture [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a) The update  of the agent’s memory translates into an update of the phase shifters ⃗θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' b)  Any unitary U can be implemented with a regular mesh of Mach-Zehnder  interferometers (MZI), consisting of two 50:50 beam splitters (BS, blue) and  two phase shifters (θ1, θ2) [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' c) When θ = π all light entering a MZI  is reflected (bar state), while when θ = 0 it is transmitted (cross state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' d)  Square, universal architectures can be carved inside larger ones by tuning the  phases around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In this panel, a universal 4 × 4 circuit (in/out modes are  labeled) is carved in a larger mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Tilted elements represent tunable phase  shifters (those inside the carved structure are in fuchsia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' BSs with arbitrary  transmissivities (orange) are implemented using tunable MZIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Light can be  confined to the pink waveguides by setting the green optical elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 3 and App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' F) [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The variational parameters ⃗θ  correspond to the phase settings that control U k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In the fully coherent case, without additional attenuators,  one can set Σ = I in the singular value decomposition,  meaning that the U k become unitary transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In  this case, due to unitarity constraints and since each U k  acts as an adjacency matrix, additional edges between clips  are created that are not present in the original ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This  limitation represents an additional departure from classical  PS, which can open up interesting opportunities in the context  of learning, for instance leveraging the constraints to improve  generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Percept encoding — We associate each percept with a  photon input in a spatial mode of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The size of the  Hilbert space grows linearly in the number of percepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  ECM structure — We associate each clip with one optical  mode of the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The unnormalized transition prob-  abilities hij that describe the transitions ci → cj are now re-  placed by transition amplitudes between optical modes, de-  scribed by a complex amplitude U k  ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Our chosen architecture is especially useful for reasons that  go beyond its universality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' noise resilience and compactness  5  Element  Classical PS  Quantum PS (→ Optical implementation)  Decision making  Random walk of one excitation  Quantum walk of a single excitation (photon)  Agent’s memory  Directed graph of clips  Variational (optical) circuit  Clips  Vertices of the graph  (Optical) mode operators  Clip-to-clip transitions  h-values  Quantum scattering amplitudes  Input/Output encoding  Percept/Action clips  Input/Output modes  Learning mechanism  Update of h-values via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (2)  Update of variational parameters (phase shifters) via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (8)  TABLE I: Links between the main elements of projective simulation (PS) and its proposed optical quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  [33], since it also promises to support key features of PS:  Clip creation/deletion: new clips (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=', optical modes)  can be added during the learning process, while clips  that are rarely traversed can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Edge creation/deletion: the random walk can be manip-  ulated by creating or deleting links between clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Clip composition: fictitious clips can also be created, by  random variation or merging of existing clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  We observe that this mesh allows one to carve substructures  that are in turn universal for unitary transformations, by  adjusting the surrounding phases (without the need to modify  the hardware;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  This means that the Hilbert  space describing the ECM can be dynamically enlarged  to accommodate new clips, and the U k  ji can be controlled  by intervening on the regions of the circuit that implement U k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Action decoding — In PS, the decision-making process  ends when an action clip is hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In the proposed implemen-  tation, this corresponds to detecting photons from one of the  output modes that correspond to an action clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Whenever a  photon is detected in a mode that does not correspond to an ac-  tion, that event is discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In the future, these photons could  be routed back into the circuit, or associated with mechanisms  of suspension of belief [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  TRAINING THE ARCHITECTURE  In this section, we discuss two algorithms to train the quan-  tum optical PS agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To this end, we first recall that the stan-  dard update rule in PS has no straightforward quantum ana-  logue: while only one path is taken by the classical excitation  and rewarded by the update rule (technically, the h-values as-  sociated with its edges), all paths from percept to action are  visited by the quantum excitation, and the very notion of path  loses its meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Hence, also motivated by the advances in  quantum machine learning, here we take a different, varia-  tional approach to update the parameters ⃗θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Variational approach to mimic classical PS  The first algorithm to train the quantum PS agent is based  on a variational optimization of a loss function L inspired by  classical standard PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Since a quantum circuit does not have access to the inter-  mediate states of the computation, the update rule we propose  adapts that of a classical 2-layer ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This does not mean  that the quantum agent is necessarily 2-layered (in fact, the  architecture supports multi-layer ECMs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  rather, it means that we update the observable transition prob-  abilities psa(⃗θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In turn, also the intermediate transition am-  plitudes in the circuit are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' However, the update rule of  PS requires the use of unnormalized h-values instead of (nor-  malized) probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Thus, let us first introduce the classical  normalization factor of the h-values hs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' hs := �  a hsa,  from which we get hsa = psahs and the classical 2-layer PS  update rule  p(t+1)  sa  (⃗θ)h(t+1)  s  = 1 + (1 − γ)  �  p(t)  sa h(t)  s  − 1  �  + g(t)  sa R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (7)  A loss function L that implements Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (7) is  L(⃗θ(t+1),⃗h(t+1)  s  ) =  �  s  D  �  p(t+1)  sa  (⃗θ)h(t+1)  s  − 1, (1 − γ)  �  p(t)  sa h(t)  s  − 1  �  + g(t)  sa R  �  (8)  Here, ⃗θ(t+1) and ⃗h(t+1)  s  are variational parameters, while D  is a suitable distance measure such as the Kullback-Leibler  divergence (after rearranging the arguments of D), R is the  reward, and gsa is the glow factor described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The  expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (3) is still valid in the proposed quantum  extension, since the update rule underlying the loss function  is taken from a classical two-layer PS, and both input (s) and  output (a) are classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' However, we emphasize again that  the quantum ECM itself is multi-layered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Note that only the first argument of D in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (8) contains  tunable parameters, while the right entry contains the target  values calculated from the previous state of the agent, as a  6  reference, similar to deep-Q learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Similarly, one can use  experience replay [35] to collect rewards and glows for a batch  of percept-action pairs, and add up or merge the loss functions  in appropriate ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' That is because continuous online learn-  ing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' updating after each action) has the disadvantage that  the unitarity constraints will also affect the transition proba-  bilities of undetected percept-action pairs, without taking their  rewards into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In the following, we discuss suitable heuristic simplifica-  tions of this loss function, which make it more amenable to an  actual implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' First, by approximating h(t)  s  ≈ hs and  introducing an effective reward rs :=  R  hs in the exact update  rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (7), we obtain  p(t+1)  sa  (⃗θ) − 1  hs  ≈ (1 − γ)  �  p(t)  sa − 1  hs  �  + g(t)  sa rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (9)  Without a reward (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' rs = 0), the forgetting mechanism  after the approximation seeks to reset p(t)  sa to  1  hs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' However,  in information theory, the no-information distribution is the  uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This motivates us to replace  1  hs with a  uniform distribution  1  |A|,  p(t+1)  sa  (⃗θ) =  1  |A| + (1 − γ)  �  p(t)  sa − 1  |A|  �  + g(t)  sa rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (10)  So far, we did not make any choices concerning the source of  the rewards R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' For our simplifications, we now take the point  of view that the reward R is designed by a programmer to  achieve an intended agent behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Then, a further simplifi-  cation can be made by directly modelling the effective reward  r, rather than R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Therefore, we replace rs with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' With this  change in mind, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (10) now looks like an analogue of the  exact update rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (7), but for probabilities instead of  h-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To ensure that the agent updates get smaller over  time, it is recommended to use an “annealing schedule” for  the reward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' to put an additional time dependence into the  reward of the form r := f(t) · r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here, f(t) is a function that  monotonically decreases from 1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This allows the training  algorithm to make finer and finer updates to the probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In order to stay as close as possible to classical PS, the up-  date rule would have to be applied to all percept-action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  This would require implementing the forgetting mechanism  (γ > 0) for unvisited percept-action pairs, or enforcing the  transition probability of unvisited pairs to remain unperturbed  (γ = 0), leading to a complicated loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To avoid  this overhead, we include only the actually observed percept-  action pairs in the loss function, while the transition probabil-  ities of unobserved pairs are allowed to change according to  the unitarity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Combining all of the above simplifi-  cations, we obtain the loss function  L(⃗θ) =  �  (s,a) | g(t)  sa >0  D  �  p(t+1)  sa  (⃗θ) ,  (11)  C  � 1  |A| + (1 − γ)  �  p(t)  sa − 1  |A|  �  + g(t)  sa r  � �  where C is a (smooth) cutoff function to enforce that proba-  bilities are in [0, 1], and only (s, a)-pairs with non-zero glow  are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Variational approach based on causal diamonds  In the previous section, we implicitly assume that at each  step one tunes all phase shifters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=', parameters ⃗θ) in the cir-  cuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Indeed, this is possible and can be beneficial to train the  agent faster and with a better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' At the same time,  updating all phases at each step might not be necessary for the  following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In a square architecture with |C| input/output modes, the  number of phase shifters is quadratic in |C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' However, for a  fixed transition probability psa(⃗θ), one can see that only some  phases play a role: Those at the intersection of the future and  past light cones of s and a, respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In the  literature on relativity, this intersection is called causal dia-  mond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A number of considerations follow:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To update psa(⃗θ), one could simply tune the phase  shifters inside the causal diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' For light to reach a from s, what matters is that light  does not leak out of their causal diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Hence, one  can also just tune the O(|C|) phases that lie on the sur-  face of the causal diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Since leakage only occurs on the surface of the past  light cone, it is sufficient to focus on its intersection  with the causal diamond (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We call these  phase shifters leaking nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Overall, in terms of reconfiguration time, stability and  power consumption, the above observations allow for progres-  sively simpler experimental requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The price to pay is  that the surfaces of different causal diamonds (for different  (s, a) pairs) intersect and, for this reason and due to unitarity  constraints, updating one transition probability also affects the  other transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Also, the fewer phase shifters are tuned, the  smaller the set of values that solve the optimization problem  when multiple (s, a) pairs are rewarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The desired trade-off  can be found with the experimental conditions in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  We consider two main strategies to tune the sets of relevant  phases described by points (i-iii) above: (a) phases are in-  dependently updated by gradient descent, and (b) phases are  updated sequentially (from left to right, each phase in a layer  is set to a new value before the phases in the next layers are  adjusted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The latter strategy is possible because we assume  that light propagates only forward, and it leads to a faster and  smoother training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  A quantitative comparison between the  two strategies, and a study of the most suitable one with multi-  ple rewarded percept/action pairs, can be done with our shared  package, keeping experimental requirements in mind [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' An  illustrative analysis for strategy (b) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Addi-  tional information can be found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' E 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Importantly, we point out that this approach is not restricted  to the regular mesh considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Rather, similar  considerations also apply to circuits with arbitrary mode con-  nectivity and even higher-level structures in the ECM [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  7  a  c  b  Steps  psa  0  250  500  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='75  1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 4: Update via causal diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To reinforce a transition probability psa in a 2-layer PS, it is not necessary to update all phase shifters at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  a) Only the phases inside the causal diamond of (s, a) influence psa (colored nodes indicate the surface of the causal diamond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Since light can only leave the  causal diamond from the purple nodes (leaking nodes), it is sufficient to tune their phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' b) The approach works also with multiple transitions, by iteratively  tuning different sets of leaking nodes (here shown with five different colors, including gray for the overlaps, and representing pairs of adjacent phase shifters  and beam splitters as a single node), even if the updates compete with each other (some leaking nodes belong to multiple causal diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' c) The agent learns  to maximize psa for the four (s, a) pairs in panel (b), by tuning only the leaking nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In some cases, it may be necessary to tune more nodes to maximize all  probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Each percept/action is encoded in a pair of adjacent input/output modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Direct updating via a Gram-Schmidt process  The previous sections consider a variational approach to the  optimization problem, where phases are gradually adjusted  to produce a new transformation that behaves better under a  given figure of merit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We now consider a complementary ap-  proach, where we first update the unitary evolution U of the  ECM, and then use a well-known decomposition [33] to re-  trieve the ⃗θ that produce the desired U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In this case, one di-  rectly adjusts the matrix elements of U to implement an up-  date rule such as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (7) or (11): the new matrix is no longer  unitary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' however, a Gram-Schmidt orthonormalization (GSO)  process yields a new U ′ that approximates the intended U ′  as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  We add a few remarks on this approach (see also App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' E 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  First, if U ′  as is close to the old value Uas, one step of GSO  only causes a small update of the other entries of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In this  sense, this approach allows for a smooth, flexible and con-  trolled update of U (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Also, both the GSO and the  unitary decomposition are efficient and relatively fast subrou-  tines, which could even be faster than collecting statistics for  other variational approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In other words, the overhead  is outsourced to the classical CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The limitation of this ap-  proach is that it assumes perfect knowledge of U and a precise  control of the experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The impact of imperfec-  tions can be estimated using the shared package [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  QUANTUM VERSUS CLASSICAL PROJECTIVE  SIMULATION IN A TRANSFER-LEARNING SCENARIO  In this section, we apply the proposed quantum PS to an  actual learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The PS update rule of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (7)  and (8) has already been used in various applications [13–  16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here, we consider a scenario which pronounces the dif-  ferences between classical and quantum PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We consider a  transfer-learning scenario [36]: a machine learning setting  where a model trained for one task can solve another task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The scenario we choose is that of Eva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' [37] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  0  25  50  75  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='5  1  p  ∼  Steps  sa  p  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 5: Update via a Gram-Schmidt process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We can directly update  the unitary transformation U that describes the ECM using a Gram-Schmidt  orthonormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here, a learning curve (purple) is shown for a single  rewarded (s, a) pair in a 2-layer PS described by a 10 × 10 Haar-random U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  All other transition probabilities (other colors), from s and to a, decrease to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Curves correspond to a single agent, without average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Inset: (|Uij|2)ij  after training (arrows indicate (s, a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In this scenario, one considers a list of particles character-  ized by discretized observables, and asks whether two observ-  ables have two particular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To answer these questions,  or tasks, in our adapted version of this thought experiment the  agent attempts to build a meaningful representation of the par-  ticles in two stages (corresponding to the middle layer and a  task layer, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 6a): In stage 1, the agent learns a represen-  tation from the percept layer to the middle layer, where each  node corresponds to one observable and one value of that ob-  servable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' While Eva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' intended that this representation  is learned implicitly, in our case we directly train the middle  layer to achieve this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In stage 2, the connec-  tions from percept to middle layer are kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The task  layer has two nodes, which correspond to yes/no-answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' For  each question (task), we train a separate set of connections  from the middle layer to the corresponding task layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  8  a  c  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  1,2,3  1,2,3  1,2,3  Percept 𝑠  Values  Tasks  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Values  1 … 3  Not used  Yes  No  𝑠  𝑈  Tasks  1 … 3  1 … 3  𝑠  𝑣  𝑒  Value  (27)  b  (9)  d  Aq  c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Experiments  1  27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='95  1  Steps  Aq  A  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 6: A transfer-learning scenario for PS agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a) Structure of the  ideal, classical ECM considered in the transfer-learning scenario in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Panel adapted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' b) Percepts s consist of the values of three ob-  servables which characterize particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The experiment consists of two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (i) In the first stage, the PS agent is set up to learn a middle layer of nodes,  in which each node corresponds to one observable and one value of that ob-  servable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (ii) In the second stage, this middle layer is kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The task  layer is exchangeable and it represents one yes/no question about pairs of ob-  servables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Since the excitation of the classical ECM is localized, in the middle  layer the classical agent can only use the value of one observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Interference  allows the quantum PS agent to combine knowledge of several observables at  once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' c) Layout of the quantum PS agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To fulfill the unitarity constraints,  the connection from percept to middle layer is an interferometer tree with 9  outputs (one tree per percept).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The connection from the middle layer to the  task layer is a 9 × 9 square architecture, shared by all percepts but different  for each task layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Two output modes correspond to the yes/no answers, the  others are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' d) Prediction accuracy AQ of the quantum PS agent  for each of the 27 experiments (yes/no question).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Inset: Learning curve for  a single experiment (dark orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A classical agent can achieve at most an  accuracy AC = 8/9 (horizontal line, for both plots), always lower than AQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The contrast between classical and quantum learning is  rooted in the way the decision-making process handles the  structured representation of the PS agents: Since the exci-  tation of the classical PS agent is localized, it can only use  knowledge about one observable and, therefore, it cannot  reliably answer questions involving two observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The  quantum PS agent, however, can use interference to combine  knowledge about two observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  To be more specific, Eva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' [37] consider three observ-  ables, each of which can take three values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We label these  observables and their values 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Each percept can be writ-  ten as (v0, v1, v2) with vj ∈ {0, 1, 2}, and vj is the value of  observable Oj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In total, there are 33 = 27 percepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In ad-  dition to the percept layer, there is a middle layer and a final  layer (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The purpose of the middle layer is to explic-  itly represent the value of each observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' More specifically,  this layer contains 32 = 9 nodes, each corresponding to one  observable-value pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' For each percept (v0, v1, v2), the idea is  that the agent learns to visit the middle clips (Oj, vj) with uni-  form probability 1  3, while not visiting the other middle clips  (Oj, v) corresponding to wrong values v of the observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  For  the  proposed  quantum  PS,  we  represent  the  correct  middle-layer  state  of  percept  (v0, v1, v2)  as  1  √  3 (|v0⟩ + |3 + v1⟩ + |6 + v2⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  For example,  (2, 1, 2)  is represented by  1  √  3 (|2⟩ + |4⟩ + |8⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Since these states can  be non-orthogonal for different percepts, for each percept we  use a separate binary tree (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 6b) with 9 output leaves  (27 such trees in total), instead of the unitaries U k shown in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  To make a fair comparison to a classical PS agent, we con-  sider the following training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Each of the 27 percepts is sam-  pled with uniform probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Then the middle layer is mea-  sured to output one (observable, value) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Depending on  whether the pair is predicted right or wrong, a fixed reward  ±r is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We use the simplified loss function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (11)  without forgetting-mechanism (γ = 0), and only reward the  current percept-action pair (η = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To enforce that all relative  complex phases are close to zero, we add the ℓ1 norm of these  phases to the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Since we use separate binary trees  for each percept, and the goal of the tree is just to prepare one  simple state, the middle layer can be trained to achieve the  intended representation essentially perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  After the middle layer is trained, we fix its weights, mean-  ing that we will not adjust the binary trees for anything that  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Now, we introduce the final layer, which we call the  task layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The task layer represents yes-no-questions about  the values of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Our transfer learning scenario re-  quires that the task layer be exchangeable, while we keep the  middle layer fixed and independent of the task layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The task  layers we consider here pick two observables O, O′ and ask  whether these observables have specific values vO and vO′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  There are 27 =  �3  2  �  32 such task layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The quantum PS  agent uses a 9x9 square architecture for each task layer, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We emphasize that the same task unitary  is used for each percept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Only two output modes are used,  which correspond to yes/no-answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The 27 percepts are  sampled uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 6d) shows the achieved (weighted)  accuracies for all two-observable task layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Our agent con-  sistently achieved over 97% (weighted) accuracy for all task  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The analysis was implemented with PyTorch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' details  are provided in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  While the proposed quantum PS agent can achieve near per-  fect transfer learning, one can use analytical reasoning to see  that the classical PS agent cannot achieve near perfect accu-  racy: Since the classical excitation is always localized in one  node, in the middle layer the classical PS agent can at most  know the value of one relevant observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Therefore, the best  thing the agent can do is to guess about the other observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Since two out of three possible values of that other observable  lead to a right no-answer, the best policy the classical PS agent  can represent is to always answer no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Hence, the performance  of the classical PS agent is upper bounded by 8  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  9  DISCUSSION  In this work, we introduce a variational approach to quan-  tum reinforcement learning that can be implemented with in-  tegrated photonic quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  To this end, we  started from preliminary results by some of the authors [10],  based on the projective simulation (PS) learning model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  This allowed us to establish a map between the key elements  of classical PS and its counterparts in the quantum domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  At the core of this analogy lies the random walk of a physi-  cal excitation, which becomes a quantum walk in the agent’s  memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We then discussed an approach to implement the de-  cision making process and training of the quantum PS agent  using photonic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  To this end, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III we introduced several variational  methods to train the agent, all compatible with, or tailored to,  a quantum optical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The variational parame-  ters are represented by the phases in the optical architecture,  given by the canonical mesh of Mach-Zehnder interferome-  ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' More specifically, we proposed a variational method that  is general and can accommodate custom loss functions in-  side the update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In particular, it allows to draw a close  connection to the update rule of classical PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We proposed a  modification of the variational training that exploits features  of the interferometer implementation to ease the experimental  requirements (by reducing the number of parameters that need  to be updated during the learning process), which makes train-  ing the architecture more efficient and flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Furthermore,  we proposed an alternative method for updating that consists  of a deterministic and exact algorithm to update the overall  unitary transformation that describes the quantum PS agent’s  memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' All algorithms can also be combined to refine the  update or to address more complex structures in the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  As a specific numerical experiment that demonstrates im-  portant differences between classical and quantum PS, we  considered a transfer learning scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We show how quan-  tum interference allows the quantum PS agent to combine in-  formation from several clips, while the classical PS agent only  has access to the information of one clip at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  This work also sets the stage for a number of follow-up  investigations, which we outline in the Appendix with addi-  tional numerical analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In particular, Apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' B and C dis-  cuss the potential offered by multi-photon (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=', multi-percept)  and multi-frequency quantum information processing, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Finally, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' D takes a first step towards a notion of  interpretability in quantum artificial intelligence, an issue that  has attracted more and more attention in the last few years and  that is especially relevant within PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' All results can be tested  with a package we release online [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This project has received funding from the European  Union’s Horizon 2020 research and innovation program un-  der the Marie Skłodowska-Curie grant agreement, Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  801110 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 885567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' It reflects only the author’s view,  the EU Agency is not responsible for any use that may be  made of the information it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' ESQ has received fund-  ing from the Austrian Federal Ministry of Education, Science  and Research (BMBWF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This work was supported in part by  the Austrian Science Fund (FWF) through the SFB BeyondC  F7102 and the Volkswagen Foundation (Az:97721).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  This  work was also co-funded by the European Research Coun-  cil (ERC) under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 101055129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Views and opinions  expressed are however those of the authors only and do not  necessarily reflect those of the European Union or the Euro-  pean Research Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Neither the European Union nor the  granting authority can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Jumper, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Evans, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Pritzel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Green, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Figurnov,  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Ronneberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Tunyasuvunakool, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
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+page_content=' Meyer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
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+page_content=' Bal-  lard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
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+page_content=' Romera-Paredes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
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+page_content=' Sifre,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
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+page_content=' Pan-  neershelvam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Lanctot, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Dieleman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Grewe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Nham,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Kalchbrenner, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Sutskever, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
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+page_content=' Gutin, Digraphs: Theory, Algorithms  and Applications (Second Edition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Springer London, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  11  a  b  c  C1  C4  C2  C3  C5  C6  C7  C8  C9  C1  C4  C2  C3  C5  C6  C7  C8  C9  C1  C4  C3  C7  C5  C6  C8  C9  C2  C4  C5  C6  C7  C9  C8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 7: Simulating a quantum ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a) Directed acyclic graph describ-  ing the ECM shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 2, where each vertex corresponds to a clip ck (here  including percepts and actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' b) Vertices, labeled as in panel (a), are rear-  ranged in a layered structure and recolored depending on the order in which  the corresponding unitary acts on the excitation (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' H for a discussion  on the ordering of the unitaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Some unitaries that appear consecutively  in the ordering can be seen to commute trivially and are thus grouped in the  same layer with the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' c) Subgraphs corresponding to percepts c1  and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Vertices are colored as in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  APPENDIX  Appendix A: Package overview  The above considerations can be tested and explored with a  Mathematica package we make available online [12, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Below we describe its main functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Function 1 allows to study the structure of the quantum  ECM proposed in this work, starting from the directed acyclic  graph (DAG) that describes the connectivity of the ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' It  can be used to test the learning model and to develop exten-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This function operates at an abstract level and, specif-  ically, does not presume an optical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Func-  tions 2-4 below can interact with its output by manipulating  the transformations associated with each vertex of the DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Simple examples and details can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Function 1 — InitializeQuantumECM  Input: The directed acyclic graph (DAG) G describing an ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Output: An equivalent G with explicit layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' the vertices of G that cor-  respond to percepts and actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' all subgraphs of G that are reachable  from each percept (where quantum walks take place);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a description of  the clip-to-clip connectivity in G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' extra information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Functions 2-4 have been adapted from a package from  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' They make it possible to manipulate unitary trans-  formations on a linear-optical circuit, with a focus on the  canonical decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' These routines serve to instanti-  ate the abstract representation of a quantum ECM, and al-  0  20  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='5  1  psa  (α)  α  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='3  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  Steps  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 8: Combining causal diamonds and Gram-Schmidt process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The  Gram-Schmidt orthonormalization can be applied to the Uk associated with  an arbitrary path in the ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In this example, we reinforce the transition  probability psa by updating the Uk along the longest path (1,4,5,20) that  connects a percept (s: green, k = 1) to an action (a: red, k = 20) for  the ECM shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here, we assume that at each step the same  (s, a) gets rewarded, for three different factors α (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' E 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Inset: The  method based on causal diamonds can be extended to the ECM graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' While  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III B the analyses zoomed in the structure of each Uk, one can also  zoom out and reinforce psa by tuning the leaking nodes (“leaking Uk”) of (s,  a) along the quantum walk (black vertices, k ∈ {5, 6, 14}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Other percepts  (actions) are colored in light green (light red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  low to study arbitrary circuits as graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' They handle the ⃗θ  parametrization in two formats, one for canonical architec-  tures and one for arbitrary connectivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Function 2 — UnitaryDecomposition  Input: A unitary U and the choice of an algorithm for its decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Output: For canonical architectures [33, 34], it decomposes U into phases  and transmissivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' If beam splitters are implemented as Mach-Zehnder  interferometers, it returns the appropriate phase settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Function 3 — GetUfromParameters  Input: Parameters in the format handled by Function 2, or in a layer-wise  format that allows arbitrary connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Output: Unitary generated by any of three methods with the input parame-  ters (see Function 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Optional noise and/or losses can be set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Function 4 — GetGraph  Input: Information on the mode connectivity in the circuit, in the format  handled by Functions 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Output: Graph representation of the input circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Functions 5-6 enable the update based on the causal  diamond for a 2-layer PS (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III B and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 4) and multi-  layer PS (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' They make use of the graph representa-  tions given by Functions 1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Function 7 implements the  update based on the Gram-Schmidt process (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Function 5 — FindLeakingNodes(AnyDAG)  Input: Graph G describing an optical circuit / a quantum ECM (see Func-  tion 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a percept/action pair (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Output: Vertices of G in the causal diamond and leaking nodes of (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  12  Function 6 — UpdateCausalDiamond  Input: Layered parametrization of the circuit (partially pre-computed for a  faster evaluation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a percept/action pair (s, a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' phase shift δθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Output: New layered parametrization and the corresponding unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Function 7 — UpdateGramSchmidt  Input: A unitary U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a percept/action pair (s, a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a rescaling factor α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Output: Updated U after one Gram-Schmidt process around Uas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Appendix B: Multi-photon quantum walks  In this work, we replaced the excitation at the core of the  decision making process of PS with a single-photon excitation  in a quantum walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In this picture, a photon is always injected  into the mode that corresponds to a percept, and the new ac-  tion is given by the output mode the photon is measured in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' An  interesting extension of this framework considers the simulta-  neous injection of multi-photon states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A quantum PS agent  equipped with this feature could be able to process multiple  percepts at the same time, or a percept with a more complex  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Furthermore, information encoded in n indistin-  guishable photons injected in n different input modes would  interfere at the level of quantum amplitudes, potentially open-  ing up advantages with respect to the classical scheme, where  such a feature is currently and independently under investiga-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Foreseen potential advantages in this sense are: (i) the  advantages inherited by a multi-excitation classical PS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (ii) the  exponentially larger (in n) percept/action (Hilbert) space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (iii)  quantum computational advantages related to the hardness of  simulating the dynamics [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Appendix C: Quantum walks and frequency encoding  An interesting possibility offered by an optical implemen-  tation is to use additional degrees of freedom of photons, be-  sides the spatial modes, to encode information and/or to train  multiple agents in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In particular, frequency appears  to be a convenient candidate since the operation of integrated  photonic circuits is wavelength-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In fact, both beam  splitters’ and phase shifters’ transformations depend on the  wavelength as follows [41]:  U bs  λ ∝  �  cos π  4λ − sin π  4λ  sin π  4λ  cos π  4λ  �  ,  U ps  λ (P) ∝  �1  0  0 ei P  λ  �  (C1)  where P is the (suitably rescaled) power setting that controls  the phase shifter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' With these simplified expressions, we can  already get an idea about the average impact of frequency en-  coding on unitary transformations U of increasing size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Re-  sults for this analysis are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We see that, as  expected since the architecture consists of hundreds of opti-  cal elements, a tiny variation in wavelength λ leads to signifi-  cantly different effective Us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This means that photons injected  with these deviations would undergo different Us, possibly  implementing quantum walks in different quantum ECMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  a  b  p  Size of U  Steps  sa  (λ)  F  δ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='01  0  250  500  750  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='015  20  30  40  50  60  70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='95  1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 9: Wavelength-dependent quantum walks in the agent’s mem-  ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The wavelength dependence of beam splitters and phase shifters turn a  limitation of optical technologies into a potential advantage (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a)  Fidelity of random effective unitary transformations U of increasing size, as  perceived by photons with different separation in wavelength λ (see legend),  averaged over 100 pairs of random unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Each U is numerically generated  by uniformly sampling power settings in the phase shifters (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' C1) of a  square architecture where beam splitters are implemented as tunable Mach-  Zehnder interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Standard deviations are smaller than the point size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  b) Four learning curves for a 2-layer PS agent based on a 10×10 square archi-  tecture (four rewarded percept/action pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' each percept/action is encoded in  a pair of adjacent input/output optical modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The figure of merit optimized  by the variational approach based on causal diamonds (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III B) is the  geometric mean of the transition probability p(λ)  sa for three values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Finally, we investigate whether a variational approach can  work also while additional information is encoded in the fre-  quency degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To do so, we simulate quantum  learning agents that attempt to solve the same RL task (same  rewarded percept/action pairs) using different λs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Inciden-  tally, this analysis is similar to, but more sophisticated than,  the one presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 9b,  using the approach based on the causal diamond in a 2-layer  PS (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We see that the agent learns to map each  of five percepts to the corresponding rewarded action, at the  same time for three different λs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Importantly, all phase set-  tings are the same for all λ, that is, the phase configuration  found by the algorithm operates well (as shown) when input  with different λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The above considerations suggest that there is a large poten-  tial in the use of additional degrees of freedom that influence  the quantum walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Foreseen potential advantages of this as-  pect are: (i) extra room for more complex RL mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (ii)  parallelism in the agent’s decision making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  13  Appendix D: Towards interpretable quantum artificial  intelligence  While neural networks show excellent performance, their  black-box-like nature can be problematic for some purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  For example, in delicate decisions that directly impact hu-  mans, one may wish to have a justification before following  the advice of an enigmatic machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Similar concerns apply  in science where, rather than developing an oracle, one might  aim to acquire a deeper understanding of a given phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  These issues have recently sparked interest in interpretable  and explainable artificial intelligence [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' PS offers one vi-  able framework towards this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Differently from neural  networks [17], in classical PS an interpretation can be found  by studying the path taken from percept to action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' However,  this is no longer true in the quantum domain, where we lose  the notion of observable path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Since in a quantum walk all al-  lowed paths contribute to the decision making process, a quan-  tum PS agent loses interpretability in the classical sense (there  is no longer a unique path that can be retraced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  A notion of interpretability that is applicable to quantum  agents is currently being explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here we make a first step  in this direction by observing that a notion of partial trace can  still be recovered, if one studies the likelihood that a photon  (or more photons, as in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' B) would be measured in any  given mode of the circuit (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This information can  be retrieved by classically simulating the single-photon quan-  tum walk, since all phase settings are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This possibility  still holds for multi-photon quantum walks despite the com-  putational complexity of the task, which becomes practically  unfeasible only from n ∼ 30 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Appendix E: Variational algorithms  In this section, we provide additional information on the  two variational algorithms presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The following algorithms naturally take into account the  unitarity constraint inherent to the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We point out  that this constraint is not necessarily a limitation and, rather,  it could be advantageous to improve the agent’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In fact, since a machine learning ansatz with more freedom is  also more prone to overfitting, it is reasonable to believe that  a specialized ansatz with restrictions and symmetries might  generalize better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A reduced number of degrees of freedom  might also help during the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Update via Gram-Schmidt process  Here we outline the update via a Gram-Schmidt process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The algorithm consists of four steps (i-iv):  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Compute U (t+1)  a0s0  from the current U (t)  as according to a  desired update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here, s0 and a0 are the observed  percept and action of the current round, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In the examples in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III C, we considered a constant  rescaling factor α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' U (t+1)  a0s0  = α U (t)  a0s0, which acts  like a learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Normalize the row U (t+1)  a0,• .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Use GSO on the other rows to obtain a new unitary, in  the following way: For each row a ̸= a0, define  ˜U (t+1)  as  := U (t)  as −  �  ˜s∈S  �  ˜a∈Aa  U (t+1)  ˜a˜s  U (t)  a˜s U (t+1)  ˜as  ,  (E1)  where the overbar denotes the complex conjugation,  and Aa is the set of rows ˜a for which we have already  defined U (t+1)  ˜a,•  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' the previous ˜a in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Deter-  mine U (t+1)  a,•  from ˜U (t+1)  a,•  via normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Use the decomposition [33] of canonical square archi-  tectures to find the phase settings that implement the  new unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The decomposition is exact and requires a  number of operations quadratic in the size of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The main advantage of this approach is that it allows one  to control any theoretical aspect of the learning process, from  the update rule to the learning speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' One limitation is that  it assumes perfect knowledge and control of the experimental  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Appending to it one step of variational fine-tuning, as  the one we describe in the following section, can help mitigate  this issue and speed up the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Update via causal diamonds  Here we provide additional considerations on the update  based on causal diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The advantage of this approach  is that it is efficient (only few phase shifters are adjusted at  each step) and that it is variational (it takes into account the  imperfect control over the experimental settings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  We consider two main approaches (a,b) to update a given  subset Θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' , θN) of N phase shifters at each training  step, albeit more sophisticated strategies can be devised and  tailored to the hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' These phases can correspond to the  set of leaking nodes described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III B, or to a larger set  selected via other criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Gradient ascent — Each phase θ ∈ Θ is independently  tuned to maximize a predefined figure of merit F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Sequential — Each phase θ ∈ Θ, in the order given by  the light propagation, is tuned to independently maxi-  mize a figure of merit F, and is immediately set to the  new value before the next phase is probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The latter approach (b) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 8) is usually  faster than (a), since each phase update starts from a progres-  sively higher value of F, and the individual updates within a  training step do not sabotage each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' However, an update  based on gradient ascent as in (a) can be beneficial to escape  local minima, which are more likely to cause problems in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Appendix F: Optical architecture  In this section we review the basics behind the operation of  the optical architecture discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' II B,  14  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 10: Partial traceability in the decision making process of quantum PS agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' a-c) Three quantum walks, associated with as many 2-layer ECMs,  are numerically simulated with a square optical architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Each panel shows the probability P to measure a photon in one of the output modes before training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Each column (label: Layers) displays the squared column (for a fixed percept) of the unitary matrix obtained by adding a new layer of optical elements to the  square architecture, from left (input layer) to right (full circuit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The rightmost column of each plot corresponds to the output probabilities of a quantum walk  over the full ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' d-f) Same analysis repeated after training with the variational approach based on the causal diamond (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Panels (d), (e) and (f)  correspond to the evolutions in panels (a), (b) and (c) after training, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We observe that a notion of partial trace emerges inside the circuit, which  might be leveraged to study interpretability in quantum learning agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Analogous results (not shown) are found for the numerical simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  we proposed to consider the square canonical layout [33] for  several reasons, above all because it supports the implemen-  tation of any unitary transformation U, with no modifications  at the hardware level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This is possible because any m × m  U can be decomposed as U = D �  b Ub, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=', the product of  a diagonal matrix D (whose elements are complex and have  modulo 1) and m(m − 1)/2 unitaries Ub implementing com-  plex rotations in the plane spanned by two adjacent modes:  Ub =  �  �  �  �  �  �  �  �  �  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
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+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 1  �  �  �  �  �  �  �  �  �  ,  (F1)  where the uij (i, j ∈ {1, 2}) describe the transformation act-  ing on the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  This 2 × 2 transformation is im-  plemented by means of a pair of phase shifter (θ1) and im-  balanced beam splitter, whose transformation U BS is imple-  mented as a Mach-Zehnder interferometer with tunable phase  shifter (θ2):  U BS(θ2) = 1  2  �  1 i  i 1  � �  eiθ2 0  0  1  � �  1 i  i 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (F2)  Overall, the proposed ansatz can be implemented using  well-established, integrated photonic components that only  involve tunable phase shifters and fixed, balanced beam split-  ters (directional couplers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Appendix G: Numerical details of the transfer learning scenario  In this section, we report additional details on the numerical  analysis described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' IV, performed with Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Both classical and quantum PS agents have three layers:  percept, middle and task layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' While in classical PS each  percept s is represented by a vertex in the ECM graph, in  this analysis the quantum PS agent associates with it a (linear-  optical) binary tree [10], to overcome the limitation due to  the unitarity constraints in the middle layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' All its phases  are initialized to π  4 to yield a balanced output distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The classical PS agent is expected to pick an observable uni-  formly at random, and then send the excitation to the mid-  dle clip with the right value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' For the quantum PS agent, we  wish that the percept (v0, v1, v2) is represented by the state  1  √  3  �  |v0⟩ + |3 + v1⟩ + |6 + v2⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To this end, we first train  the binary trees, considering a RL scenario where percepts are  sampled i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We use the simplified loss function  LPS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (11), with the Kullback-Leibler divergence applied  to the probability distribution {psa, 1 − psa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Furthermore,  we use ReLU functions to cut off the target values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (11)  such that they are in [0, 1]:  p(t+1)  sa  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='0 − ReLU  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='0 − ReLU(p(t)  sa + r)  �  (G1)  As a reward, we assign r := ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='1 depending on whether or  not the agent predicts the right value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To ensure that each ob-  servable is picked uniformly, we implement a curiosity mech-  anism by including the Shannon neg-entropy of the observ-  15  ables in the loss function:  LShannon = log(3) +  2  �  j=0  psOj log psOj  (G2)  where psOj is the probability of the j-th observable Oj, given  a percept s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' To enforce that the relative phases among the  modes of the middle layer vanish, we also add to the loss  function the ℓ1 norm of the complex phases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Lphase :=  �8  m=0 |Φsm|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here, Φsm is the phase in the middle-layer  mode m when percept s is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The full loss function  for the middle layer is then:  Lfull = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='0 LPS + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='0 LShannon + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='0 Lphase  (G3)  We use experience replay, and add up the loss functions of a  batch of 600 sampled percepts before running 10 optimizer  steps (Adam optimizer [44]) and wiping the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The learn-  ing rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='01, decreased to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='001 once the accuracy  of the last batch was at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Training stops when the  accuracy of the middle layer exceeds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='99 for 10 consecu-  tive rounds and, at the same time, the ℓ1-norm of the phases  weighted by psm is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Once the middle layer is trained, we fix its phase shifters so  that the trees do not change anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Now, we introduce the  task layer, which is exchangeable and independent of the mid-  dle layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' While Eva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' only considered single-observable  experiments, our task layers pick two observables (O0, O1)  and ask whether they have specific values vO0 and vO1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The quantum PS agent uses one 9x9 interferometer for each  task layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Again, we use experience replay, with a batch size  of 500 (much larger than the number of percepts) and train the  agent for 40000 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Since our tasks ask for specific val-  ues of two observables simultaneously, only 1  9 of the percepts  will correspond to a yes-answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Hence, in addition to the raw  accuracy, we use a weighted accuracy where yes-answers are  given an additional weight 8 (otherwise the agent could just  always answer no, which would yield a seemingly good ac-  curacy of 8  9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The loss function is the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (8), with a  reward equal to ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='1 for a right or wrong answer, and an addi-  tional factor 8 if the right answer is yes, the Kullback-Leibler  divergence for {psa, 1 − psa}, ReLU functions to cut off the  target values and neither glow nor forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We normalize  the probability of the yes- and no-mode of the task layer to  one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' we post-select on getting one of these two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  For the learning rate and number of optimizer steps we use  the same settings as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Overall, the quantum PS agent consistently achieves over  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='97 accuracy in all two-observable tasks (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Appendix H: Classical-quantum equivalence of ECM graphs  In this section we formalize the connection between the di-  rected graph of a classical ECM and the quantum walks that  derive from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We refer to this set of quantum walks (from  input to output) as unitary routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We begin by introducing  some basic elements from graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Definition 1 (Topological ordering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A topological ordering  oG of a directed graph G = (V, E) is a map, o : V →  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' , |V |}, which defines a total ordering of vertices of G  such that for every directed edge (vj, vk) from vertex vj to ver-  tex vk, vj comes before vk in the ordering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=', o(vj) < o(vk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Let OG be the set of all topological orderings of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  An example of topological ordering is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Directed acyclic graphs (DAGs) are directed graphs that do  not contain loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' They can be defined as follows [45]:  Definition 2 (Directed acyclic graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A directed graph is  acyclic if and only if it admits a topological ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Thus, the graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 11 is an example of a DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' It is worth  noticing that the topological ordering of a DAG such as in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 11 is not unique, because swapping consecutive vertices  (in the topological ordering) which are not connected by an  edge yields another valid topological ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Below we recall two additional, useful definitions:  Definition 3 (Ordered DAG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' An ordered DAG ⃗G is a  DAG (V, E) with V  = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' , |V |} such that o : V  →  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' , |V |}, o(j) = j is a topological ordering of (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  For any DAG G = (V, E) with topological ordering o we  denote the induced ordered DAG by  ⃗G(o) =  �  {o(vi) | vi ∈ V }i,  (H1)  {(o(vj), o(vk)) | (vj, vk) ∈ E}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Let ocDAG be the set of all ordered and connected DAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  𝑐4  𝑎1  𝑠2  𝑐5  𝑎2  𝑐3  𝑐2  𝑐1  𝑠1  𝑠1  𝑠2  𝑐1  𝑐2  𝑐3  𝑐4  𝑐5  𝑎1  𝑎2  𝑐4  𝑎1  𝑠2  𝑐5  𝑎2  𝑐2  a  b  𝑠1  𝑐3  𝑈𝑐5𝑠1  5  𝑈𝑐2𝑠1  2  𝑈𝑐1𝑠1  1  𝑐1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 11: a) Directed acyclic graph shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 2 (top) and one of its  topological orderings (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Notice that, for the sake of simplicity, all ver-  tices had already been labeled with subscripts in ascending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' b) Fulfilling  unitarity constraints in the mode-mixing transformations Uk in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' (H3) may  generate edges (blue) that are not present in the classical ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  16  Object  Intuition  Routing  Sequence of parametrized unitary transformations,  constructed from a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Route  A specific sequence of the routing, where all param-  eters have been fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Router  The map from Go to its routing  TABLE II: Summary of the main quantities introduced in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Definition 4 (Parents and children).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Let G = (V, E) be a  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' For any vertex vi ∈ V , let PG(vi) be the set of parents  of vi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=', PG(vi) = {vj|vj ∈ V, ∃(vj, vi) ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' For any  vertex vi ∈ VG, let CG(vi) be the set of children of vi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=',  CG(vi) = {vj|vj ∈ V, ∃(vi, vj) ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Equipped with the above definitions, we proceed by defin-  ing an ECM graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Here we focus on ECMs described by  connected DAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Definition 5 (ECM graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' An ECM graph is a connected  DAG G = (V, E) whose vertices are called either percepts,  S = {vj|vj ∈ V, PG(vj) = ∅}, actions A = {vj|vj ∈  V, CG(vj) = ∅}, or intermediate clips C = V \\(S ∪ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  The quantum version of a classical ECM graph is defined  via a unitary routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We refer to Table II for an overview of  the terminology introduced in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Definition 6 (Unitary route).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Let G = (V, E) be an ECM  graph and o a topological ordering of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' A unitary route of  the induced ordered DAG ⃗G(o) is a sequence of |V | × |V |  unitaries, ordered in ascending order of their superscripts,  route( ⃗G(o)) =  �  U o(vi)�  vi∈C∪A ,  (H2)  whose matrix elements are given by  U i  jk =  �  ui  jk  if j, k ∈  �  {i} ∪ P ⃗G(o)(i)  �  δjk  else,  (H3)  where the ui  jk ∈ C are only subject to unitarity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  For example, with reference to the example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 11  and the topological ordering (o(s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=', o(a2)) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=', 9),  PG(c2) = (s1, s2) and P ⃗G(o)(4) = (1, 2), hence  U 4 =  �  �  �  �  �  �  �  �  �  �  �  u4  11 u4  12 0 u4  14  0  · · 0  u4  21 u4  22 0 u4  24  0  0  1  0  u4  41 u4  42 0 u4  44  0  0  0  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' 0  0  0  · ·  0  1  �  �  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  (H4)  Definition 7 (Unitary routing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Let routing(⃗G(o)) be the  set of all unitary routes of ⃗G(o), generated by the freedom in  choosing the coefficients ui  jk within the constraints of unitar-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Let also ocROUTING = {routing(⃗G(o))} ⃗G(o)∈ocDAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Definition 8 (Unitary router).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' We define a unitary router as  the function  router : ocDAG → ocROUTING,  (H5)  ⃗G(o) �→ routing(⃗G(o)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Given a unitary route route( ⃗G(o)), the unitary matrix de-  scribing the whole quantum ECM is then given by  U =  �  U i∈route( ⃗G(o))  U i,  (H6)  where the product is ordered from left to right in descending  order of the superscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Note that here the superscript i refers  to the position of the vertices in the ordering o, while in the  main text the superscript refers to an unspecified ordering, and  that in general the U i do not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  If an ECM graph G admits more than one topological or-  dering, G can have different unitary routings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Nevertheless,  there is the following one-to-one quantum-classical corre-  spondence:  Lemma 9 (Quantum-classical correspondence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' The func-  tion router is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  To show bijection we will first show injectivity,  that is, we must show that router(⃗G(o)) = router( ⃗H(o′)) ⇒  ⃗G(o) = ⃗H(o′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' According to the definition of routing, a ⃗G(o)  has routing(⃗G(o)) only if (i) it has vertices V = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' , n},  where n is the number of rows (or columns) of each U i in any  of the routes in routing(⃗G(o)), and (ii) it has directed edges  E = {(j, k) | ∃ route( ⃗G(o)) ∈ routing(⃗G(o)) and ∃ U k ∈  route( ⃗G(o)) such that U k  jk ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This completely determines  ⃗G(o) as ⃗G(o) = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Thus router(⃗G(o)) = router( ⃗H(o′))  can only be true if ⃗G(o) = ⃗H(o′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  Surjectivity is trivial because the codomain of router is de-  fined as its range, which completes the proof that router is  bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' □  While a one-to-one correspondence exists on the level of  ordered DAGs ⃗G(o) and unitary routings, there is an interest-  ing freedom on the level of ECM graphs and/or unitary routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  In the latter case, a unitary route is not sufficient to uniquely  identify a ⃗G(o) because if ui  jk = ui  kj = 0 for some j ̸= k, the  corresponding directed edge (j, k) in ⃗G(o) cannot be recov-  ered just from the unitary route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Instead, one would recover  a ⃗G(o)′ without that edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' Classically, this corresponds to the  case of a vanishing h-value, hjk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' In the former case, it is  worth noticing that ECM graphs are defined without ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content='  However, ordering matters for unitary routes since in general  the U is do not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
+page_content=' This represents an interesting depar-  ture from classical ECMs: different orderings can give rise to  different unitary routings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFRT4oBgHgl3EQf4jgk/content/2301.13669v1.pdf'}
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+1
+Accuracy-Guaranteed Collaborative DNN Inference
+in Industrial IoT via Deep Reinforcement Learning
+Wen Wu, Member, IEEE, Peng Yang, Member, IEEE, Weiting Zhang, Student Member, IEEE,
+Conghao Zhou, Student Member, IEEE, and Xuemin (Sherman) Shen, Fellow, IEEE
+Abstract—Collaboration among industrial Internet of Things
+(IoT)
+devices
+and
+edge
+networks
+is
+essential
+to
+support
+computation-intensive deep neural network (DNN) inference
+services which require low delay and high accuracy. Sampling
+rate adaption which dynamically configures the sampling rates
+of industrial IoT devices according to network conditions, is
+the key in minimizing the service delay. In this paper, we
+investigate the collaborative DNN inference problem in industrial
+IoT networks. To capture the channel variation and task arrival
+randomness, we formulate the problem as a constrained Markov
+decision process (CMDP). Specifically, sampling rate adaption,
+inference task offloading and edge computing resource allocation
+are jointly considered to minimize the average service delay while
+guaranteeing the long-term accuracy requirements of different
+inference services. Since CMDP cannot be directly solved by
+general reinforcement learning (RL) algorithms due to the
+intractable long-term constraints, we first transform the CMDP
+into an MDP by leveraging the Lyapunov optimization technique.
+Then, a deep RL-based algorithm is proposed to solve the MDP.
+To expedite the training process, an optimization subroutine
+is embedded in the proposed algorithm to directly obtain the
+optimal edge computing resource allocation. Extensive simulation
+results are provided to demonstrate that the proposed RL-based
+algorithm can significantly reduce the average service delay while
+preserving long-term inference accuracy with a high probability.
+Index Terms—Sampling rate adaption, inference accuracy,
+collaborative DNN Inference, deep reinforcement learning.
+I. INTRODUCTION
+With the development of advanced neural network tech-
+niques and ubiquitous industrial Internet of Things (IoT)
+devices, deep neural network (DNN) is widely applied in
+extensive industrial IoT applications, such as facility moni-
+toring and fault diagnosis [1]. Industrial IoT devices (e.g.,
+vibration sensors) can sense the industrial operating envi-
+ronment and feed sensing data to a DNN, and then the
+DNN processes the sensing data and renders inference results,
+namely DNN inference. Although DNN inference can achieve
+high inference accuracy as compared to traditional alternatives
+(e.g., decision tree), executing DNN inference tasks requires
+W. Wu, C. Zhou, and X. Shen are with the Department of Electrical
+and Computer Engineering, University of Waterloo, 200 University Av-
+enue West, Waterloo, ON N2L 3G1, Canada (email: {w77wu, c89zhou,
+sshen}@uwaterloo.ca).
+P. Yang (Corresponding author) is with the School of Electronic Information
+and Communications, Huazhong University of Science and Technology,
+Wuhan, 430074, P.R. China (email: yangpeng@hust.edu.cn).
+W. Zhang is with the School of Electronic and Information Engi-
+neering, Beijing Jiaotong University, Beijing 100044, P.R. China (email:
+17111018@bjtu.edu.cn).
+extensive computation resource due to tremendous multiply-
+and-accumulation operations [2]. A device-only solution that
+purely executes DNN inference tasks at resource-constrained
+industrial IoT devices, becomes intractable due to prohibitive
+energy consumption and a high service delay. For example,
+processing an image using AlexNet incurs up to 0.45 W energy
+consumption [3]. An edge-only solution which purely offloads
+large-volume sensing data to resource-rich edge nodes, e.g.,
+access point (AP), suffers from an unpredictable service delay
+due to time-varying wireless channel [4]. Hence, neither a
+device-only nor an edge-only solution can effectively support
+low-delay DNN inference services.
+Collaborative
+inference,
+which
+coordinates
+resource-
+constrained industrial IoT devices and the resource-rich
+AP, becomes a de-facto paradigm to provide low-delay and
+high-accuracy inference services [5]. Within the collaborative
+inference, sensing data from industrial IoT devices can be
+either processed locally or offloaded to the AP. At industrial
+IoT devices, light-weight compressed DNNs (i.e., neural
+networks are compressed without significantly decreasing
+their performance) are deployed due to constrained on-board
+computing capability, which saves computing resource at the
+cost of inference accuracy [6], [7]. At the AP, uncompressed
+DNNs are deployed to provide high-accuracy inference
+services at the cost of network resources. Through the
+resource allocation (e.g., task offloading) between industrial
+IoT devices and the AP, the overall service performance can
+be enhanced.
+However, the sampling rate adaption technique that dynam-
+ically configures the sampling rates of industrial IoT devices,
+is seldom considered. Through dynamically adjusting the sam-
+pling rates according to channel conditions and AP’s workload,
+sensing data from industrial IoT devices can be compressed,
+thereby reducing not only the offloaded data volume, but also
+task computation workload. In our experiments, we implement
+AlexNet to conduct bearing fault diagnosis based on the
+collected bearing vibration signal from dataset [8].1 As shown
+in Fig. 1, inference accuracy grows sub-linearly with the
+sampling rate. For example, when the sampling rate increases
+from 18 KHz to 24 KHz, the accuracy increases from 95%
+to 98.7%. Hence, when the channel condition is poor or
+edge computation workload is heavy, decreasing the sampling
+rate can reduce the offloaded data volume and requested
+computation workload, thereby reducing the service delay at
+1The experiment is conducted on an open-source dataset [8]. This dataset
+collects the vibration signal of drive end bearings at a sampling rate of
+48 KHz, and there are 10 types of possible faults.
+arXiv:2301.00130v1  [eess.SY]  31 Dec 2022
+
+2
+the cost of limited inference accuracy. When channel condition
+is good and edge computation workload is light, increasing the
+sampling rate can help deliver a high-accuracy service with
+an acceptable service delay. Hence, sampling rate adaption
+can effectively reduce the service delay, which should be
+incorporated in the collaborative DNN inference.
+The sampling rate adaption and resource allocation for
+collaborative DNN inference are entangled with the following
+challenges. Firstly, due to time-varying channel conditions and
+random task arrivals, sampling rate and resource allocation
+should be dynamically adjusted to achieve the minimum
+service delay. Minimizing the long-term service delay requires
+the stochastic information of network dynamics. Secondly, in
+addition to minimizing the service delay, the long-term accu-
+racy requirements should be guaranteed for different inference
+services. The long-term accuracy performance is determined
+by decisions of sampling rate adaption and resource allocation
+over time, and hence the optimal decisions require future
+network information. To address the above two challenges, a
+reinforcement learning (RL) technique is leveraged to interact
+with the unknown environment to capture the network dynam-
+ics, and then a Lyapunov optimization technique is utilized
+within the RL framework to guarantee the long-term accuracy
+requirements without requiring future network information.
+In this paper, we investigate the collaborative DNN infer-
+ence problem in industrial IoT networks. Firstly, we formu-
+late the problem as a constrained Markov decision process
+(CMDP) to account for time-varying channel conditions and
+random task arrivals. Specifically, sampling rates of industrial
+IoT devices, task offloading, and edge computation resource
+allocation are optimized to minimize the average service delay
+while guaranteeing the long-term accuracy requirements of
+multiple services. Secondly, since traditional RL algorithms
+target at optimizing a long-term reward without considering
+policy constraints, they cannot be applied to solve CMDP with
+long-term constraints. To solve the problem, we transform the
+CDMP into an MDP via the Lyapunov optimization technique.
+The core idea is to construct accuracy deficit queues to
+characterize the satisfaction status of the long-term accuracy
+constraints, thereby guiding the learning agent to meet the
+long-term accuracy constraints. Thirdly, to solve the MDP,
+a learning-based algorithm is developed based on the deep
+deterministic policy gradient (DDPG) algorithm. Within the
+learning algorithm, to reduce the training complexity, edge
+computing resource allocation is directly solved via an op-
+timization subroutine based on convex optimization theory,
+since it only impacts one-shot delay performance according
+to theoretical analysis. Extensive simulations are conducted
+to validate the effectiveness of the proposed algorithm in
+reducing the average service delay while preserving the long-
+term accuracy requirements.
+Our main contributions in this paper are summarized as
+follows:
+• We formulate the collaborative DNN inference problem
+as a CMDP, in which the objective is to minimize the
+average service delay while guaranteeing the long-term
+accuracy constraints;
+6
+12
+18
+24
+Sampling Rate (KHz)
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Inference Accuracy
+AlexNet
+Fitted curve
+Fig. 1.
+Inference accuracy with respect to sampling rates on the bearing
+vibration dataset [8].
+• We transform the CMDP into an MDP via the Lyapunov
+optimization technique which constructs accuracy deficit
+queue to characterize the satisfaction status of the long-
+term accuracy constraints;
+• We propose a deep RL-based algorithm to make the
+optimal sampling rate adaption and resource allocation
+decisions. To reduce the training complexity, an optimiza-
+tion subroutine is embedded in the proposed algorithm for
+the optimal edge computing resource allocation.
+The remainder of this paper is organized as follows. Sec-
+tion II reviews related works. The system model and problem
+formulation are presented in Section III. Section IV proposes
+a learning-based solution. Simulation results are given in
+Section V. Finally, Section VI concludes this paper.
+II. RELATED WORK
+DNN inference for resource-constrained industrial IoT de-
+vices has garnered much attention recently. A device-only
+solution aims to facilitate DNN inference services resorting
+to on-board computing resources. To reduce the computational
+complexity, DNN compression techniques are applied, such as
+weight pruning [6] and knowledge distillation [9]. Considering
+the widely-equipped energy-harvesting functionality in IoT
+devices, Gobieski et al. designed a light-weight DNN inference
+model, which can dynamically compress the model size in
+order to balance inference accuracy and energy efficiency [2].
+In another line of research, edge-assisted DNN inference solu-
+tions can provide high-accuracy inference services by utilizing
+powerful edge computing servers. To facilitate low-delay and
+accurate DNN-based video analytics, Yang et al. proposed
+an online video quality and computing resource allocation
+strategy to maximize video analytic accuracy [10]. Another
+inspiring work proposed a novel device-edge collaborative
+inference scheme, in which the DNN model is partitioned and
+deployed at both the device and the edge, and intermediate
+results are transferred via wireless links [5]. The above works
+can provide possible resource allocation solutions to enhance
+DNN inference performance. Different from existing works,
+our work takes the sampling rate adaption of industrial IoT
+devices into account, aiming at providing accuracy-guaranteed
+inference services in dynamic industrial IoT networks.
+RL algorithms have been widely applied in allocating net-
+work resources in wireless networks, such as service migra-
+
+3
+Service 1
+Service 2
+Service M
+Compressed 
+DNN
+Uncompressed 
+DNN
+AP
+Edge Server
+Industrial IoT 
+devices
+Sensing 
+data
+Sampling
+rate
+Fig. 2. The collaborative DNN inference framework for industrial IoT devices.
+tion in vehicular networks [11], network slicing in cellular
+networks [12], content caching in edge networks [13], and
+task scheduling in industrial IoT networks [14]. Hence, RL
+algorithms are considered as plausible solutions to manage
+network resources for DNN inference services. However,
+DNN inference services require minimizing the average delay
+while satisfying the long-term accuracy constraints. Tradi-
+tional RL algorithms, e.g., DDPG, can be applied to solve
+MDPs, in which learning agents seek to optimize a long-
+term reward without policy constraints, while they cannot deal
+with constrained long-term optimization problems [15], [16].
+Our proposed deep RL-based algorithm can address long-
+term constraints within the RL framework by the modification
+of reward based on the Lyapunov optimization technique.
+In addition, an optimization subroutine is embedded in our
+algorithm to further reduce the training complexity.
+III. SYSTEM MODEL AND PROBLEM FORMULATION
+A. Network Model
+As shown in Fig. 2, we consider a wireless network with
+one AP to serve multiple types of industrial IoT devices.
+The AP is in charge of collecting network information and
+resource orchestration within the network. Consider M types
+of inference services, denoted by a set M, such as facility fault
+diagnosis and facility monitoring services. Taking the facility
+fault diagnosis service as an example, vibration sensors in-
+stalled on industrial IoT devices sense the operating conditions
+at a sampling rate, and feed the sensed vibration signal into a
+DNN, then the DNN diagnoses the facility fault type. The set
+of industrial IoT devices subscribed to service m is denoted
+by Nm, and the set of all industrial IoT devices is denoted by
+N = ∪m∈MNm. In the collaborative inference framework,
+two types of DNNs are deployed. One is a compressed DNN,
+which is deployed at industrial IoT devices. The compressed
+DNN can be implemented via the weight pruning technique,
+which prunes less-important weights to reduce computational
+complexity while maintaining similar inference accuracy [6].
+The other is an uncompressed DNN, which is deployed at the
+AP. In this way, M types of uncompressed DNNs share the
+edge computing resource to serve different inference requests.
+Important notations are summarized in Table I.
+Table I
+SUMMARY OF NOTATIONS.
+Notation
+Description
+Am
+Achieved instantaneous accuracy of service m
+Ath
+m
+Long-term accuracy requirement of service m
+Bt
+Local computing queue backlog in time slot t
+c
+Computing resource allocation decision vector
+D
+Service delay
+L (·)
+Lyapunov function
+o
+Task offloading decision vector
+Qt
+Edge computing queue backlog in time slot t
+V
+Parameter to balance delay and accuracy requirement
+X
+Sampling rate selection decision matrix of all devices
+Zt
+Accuracy deficit queue backlog in time slot t
+ξn
+Raw task data size of device n
+ηm
+Task computation intensity of service m
+λn
+Average task arrival rate of device n
+ζ
+�
+xt
+n
+�
+Task data size of device n in time slot t
+Ψt
+Amount of dropped tasks in computing queues
+The collaborative DNN inference framework operates in a
+time-slotted manner. Let t denote the time index, where t ∈
+T = {1, 2, ..., T}. The detailed procedure is given as follows.
+1) Sampling rate selection: Industrial IoT devices first
+select their sampling rates according to channel condi-
+tions and computation workloads. The set of candidate
+sampling rates is denoted by K = {θ1, θ2, ..., θK},
+where θK denotes the raw sampling rate. We assume
+the sampling rate in K increases linearly with the index,
+i.e., θk = kθK/K. Let Xt denote the sampling rate
+decision matrix in time slot t, whose element xt
+n,k = 1
+indicates industrial IoT device n ∈ N selects the k-th
+sampling rate.
+2) Task processing: The sensing data from industrial IoT
+devices within a time slot is deemed as a computation
+task, which can be either offloaded to the AP or executed
+locally. Let ot ∈ R|N |×1 denote the offloading decision
+vector in time slot t, whose element ot
+n = 0 indicates
+offloading the computation task from industrial IoT
+device n. Otherwise, ot
+n = 1 indicates executing the
+computation task locally.
+B. Service Delay Model
+A computation task can be either processed locally or
+offloaded to the AP. In what follows, we analyze the service
+delay in these two cases.
+1) Executing locally: Let λt
+n denote the task arrival rate
+of the n-th industrial IoT device in time slot t, which is
+assumed to follow a general random distribution. The raw
+data size of the generated tasks at the n-th device is denoted
+by ξt
+n = λt
+nνm, ∀n ∈ Nm, where νm denotes the raw data
+size of a task for service m. After the sampling rate is
+selected, the data size of the generated task is represented
+by ζ (xt
+n) = �K
+k=1 xt
+n,kξt
+nk/K, where xt
+n = {xt
+n,k}k∈K is
+the sampling rate selection decision vector of the n-th device.
+When the inference task is processed locally by a compressed
+
+((o)4
+DNN, the service delay includes the queuing delay in the local
+computing queue and task processing delay, which is given by
+dt
+n,l = ot
+nηm,c (Bt
+n + ζ (xt
+n))
+fn
+, ∀n ∈ Nm,
+(1)
+where fn is the CPU frequency of the n-th industrial IoT
+device, and ηm,c denotes the computation intensity of the
+compressed DNN for the m-th service. Here, Bt
+n is the
+backlogged computation tasks (in bits) in the local computing
+queue, which is updated via
+Bt+1
+n
+= min
+��
+Bt
+n + ot
+nζ
+�
+xt
+n
+�
+− fnτ
+ηm,c
+�+
+, Bmax
+n
+�
+,
+(2)
+where [x]+ = max{x, 0}, Bmax
+n
+is the capacity of the local
+computing queue, and τ is the duration of a time slot. Tasks
+will be dropped if the local computing queue is full. Let
+Ψt
+b,n = max
+�
+Bt
+n + ot
+nζ
+�
+xt
+n
+�
+− fnτ
+ηm,c
+− Bmax
+n
+, 0
+�
+(3)
+denote the amount of the dropped tasks in the local computing
+queue of device n. Here, Ψt
+n,b > 0 indicates that an event of
+local computing queue overflow occurs at the n-th device,
+and the corresponding penalty will be incurred to avoid queue
+overflow.
+2) Offloading to AP: When a task is offloaded to the AP,
+it will be processed by an uncompressed DNN. The service
+delay consists of task offloading delay, queuing delay in the
+edge computing queue, and task processing delay, which are
+analyzed respectively as follows.
+• Task offloading delay: For the n-th industrial IoT device,
+the offloading delay is given by
+dt
+n,o = (1 − ot
+n) ζ (xt
+n)
+Rtn
+,
+(4)
+where transmission rate between the n-th industrial IoT
+device and the AP, Rt
+n, is given by
+Rt
+n = W
+N log2
+�
+1 + PT G(Ht
+n)
+Nfσ2
+�
+.
+(5)
+Here, W, PT , G(Ht
+n), and Nf represent the system band-
+width, transmit power, channel gain, and noise figure, re-
+spectively. σ2 = NoW/N denotes the background noise
+where No is thermal noise spectrum density. Channel gain
+G(Ht
+n) varies in terms of channel state Ht
+n. Based on
+extensive real-time measurements, channel state Ht
+n can
+be modeled with a finite set of channel states H [17]. The
+evolution of channel states is characterized by a discrete-
+time and ergodic Markov chain model, whose transition
+matrix is P ∈ R|H|×|H|.
+• Task processing delay: The tasks from all industrial IoT
+devices subscribed to the m-th service are placed in the
+edge computing queue for the m-th service. The amount
+of aggregated tasks is given by �
+n∈Nm (1 − ot
+n) ζ (xt
+n).
+The computing resource is dynamically allocated among
+multiple services at the AP according to service task
+arrivals, which can be realized via containerization tech-
+niques, such as Dockers and Kubernetes [18]. Let ct ∈
+RM×1 denote the computing resource allocation decision
+vector in time slot t. Each element 0 ≤ ct
+m ≤ 1 denotes
+the portion of computing resource allocated to the m-th
+service. Hence, the processing delay is given by
+dt
+n,p = ηm,u (1 − ot
+n) ζ (xt
+n)
+ctmfb
+, ∀n ∈ Nm,
+(6)
+where fb is the CPU frequency of the computing server
+at the AP, and ηm,u denotes the computation intensity
+of processing the m-th service task by the uncompressed
+DNN. Note that ηm,u > ηm,c, since the uncompressed
+DNN consumes more computing resource.
+• Queuing delay: The queuing delay consists of two com-
+ponents: (i) the time taken to process backlogged tasks
+in the edge computing queue, which is given by
+dt
+n,q = Qt
+mηm,u
+ctmfb
+, ∀n ∈ Nm.
+(7)
+Here, Qt
+m denotes the edge computing queue backlog for
+the m-th service in time slot t, which is updated according
+to
+Qt+1
+m
+= min
+��
+Qt
+m + at
+m − ct
+mfbτ
+ηm,u
+�+
+, Qmax
+m
+�
+.
+(8)
+Here, at
+m = �
+n∈Nm (1 − ot
+n) ζ (xt
+n) and Qmax
+m
+denotes
+the capacity of the m-th edge computing queue. Similar
+to that in local computing queues, tasks will also be
+dropped if the edge computing queue is full, and the
+amount of dropped tasks for the m-th edge computing
+queue is given by
+Ψt
+q,m = max
+�
+Qt
+m + at
+m − ct
+mfbτ
+ηm,u
+− Qmax
+m
+, 0
+�
+.
+(9)
+Here, Ψt
+q,m > 0 indicates that an event of edge computing
+queue overflow occurs; and (ii) average waiting time
+among all newly arrived tasks until the task of industrial
+IoT device n is processed, which is given by
+dt
+n,w =
+ηm,u
+�
+i̸=n,i∈Nm (1 − ot
+i) ζ (xt
+i)
+2ctmfb
+.
+(10)
+Here, �
+i̸=n,i∈Nm (1 − ot
+i) ζ (xt
+n) denotes the amount of
+aggregated tasks except the task of industrial IoT de-
+vice n.
+Taking both local execution and offloading into account, the
+service delay in time slot t is given by
+D
+�
+Xt, ot, ct�
+=
+�
+n∈N
+�
+dt
+n,l + dt
+n,o + dt
+n,p + dt
+n,q + dt
+n,w
+�
++ wp
+� �
+n∈N
+1{Ψt
+b,n>0} +
+�
+m∈M
+1{Ψt
+q,m>0}
+�
+,
+(11)
+where 1{x} = 1 and wp > 0 are the indicator function and
+the positive unit penalty cost for queue overflow, respectively.
+The first term represents the experienced delay to complete all
+tasks in time slot t. The second term represents the penalty for
+potential overflow events in local and edge computing queues.
+
+5
+C. Inference Accuracy Model
+The inference accuracy depends on the sampling rate of a
+task and the type of DNN that executes a task. Firstly, we
+characterize the relationship between the inference accuracy
+and the sampling rate, which is specified by accuracy function
+g(θk), ∀θk ∈ K. Specifically, we implement a DNN inference
+algorithm, i.e., AlexNet [19], and apply the AlexNet to diag-
+nose facility fault type based on the collected bearing vibration
+signal from the dataset [8], and then measure the accuracy
+function values with respect to sampling rates, as shown
+in Fig. 1. Secondly, the relationship between the inference
+accuracy and the type of DNN is also characterized via
+experiments. Here, hm,c and hm,u represent the inference
+accuracy of the compressed DNN and the uncompressed DNN
+for the m-th service, respectively. Note that, hm,c < hm,u,
+as an uncompressed DNN achieves higher fault diagnosis
+accuracy.
+Since the DNN model selection (i.e., task offloading de-
+cision) and the sampling rate selection are independent,
+inference accuracy is the product of the accuracy value
+with respect to the selected sampling rate and the accu-
+racy value with respect to the selected DNN type, i.e.,
+g
+��
+k∈K xt
+n,kθk
+�
+(ot
+nhm,c + (1 − ot
+n) hm,u). Hence, the av-
+erage inference accuracy for the m-th service in time slot t
+can be given by
+Am
+�
+Xt, ot�
+=
+�
+n∈Nm
+1
+|Nm|g
+��
+k∈K
+xt
+n,kθk
+�
+·
+�
+ot
+nhm,c +
+�
+1 − ot
+n
+�
+hm,u
+�
+.
+(12)
+Note that the model can be readily extended to cases when
+other inference methods are adopted, since the accuracy values
+with respect to sampling rates and DNN types are obtained via
+practical experiments.
+D. Problem Formulation
+DNN inference services require not only minimizing service
+delay, but also guaranteeing their long-term accuracy require-
+ments, which can be modeled via a CMDP. Its action, state,
+reward, and state transition matrix are defined as follows:
+• Action: The action includes the sampling rate selection,
+task offloading, and edge computing resource alloca-
+tion decisions, i.e., ˆat = {Xt, ot, ct}. Note that the
+components of the action should satisfy following con-
+straints: (1) xt
+n,k ∈ {0, 1} constrains the sampling rate
+selection decision; (2) ot
+n ∈ {0, 1} requires the binary
+task offloading decision; and (3) �
+m∈M ct
+m ≤ 1 and
+0 ≤ ct
+m ≤ 1 constrain a continuous computing resource
+allocation decision.
+• State: The state includes local computing queues backlog
+of industrial IoT devices Bt
+n, edge computing queues
+backlog Qt
+m, channel conditions of industrial IoT devices
+Ht
+n, and the raw data size of the generated tasks at
+industrial IoT devices ξt
+n, i.e.,
+ˆst ={{Bt
+n}n∈N , {Qt
+m}m∈M, {Ht
+n}n∈N , {ξt
+n}n∈N }.
+(13)
+The queue backlogs, i.e., {Bt
+n}n∈N and {Qt
+m}m∈M,
+adopt a unit in bits, which result in large state space,
+especially for a large number of industrial IoT devices.
+• Reward: The reward is designed to minimize the service
+delay in (22) in time slot t, which is defined as ˆrt =
+−D (Xt, ot, ct) .
+• State transition probability: State transition probability is
+given by
+Pr
+�
+ˆst+1|ˆst, ˆat�
+=
+�
+n∈N
+Pr
+�
+Bt+1
+n
+|Bt
+n, xt
+n,k, ot
+n
+�
+·
+�
+m∈M
+Pr
+�
+Qt+1
+m |Qt
+m, Xt, ot�
+·
+�
+n∈N
+Pr
+�
+Ht+1
+n
+|Ht
+n
+�
+·
+�
+n∈N
+Pr
+�
+ξt+1
+n
+|ξt
+n
+�
+.
+(14)
+The equality holds due to the independence of different
+state components. The first two components are gov-
+erned by the evolution of local computing queues and
+edge computing queues in (2) and (8), respectively. The
+third component is evolved according to the discrete-
+time Markov chain of channel conditions, and the last
+component is governed by the memoryless task arrival
+pattern. Note that each of those state components only
+depends on its previous state components, which means
+the state transition is Markovian.
+Our goal is to find a stationary policy π ∈ Π that dynam-
+ically configures sampling rates selection Xt, task offload-
+ing ot, and edge computing resource allocation ct according
+to state ˆst, to minimize the service delay while guaranteeing
+long-term inference accuracy requirements {Ath
+m}m∈M, which
+is formulated as the following problem:
+P0 : min
+π∈Π
+lim
+T →∞
+1
+T
+T
+�
+t=1
+Eπ
+�
+D
+�
+Xt, ot, ct��
+(15a)
+s.t.
+lim
+T →∞
+1
+T
+T
+�
+t=1
+Am
+�
+Xt, ot�
+≥ Ath
+m, ∀m ∈ M. (15b)
+Here, P0 is a CMDP. Directly solving the above CMDP via
+dynamic programming solutions [15] is challenging due to
+the following reasons. Firstly, state transition probability is
+unknown due to the lack of statistic information on the channel
+condition variation and task arrival patterns of all industrial
+IoT devices. Secondly, even the state transition probability
+is known, large action space and state space that grow with
+respect to the number of industrial IoT devices incur an ex-
+tremely high computational complexity, which makes dynamic
+programming solutions intractable. Hence, we propose a deep
+RL-based algorithm to solve the CMDP, which can be applied
+in large-scale networks without requiring statistic information
+of network dynamics.
+IV. DEEP RL-BASED SAMPLING RATE ADAPTION AND
+RESOURCE ALLOCATION ALGORITHM
+As mentioned before, a CDMP problem cannot be directly
+solved via traditional RL algorithms. We first leverage the
+Lyapunov optimization technique to deal with the long-term
+constraints and transform the problem into an MDP. Then,
+
+6
+we develop a deep RL-based algorithm to solve the MDP.
+To further reduce the training complexity, an optimization
+subroutine is embedded to directly obtain the optimal edge
+computation resource allocation.
+A. Lyapunov-Based Problem Transformation
+The major challenge in solving problem P0 is to han-
+dle the long-term constraints. We leverage the Lyapunov
+technique [20], [21] to address this challenge. The core
+idea is to construct accuracy deficit queues to characterize
+the satisfaction status of the long-term accuracy constraints,
+thereby guiding the learning agent to meet the long-term
+accuracy constraints. The problem transformation procedure
+is presented as follows.
+Firstly, we construct inference accuracy deficit queues for
+all services, whose dynamics evolves as follows:
+Zt+1
+m
+=
+�
+Ath
+m − Am
+�
+Xt, ot�
++ Zt
+m
+�+ , ∀m ∈ M.
+(16)
+Here, Zt
+m indicates the deviation of the achieved instantaneous
+accuracy from the long-term accuracy requirement, whose
+initial state is set to Z0
+m = 0. Then, a Lyapunov function
+is introduced to characterize the satisfaction status of the
+long-term accuracy constraint, which is defined as L (Zt
+m) =
+(Zt
+m)2 /2 [20]–[22]. A smaller value of L (Zt
+m) indicates
+better satisfaction of the long-term accuracy constraint.
+Secondly, the Lyapunov function should be consistently
+pushed to a low value in order to guarantee the long-term
+accuracy constraints. Hence, we introduce a one-shot Lya-
+punov drift to capture the variation of the Lyapunov function
+across two subsequent time slots [20]. Given Zt
+m, the one-shot
+Lyapunov drift is defined as ∆ (Zt
+m) = L
+�
+Zt+1
+m
+�
+− L (Zt
+m),
+which is upper bounded by
+∆
+�
+Zt
+m
+�
+= 1
+2
+��
+Zt+1
+m
+�2 −
+�
+Zt
+m
+�2�
+≤ 1
+2
+��
+Zt
+m + Ath
+m − Am
+�
+Xt, ot��2 −
+�
+Zt
+m
+�2�
+= 1
+2
+�
+Ath
+m − Am
+�
+Xt, ot��2 + Zt
+m
+�
+Ath
+m − Am
+�
+Xt, ot��
+≤ Cm + Zt
+m
+�
+Ath
+m − Am
+�
+Xt, ot��
+,
+(17)
+where Cm =
+�
+Ath
+m − Amin
+m
+�2 /2 is a constant, and Amin
+m
+is the
+lowest inference accuracy that can be achieved for service m.
+The first inequality is due to the substitution of (16), and the
+second inequality is because Am (Xt, ot) ≥ Amin
+m
+.
+Thirdly, based on the Lyapunov optimization theory, the
+original CMDP of minimizing the service delay while guar-
+anteeing the long-term accuracy requirements boils down to
+minimizing a drift-plus-cost, i.e.,
+�
+m∈M
+∆
+�
+Zt
+m
+�
++ V · D
+�
+Xt, ot, ct�
+≤
+�
+m∈M
+Cm +
+�
+m∈M
+Zt
+m
+�
+Ath
+m − Am
+�
+Xt, ot��
++ V · D
+�
+Xt, ot, ct�
+,
+(18)
+where the inequality is due to the upper bound in (17). Here
+V is a positive parameter to adjust the tradeoff between the
+service delay minimization and the satisfaction status of the
+long-term accuracy constraints. The underlying rationale is
+that, if the long-term accuracy constraint is violated, i.e.,
+Zt
+m > 0, stratifying the long-term constraints by improving the
+instantaneous inference accuracy becomes more urgent than
+reducing the service delay.
+In this way, the CMDP is transformed into an MDP with
+the objective of minimizing the drift-plus-cost in each time
+slot.
+B. Equivalent MDP
+In the equivalent MDP, the action, state, reward, and state
+transition matrix are modified as follows due to the incorpo-
+ration of accuracy deficit queues.
+• Action: The action is the same as that in the CMDP, i.e.,
+at = ˆat = {Xt, ot, ct}.
+• State: Compared with the state of the CMDP, the accuracy
+deficit queue backlog of services {Zt
+m}m∈M should be
+incorporated, i.e.,
+st = {ˆst, {Zt
+m}m∈M}.
+(19)
+• Reward: The reward is modified to minimize the drift-
+plus-cost in (18) in time slot t, i.e.,
+rt = −V · D
+�
+Xt, ot, ct�
+−
+�
+m∈M
+Zt
+m
+�
+Ath
+m − Am
+�
+Xt, ot��
+.
+(20)
+Note that the constant term �
+m∈M Cm in (18) is ignored
+in the reward for brevity.
+• State transition probability: Since accuracy deficit queue
+backlogs are incorporated in the state, the state transition
+probability evolves according to
+Pr
+�
+st+1|st, at�
+= Pr
+�
+ˆst+1|ˆst, ˆat�
+·
+�
+m∈M
+Pr
+�
+Zt+1
+m |Zt
+m, Xt, ot�
+.
+(21)
+where the second term is the evolution of the accuracy
+deficit queue backlog according to (16). Note that the
+overall state transition is still Markovian.
+Then, problem P0 is transformed into the following MDP
+problem:
+P1 : min
+π∈Π
+lim
+T →∞
+1
+T
+T
+�
+t=1
+Eπ
+� �
+m∈M
+Zt
+m
+�
+Ath
+m − Am
+�
+Xt, ot��
++V · D
+�
+Xt, ot, ct��
+.
+(22)
+Similar to CMDP, solving an MDP via dynamic programming
+solutions also suffers from the curse of dimensionality due
+to large state space. Hence, we propose a deep RL-based
+algorithm to solve the MDP, which is detailed in Section IV-D.
+C. Optimization Subroutine for Edge Computing Resource
+Allocation
+Although P1 can be directly solved by RL algorithms, an
+inherent property on edge computing resource allocation can
+
+7
+be leveraged, in order to reduce the training complexity of
+RL algorithms. Through analysis on (22), the edge computing
+resource allocation is independent of the inference accuracy
+performance, and hence it only impacts the one-shot service
+delay performance. In time slot t, once task offloading and
+sampling rate selection decisions are made, the optimal com-
+puting resource allocation decision can be obtained via solving
+the following optimization problem:
+P2 : min
+ct
+D
+�
+Xt, ot, ct�
+s.t.
+�
+m∈M
+ct
+m ≤ 1
+(23a)
+0 ≤ ct
+m ≤ 1.
+(23b)
+A further analysis of (11) indicates that only the task
+processing delay and queuing delay at the AP are im-
+pacted by the edge computing resource allocation, i.e.,
+�
+n∈N
+�
+dt
+n,p + dt
+n,q + dt
+n,w
+�
+. In addition, the aggregated de-
+lay from the perspective of all devices is equivalent to the
+aggregated delay from the perspective of all services. Hence,
+the objective function in P2 can be rewritten as �
+m∈M dt
+m,
+where
+dt
+m =
+�
+n∈Nm
+�ηm,u (1 − ot
+n) ζ (xt
+n)
+ctmfb
++ Qt
+mηm,u
+ctmfb
++
+ηm,u
+�
+i̸=n,i∈Nm (1 − ot
+i) ζ (xt
+i)
+2ctmfb
+�
+(24)
+denotes the experienced delay of the m-th service. By ana-
+lyzing the convexity property of the problem, we have the
+following theorem to obtain the optimal edge computation
+resource allocation in each time slot.
+Theorem 1. The optimal edge computing resource allocation
+for problem P2 is given by
+ct,⋆
+m =
+�
+Λtm
+�
+m∈M
+�
+Λtm
+, ∀m ∈ M,
+(25)
+where
+Λt
+m =
+�
+n∈Nm
+�
+ηm,u
+�
+1 − ot
+n
+�
+ζ
+�
+xt
+n
+�
++ Qt
+mηm,u
++ηm,u
+2
+�
+i̸=n,i∈Nm
+�
+1 − ot
+i
+�
+ζ
+�
+xt
+i
+�
+�
+� .
+(26)
+Proof. Proof is provided in Appendix A.
+This optimization subroutine for the edge computing re-
+source allocation is embedded in the following proposed deep
+RL-based algorithm. In this way, the training complexity can
+be reduced, because it is no longer necessary to train the neural
+networks to obtain optimal edge computing resource allocation
+policy.
+D. Deep RL-based Algorithm
+To solve problem P1, we propose a deep RL-based al-
+gorithm, which is extended from the celebrated DDPG al-
+gorithm [23]. The main difference between DDPG and the
+Algorithm 1: Deep RL-based algorithm for sampling
+rate adaption and resource allocation
+1 Initialization: Initialize all neural networks and the
+experience replay memory;
+2 for each episode do
+3
+Reset the environment and obtain initial state s0;
+4
+for time slot t ∈ T do
+5
+Determine the sampling rate selection and task
+offloading actions {Xt, ot} by the actor
+network according to current state st;
+6
+Determine edge computing resource allocation
+action ct by (25);
+7
+Send joint action at = {Xt, ot, ct} to all
+industrial IoT devices by the AP;
+8
+Execute the joint action at industrial IoT
+devices;
+9
+Observe reward rt and new state st+1;
+10
+Store transition {st, at, rt, st+1} in the
+epxerience replay memory;
+11
+Sample a random minibatch transitions from
+the epxerience replay memory;
+12
+Train the critic and actor network by (27) and
+(28), respectively;
+13
+Update target networks by (29);
+14
+end
+15 end
+proposed algorithm is that the above optimization subroutine
+for computing resource allocation is embedded to reduce the
+training complexity. The proposed algorithm can be deployed
+at the AP, which collects the entire network state information
+and enforces the policy to all connected industrial IoT devices.
+In the algorithm, the learning agent has two parts: (a) an
+actor network, which is to determine the action based on the
+current state; and (b) a critic network, which is to evaluate
+the determined action based on the reward feedback from the
+environment. Let µ(s|φµ) and Q(s, a|φQ) denote the actor
+network and the critic network, respectively, whose neural
+network weights are φµ and φQ. As shown in Algorithm 1, the
+deep RL-based algorithm operates in a time-slotted manner,
+which consists of the following three steps.
+The first step is to obtain experience by interacting with
+the environment. Based on current network state st, the
+actor network generates the sampling rate selection and task
+offloading actions with an additive policy exploration noise
+that follows Gaussian distribution N
+�
+0, σ2�
+. The optimization
+subroutine generates the edge computation resource allocation
+action. Then, the joint action is executed at all industrial
+IoT devices. The corresponding reward rt and the next state
+st+1 are observed from the environment. The state transition
+{st, at, rt, st+1} is stored in the experience replay memory for
+actor and critic network training.
+The second step is to train the actor and critic network
+based on the stored experience. To avoid the divergence issue
+caused by DNN, a minibatch of transitions are randomly sam-
+pled from the experience replay memory to break experience
+
+8
+Table II
+SIMULATION PARAMETERS [24], [25].
+Parameter
+Value
+Thermal noise spectrum density (No)
+−174 dBm/Hz [25]
+Communication bandwidth (W)
+[5, 25] MHz
+Transmit power (PT )
+20 dBm [24]
+Average task arrival rate (λ)
+[0.6, 1] request/sec
+Noise figure (Nf)
+5 dB
+Intensity of compressed DNN (η1,c, η2,c)
+(80, 160) cycles/bit
+Intensity of uncompressed DNN (η1,u, η2,u)
+(200, 400) cycles/bit
+Device and edge server CPU frequency (fn, fb)
+(0.1, 2) GHz
+Number of Type I/II devices (N1, N2)
+5, 5
+Time slot duration (τ)
+1 second
+Balance parameter (V)
+0.05
+Unit penalty for queue overflow (wp)
+1
+Accuracy of compressed DNN (h1,c, h2,c)
+0.8, 0.8
+Accuracy of uncompressed DNN (h1,u, h2,u)
+1, 1
+Local/edge queue capacity (Bmax, Qmax)
+(3.84, 19.2) megabits
+Table III
+PARAMETERS OF THE PROPOSED RL-BASED ALGORITHM.
+Parameter
+Value
+Parameter
+Value
+Actor learning rate
+10−4
+Critic learning rate
+10−3
+Actor hidden units
+(64, 32)
+Critic hidden units
+(64, 32)
+Hidden activation
+ReLU
+Actor output activation
+Tanh
+Optimizer
+Adam
+Policy noise (σ)
+0.2
+Target update (δ)
+0.005
+Discount factor
+0.85
+Minibatch size
+64
+Replay memory size
+100,000
+Training episodes
+1, 000
+Time slots per episode
+200
+correlation. The critic network is trained by minimizing the
+loss function
+Loss
+�
+φQ�
+= 1
+Nb
+Nb
+�
+i=1
+�
+yi − Q(si, ai|φQ)
+�2 ,
+(27)
+where yi = ri + γQ′(si+1, µ′(si+1|φµ′)|φQ′), and Nb is the
+minibatch size. Here, µ′(s|φµ′) and Q′(s, a|φQ′) represent
+actor and critic target networks with weights φµ′ and φQ′.
+The actor network is trained via the policy gradient
+∇φµ ≈ 1
+Nb
+Nb
+�
+i=1
+∇aQ(si, a|φQ)|s=si,a=µ(si)∇θµµ(si|φµ)|si.
+(28)
+The third step is to update target networks. In order to ensure
+network training stability, the actor and critic target networks
+are softly updated by
+φQ′ = δφQ + (1 − δ)φQ′, φµ′ = δφµ + (1 − δ)φµ′,
+(29)
+where 0 < δ ≪ 1 denotes the target network update ratio.
+V. SIMULATION RESULTS
+A. Simulation Setup
+We consider a smart factory scenario in our simulation,
+in which industrial IoT devices, e.g., vibration sensors, are
+randomly scattered. The industrial IoT devices installed on
+industrial facilities (e.g., robot arms) sense their operating
+conditions. The sensing data is processed locally or offloaded
+to an AP in the smart factory for processing. The transmit
+power of an industrial IoT device is set to 20 dBm [24]. The
+channel condition is modeled with three states, i.e., “Good
+0
+200
+400
+600
+800
+1000
+Training Episodes
+0.25
+0.3
+0.35
+0.4
+0.45
+Service Delay (s)
+(a) Average delay performance
+0
+200
+400
+600
+800
+1000
+Training Episodes
+0.75
+0.8
+0.85
+0.9
+0.95
+Average Accuracy
+Type I Service
+Type II Service
+(b) Accuracy Performance of two services
+Fig. 3. Performance of the proposed algorithm in the training stage.
+(G)”, “Normal (N)”, and “Bad (B)”, and the corresponding
+transition matrix is given by [17]
+P =
+�
+�
+PGG
+PGN
+0
+PNG
+PNN
+PNB
+0
+PNB
+PBB
+�
+� =
+�
+�
+0.3
+0.7
+0
+0.25
+0.5
+0.25
+0
+0.7
+0.3
+�
+� .
+(30)
+Two types of DNN inference services are considered. Type
+I service: a facility fault diagnosis service to identify the fault
+type based on the collected bearing vibration signal from the
+dataset [8]. Since the duration of a time slot in the simulation
+is set to be one second, the task data size is the data volume of
+a one-second signal, which is a product of the raw sampling
+rate and the quantization bits of the signal. In the dataset,
+the bearing vibration signal is collected at 48 KHz sampling
+rate and 16 bit quantization, and hence the corresponding task
+data size is 768 kilobits. The long-term accuracy requirement
+of the service is set to 0.8. Type II service: a service extended
+from the Type I service to diagnose facility fault based on
+a low-grade bearing vibration dataset while requiring higher
+inference accuracy 0.9. The low-grade dataset collects vibra-
+tion signal at a lower sampling rate of 32 KHz, and hence
+the task data size is 512 kilobits. For both services, the task
+arrival rate of each industrial IoT device at each time slot
+follows a uniform distribution U(λ − 0.5, λ + 0.5), where λ
+is the average task arrival rate. We consider four candidate
+sampling rates for industrial IoT devices, which are 25%, 50%,
+75% and 100% of the raw sampling rate. The corresponding
+accuracy with respect to the sampling rates are 0.59, 0.884,
+0.950 and 0.987, respectively, based on extensive experiments
+
+9
+0.6
+0.7
+0.8
+0.9
+1.0
+Task Arrival Rate (req/s)
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+Service Delay (s)
+Static Configuration
+Delay Myopic
+Proposed
+Fig. 4. Service delay performance with respect to task arrival rates.
+0.6
+0.7
+0.8
+0.9
+1.0
+Task Arrival Rate (req/s)
+0.790
+0.795
+0.800
+0.805
+0.810
+Accuracy
+Type I Service
+0.6
+0.7
+0.8
+0.9
+1.0
+Task Arrival Rate (req/s)
+0.890
+0.895
+0.900
+0.905
+0.910
+Type II Service
+Fig. 5.
+Inference accuracy performance with respect to task arrival rates.
+on the bearing vibration dataset [8]. Balance parameter V is
+set to 0.05 based on extensive simulations. Other important
+simulation parameters are listed in Table II. The parameters
+of the proposed algorithm are given in Table III. The proposed
+algorithm is compared to the following benchmarks:
+• Delay myopic: Each industrial IoT device dynamically
+makes sampling rate selection and task offloading deci-
+sions by maximizing the one-step reward in (20) accord-
+ing to the network state.
+• Static configuration: Each industrial IoT device takes
+a static configuration on the sampling rate selection and
+task offloading decisions, which can guarantee services’
+accuracy requirements.
+B. Performance Evaluation
+1) Convergence of the proposed algorithm: The service
+delay performance in the training stage is shown in Fig. 3(a).
+We can clearly see that the average service delay gradually de-
+creases as the increase of training episodes, which validates the
+convergence of the proposed algorithm. In addition, Fig. 3(b)
+shows the accuracy performance for both services with respect
+to training episodes. The accuracy performance is not good at
+the beginning of the training stage, but after 1,000 episodes
+of training, the accuracy performance converges to the pre-
+determined requirements.
+2) Impact of task arrival rate: Once well-trained offline,
+we evaluate the performance of the proposed algorithm in
+the online inference. As shown in Fig. 4, we compare the
+average service delay performance of the proposed algorithm
+with benchmark schemes in terms of task arrival rates for
+5
+10
+15
+20
+25
+Bandwidth (MHz)
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+Service Delay (s)
+Static Configuration
+Delay Myopic
+Proposed
+Fig. 6. Service delay performance with respect to communication bandwidth.
+1.00
+1.05
+1.10
+1.15
+1.20
+CPU Frequency of Edge Server (GHz)
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+Service Delay (s)
+Proposed
+Proposed-fixed
+Fig. 7. Service delay in terms of CPU frequency of the edge server.
+W = 20 MHz. Each simulation point is plotted with a 95%
+confidence interval. Several observations can be obtained from
+the figure. Firstly, the service delay increases with the task
+arrival rate due to constrained communication and computing
+resources in the network. Secondly, the proposed algorithm
+significantly outperforms benchmark schemes. The reason is
+that the proposed RL-based algorithm can capture network dy-
+namics, such as the task arrival pattern and channel condition
+variation, via interacting with the environment. The learned
+knowledge is utilized to make online decisions that target at the
+long-term performance, while benchmark schemes only focus
+on the short-term performance and do not adapt to network
+dynamics. Specifically, the proposed algorithm can reduce
+the average service delay by 19% and 25%, respectively, as
+compared with delay myopic and static configuration schemes.
+As shown in Fig. 5, boxplot accuracy distribution of two
+services is presented with respect to different task arrival rates.
+The long-term accuracy requirements for two services are 0.8
+and 0.9, respectively. It can be seen that the proposed algo-
+rithm guarantees the long-term accuracy requirements of both
+services with a high probability. Specifically, the maximum
+error probability is less than 0.5%.
+3) Impact of communication bandwidth: Fig. 6 shows the
+impact of communication bandwidth on the average service
+delay. Firstly, we can see that the average service delay
+decreases as the growth of bandwidth. The reason is that
+the transmission delay is reduced when the communication
+resource becomes sufficient. In addition, the proposed al-
+
+10
+gorithm achieves good performance when the bandwidth is
+scarce. When system bandwidth is only 5 MHz, the proposed
+algorithm achieves 1.20× and 1.42× delay reduction com-
+pared with delay myopic and static configuration schemes,
+respectively, which is larger than that when the system band-
+width is 25 MHz (1.15× and 1.31×). The reason is that the
+proposed algorithm efficiently utilizes the on-board computing
+resources. Simulation results show that the proposed algorithm
+decides 47.5% computation tasks to be executed locally with
+5 MHz bandwidth, while the delay myopic benchmark only
+decides 17%. Due to the efficient resource orchestration among
+industrial IoT devices and the AP, the proposed algorithm can
+effectively reduce average service delay for both services.
+4) Impact of optimization subroutine: As shown in Fig. 7,
+we evaluate the performance of the proposed algorithm
+with the fixed computing resource allocation (referred to as
+proposed-fixed), in which the edge computing resource is
+allocated based on the average computing demand of two
+services. Compared with the proposed-fixed solution, the pro-
+posed algorithm achieves significant performance gain when
+the edge computing resource is constrained. Specifically, the
+performance gain in reducing the service delay decreases
+from 1.98× at 1 GHz CPU frequency to only 1.02× at
+1.2 GHz CPU frequency. The reason is that efficient resource
+allocation is more important in resource-constrained scenarios,
+as compared to resource-rich scenarios. The results validate
+the effectiveness of the optimization subroutine for edge
+computing resource allocation. In addition to the performance
+gain, another merit of the optimization subroutine is to reduce
+the training complexity of RL algorithms.
+VI. CONCLUSION
+In this paper, we have studied the sampling rate adaption
+and resource allocation problem for collaborative DNN infer-
+ence in industrial IoT networks. A deep RL-based algorithm
+has been developed to determine the channel variation and
+the task arrival pattern which are then exploited to provide
+accuracy-guaranteed DNN inference services. The proposed
+algorithm can optimize service delay performance on the fly,
+without requiring statistic information of network dynamics.
+The Lyapunov-based transformation technique can be applied
+to other CMDPs. For the future work, we will investigate the
+impact of device mobility on the inference performance.
+APPENDIX
+A. Proof of Theorem 1
+Firstly, the problem is proved to be a convex optimization
+problem. For brevity of notations, we omit t in the proof. With
+the definition of Λm in (26), the objective function can be
+rewritten as �
+m∈M Λm/(cmfb). The second-order derivative
+of the objective function shows 2Λm/
+�
+fbc3
+m
+�
+> 0. In addition,
+the inequality constraint is linear. Hence, the problem is a
+convex optimization problem.
+Secondly, a Lagrange function for the problem without
+considering the inequality constraints is constructed, i.e.,
+L (c, a) =
+�
+m∈M
+Λm
+cmfb
++ a
+� �
+m∈M
+cm − 1
+�
+,
+(31)
+where a denotes the Lagrange multiplier. Based on Karush-
+Kuhn-Tucker conditions [26], we have
+∂L (c, a)
+∂cm
+= − Λm
+fbc2m
++ a = 0, ∀m ∈ M.
+(32)
+By solving the above equation, we can obtain c⋆
+m
+=
+�
+Λm/afb, ∀m ∈ M. Substituting the above result into the
+complementary slackness condition �
+m∈M c⋆
+m − 1 = 0, the
+optimal value of a is given by a⋆ =
+��
+m∈M
+√Λm
+�2/fb.
+From the above equation, a⋆ takes a positive value, and
+hence {c⋆
+m}m∈M are positive values, which means constraint
+(23b), i.e., ct
+m ≥ 0, ∀m ∈ M, is automatically satisfied.
+Substituting a⋆ into the complementary slackness condition
+proves Theorem 1.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf,len=830
+page_content='1  Accuracy-Guaranteed Collaborative DNN Inference  in Industrial IoT via Deep Reinforcement Learning  Wen Wu, Member, IEEE, Peng Yang, Member, IEEE, Weiting Zhang, Student Member, IEEE,  Conghao Zhou, Student Member, IEEE, and Xuemin (Sherman) Shen, Fellow, IEEE  Abstract—Collaboration among industrial Internet of Things  (IoT)  devices  and  edge  networks  is  essential  to  support  computation-intensive deep neural network (DNN) inference  services which require low delay and high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Sampling  rate adaption which dynamically configures the sampling rates  of industrial IoT devices according to network conditions, is  the key in minimizing the service delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In this paper, we  investigate the collaborative DNN inference problem in industrial  IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' To capture the channel variation and task arrival  randomness, we formulate the problem as a constrained Markov  decision process (CMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Specifically, sampling rate adaption,  inference task offloading and edge computing resource allocation  are jointly considered to minimize the average service delay while  guaranteeing the long-term accuracy requirements of different  inference services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Since CMDP cannot be directly solved by  general reinforcement learning (RL) algorithms due to the  intractable long-term constraints, we first transform the CMDP  into an MDP by leveraging the Lyapunov optimization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Then, a deep RL-based algorithm is proposed to solve the MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  To expedite the training process, an optimization subroutine  is embedded in the proposed algorithm to directly obtain the  optimal edge computing resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Extensive simulation  results are provided to demonstrate that the proposed RL-based  algorithm can significantly reduce the average service delay while  preserving long-term inference accuracy with a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Index Terms—Sampling rate adaption, inference accuracy,  collaborative DNN Inference, deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' INTRODUCTION  With the development of advanced neural network tech-  niques and ubiquitous industrial Internet of Things (IoT)  devices, deep neural network (DNN) is widely applied in  extensive industrial IoT applications, such as facility moni-  toring and fault diagnosis [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Industrial IoT devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  vibration sensors) can sense the industrial operating envi-  ronment and feed sensing data to a DNN, and then the  DNN processes the sensing data and renders inference results,  namely DNN inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Although DNN inference can achieve  high inference accuracy as compared to traditional alternatives  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', decision tree), executing DNN inference tasks requires  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Zhou, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Shen are with the Department of Electrical  and Computer Engineering, University of Waterloo, 200 University Av-  enue West, Waterloo, ON N2L 3G1, Canada (email: {w77wu, c89zhou,  sshen}@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Yang (Corresponding author) is with the School of Electronic Information  and Communications, Huazhong University of Science and Technology,  Wuhan, 430074, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' China (email: yangpeng@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Zhang is with the School of Electronic and Information Engi-  neering, Beijing Jiaotong University, Beijing 100044, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' China (email:  17111018@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  extensive computation resource due to tremendous multiply-  and-accumulation operations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' A device-only solution that  purely executes DNN inference tasks at resource-constrained  industrial IoT devices, becomes intractable due to prohibitive  energy consumption and a high service delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' For example,  processing an image using AlexNet incurs up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='45 W energy  consumption [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' An edge-only solution which purely offloads  large-volume sensing data to resource-rich edge nodes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  access point (AP), suffers from an unpredictable service delay  due to time-varying wireless channel [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence, neither a  device-only nor an edge-only solution can effectively support  low-delay DNN inference services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Collaborative  inference,  which  coordinates  resource-  constrained industrial IoT devices and the resource-rich  AP, becomes a de-facto paradigm to provide low-delay and  high-accuracy inference services [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Within the collaborative  inference, sensing data from industrial IoT devices can be  either processed locally or offloaded to the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' At industrial  IoT devices, light-weight compressed DNNs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', neural  networks are compressed without significantly decreasing  their performance) are deployed due to constrained on-board  computing capability, which saves computing resource at the  cost of inference accuracy [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' At the AP, uncompressed  DNNs are deployed to provide high-accuracy inference  services at the cost of network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Through the  resource allocation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', task offloading) between industrial  IoT devices and the AP, the overall service performance can  be enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  However, the sampling rate adaption technique that dynam-  ically configures the sampling rates of industrial IoT devices,  is seldom considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Through dynamically adjusting the sam-  pling rates according to channel conditions and AP’s workload,  sensing data from industrial IoT devices can be compressed,  thereby reducing not only the offloaded data volume, but also  task computation workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In our experiments, we implement  AlexNet to conduct bearing fault diagnosis based on the  collected bearing vibration signal from dataset [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='1 As shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 1, inference accuracy grows sub-linearly with the  sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' For example, when the sampling rate increases  from 18 KHz to 24 KHz, the accuracy increases from 95%  to 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence, when the channel condition is poor or  edge computation workload is heavy, decreasing the sampling  rate can reduce the offloaded data volume and requested  computation workload, thereby reducing the service delay at  1The experiment is conducted on an open-source dataset [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' This dataset  collects the vibration signal of drive end bearings at a sampling rate of  48 KHz, and there are 10 types of possible faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='00130v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='SY]  31 Dec 2022  2  the cost of limited inference accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' When channel condition  is good and edge computation workload is light, increasing the  sampling rate can help deliver a high-accuracy service with  an acceptable service delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence, sampling rate adaption  can effectively reduce the service delay, which should be  incorporated in the collaborative DNN inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The sampling rate adaption and resource allocation for  collaborative DNN inference are entangled with the following  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Firstly, due to time-varying channel conditions and  random task arrivals, sampling rate and resource allocation  should be dynamically adjusted to achieve the minimum  service delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Minimizing the long-term service delay requires  the stochastic information of network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Secondly, in  addition to minimizing the service delay, the long-term accu-  racy requirements should be guaranteed for different inference  services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The long-term accuracy performance is determined  by decisions of sampling rate adaption and resource allocation  over time, and hence the optimal decisions require future  network information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' To address the above two challenges, a  reinforcement learning (RL) technique is leveraged to interact  with the unknown environment to capture the network dynam-  ics, and then a Lyapunov optimization technique is utilized  within the RL framework to guarantee the long-term accuracy  requirements without requiring future network information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  In this paper, we investigate the collaborative DNN infer-  ence problem in industrial IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Firstly, we formu-  late the problem as a constrained Markov decision process  (CMDP) to account for time-varying channel conditions and  random task arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Specifically, sampling rates of industrial  IoT devices, task offloading, and edge computation resource  allocation are optimized to minimize the average service delay  while guaranteeing the long-term accuracy requirements of  multiple services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Secondly, since traditional RL algorithms  target at optimizing a long-term reward without considering  policy constraints, they cannot be applied to solve CMDP with  long-term constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' To solve the problem, we transform the  CDMP into an MDP via the Lyapunov optimization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The core idea is to construct accuracy deficit queues to  characterize the satisfaction status of the long-term accuracy  constraints, thereby guiding the learning agent to meet the  long-term accuracy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Thirdly, to solve the MDP,  a learning-based algorithm is developed based on the deep  deterministic policy gradient (DDPG) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Within the  learning algorithm, to reduce the training complexity, edge  computing resource allocation is directly solved via an op-  timization subroutine based on convex optimization theory,  since it only impacts one-shot delay performance according  to theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Extensive simulations are conducted  to validate the effectiveness of the proposed algorithm in  reducing the average service delay while preserving the long-  term accuracy requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Our main contributions in this paper are summarized as  follows:  We formulate the collaborative DNN inference problem  as a CMDP, in which the objective is to minimize the  average service delay while guaranteeing the long-term  accuracy constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  6  12  18  24  Sampling Rate (KHz)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='0  Inference Accuracy  AlexNet  Fitted curve  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Inference accuracy with respect to sampling rates on the bearing  vibration dataset [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  We transform the CMDP into an MDP via the Lyapunov  optimization technique which constructs accuracy deficit  queue to characterize the satisfaction status of the long-  term accuracy constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  We propose a deep RL-based algorithm to make the  optimal sampling rate adaption and resource allocation  decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' To reduce the training complexity, an optimiza-  tion subroutine is embedded in the proposed algorithm for  the optimal edge computing resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Sec-  tion II reviews related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The system model and problem  formulation are presented in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Section IV proposes  a learning-based solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Simulation results are given in  Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Finally, Section VI concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' RELATED WORK  DNN inference for resource-constrained industrial IoT de-  vices has garnered much attention recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' A device-only  solution aims to facilitate DNN inference services resorting  to on-board computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' To reduce the computational  complexity, DNN compression techniques are applied, such as  weight pruning [6] and knowledge distillation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Considering  the widely-equipped energy-harvesting functionality in IoT  devices, Gobieski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' designed a light-weight DNN inference  model, which can dynamically compress the model size in  order to balance inference accuracy and energy efficiency [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  In another line of research, edge-assisted DNN inference solu-  tions can provide high-accuracy inference services by utilizing  powerful edge computing servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' To facilitate low-delay and  accurate DNN-based video analytics, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' proposed  an online video quality and computing resource allocation  strategy to maximize video analytic accuracy [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Another  inspiring work proposed a novel device-edge collaborative  inference scheme, in which the DNN model is partitioned and  deployed at both the device and the edge, and intermediate  results are transferred via wireless links [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The above works  can provide possible resource allocation solutions to enhance  DNN inference performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Different from existing works,  our work takes the sampling rate adaption of industrial IoT  devices into account, aiming at providing accuracy-guaranteed  inference services in dynamic industrial IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  RL algorithms have been widely applied in allocating net-  work resources in wireless networks, such as service migra-  3  Service 1  Service 2  Service M  Compressed  DNN  Uncompressed  DNN  AP  Edge Server  Industrial IoT  devices  Sensing  data  Sampling  rate  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The collaborative DNN inference framework for industrial IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  tion in vehicular networks [11], network slicing in cellular  networks [12], content caching in edge networks [13], and  task scheduling in industrial IoT networks [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence, RL  algorithms are considered as plausible solutions to manage  network resources for DNN inference services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' However,  DNN inference services require minimizing the average delay  while satisfying the long-term accuracy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Tradi-  tional RL algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', DDPG, can be applied to solve  MDPs, in which learning agents seek to optimize a long-  term reward without policy constraints, while they cannot deal  with constrained long-term optimization problems [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Our proposed deep RL-based algorithm can address long-  term constraints within the RL framework by the modification  of reward based on the Lyapunov optimization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  In addition, an optimization subroutine is embedded in our  algorithm to further reduce the training complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' SYSTEM MODEL AND PROBLEM FORMULATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Network Model  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 2, we consider a wireless network with  one AP to serve multiple types of industrial IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The AP is in charge of collecting network information and  resource orchestration within the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Consider M types  of inference services, denoted by a set M, such as facility fault  diagnosis and facility monitoring services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Taking the facility  fault diagnosis service as an example, vibration sensors in-  stalled on industrial IoT devices sense the operating conditions  at a sampling rate, and feed the sensed vibration signal into a  DNN, then the DNN diagnoses the facility fault type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The set  of industrial IoT devices subscribed to service m is denoted  by Nm, and the set of all industrial IoT devices is denoted by  N = ∪m∈MNm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In the collaborative inference framework,  two types of DNNs are deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' One is a compressed DNN,  which is deployed at industrial IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The compressed  DNN can be implemented via the weight pruning technique,  which prunes less-important weights to reduce computational  complexity while maintaining similar inference accuracy [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The other is an uncompressed DNN, which is deployed at the  AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In this way, M types of uncompressed DNNs share the  edge computing resource to serve different inference requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Important notations are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Table I  SUMMARY OF NOTATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Notation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Description  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Am  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Achieved instantaneous accuracy of service m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Ath  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Long-term accuracy requirement of service m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Bt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Local computing queue backlog in time slot t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Computing resource allocation decision vector  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Service delay  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='L (·)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Lyapunov function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='o  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Task offloading decision vector  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Qt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Edge computing queue backlog in time slot t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='V  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Parameter to balance delay and accuracy requirement  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='X  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Sampling rate selection decision matrix of all devices  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Zt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Accuracy deficit queue backlog in time slot t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='ξn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Raw task data size of device n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='ηm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Task computation intensity of service m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='λn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Average task arrival rate of device n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='ζ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='xt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Task data size of device n in time slot t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Ψt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='Amount of dropped tasks in computing queues  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='The collaborative DNN inference framework operates in a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='time-slotted manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Let t denote the time index, where t ∈  T = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The detailed procedure is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  1) Sampling rate selection: Industrial IoT devices first  select their sampling rates according to channel condi-  tions and computation workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The set of candidate  sampling rates is denoted by K = {θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', θK},  where θK denotes the raw sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' We assume  the sampling rate in K increases linearly with the index,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', θk = kθK/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Let Xt denote the sampling rate  decision matrix in time slot t, whose element xt  n,k = 1  indicates industrial IoT device n ∈ N selects the k-th  sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  2) Task processing: The sensing data from industrial IoT  devices within a time slot is deemed as a computation  task, which can be either offloaded to the AP or executed  locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Let ot ∈ R|N |×1 denote the offloading decision  vector in time slot t, whose element ot  n = 0 indicates  offloading the computation task from industrial IoT  device n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Otherwise, ot  n = 1 indicates executing the  computation task locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Service Delay Model  A computation task can be either processed locally or  offloaded to the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In what follows, we analyze the service  delay in these two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  1) Executing locally: Let λt  n denote the task arrival rate  of the n-th industrial IoT device in time slot t, which is  assumed to follow a general random distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The raw  data size of the generated tasks at the n-th device is denoted  by ξt  n = λt  nνm, ∀n ∈ Nm, where νm denotes the raw data  size of a task for service m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' After the sampling rate is  selected, the data size of the generated task is represented  by ζ (xt  n) = �K  k=1 xt  n,kξt  nk/K, where xt  n = {xt  n,k}k∈K is  the sampling rate selection decision vector of the n-th device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  When the inference task is processed locally by a compressed  ((o)4  DNN, the service delay includes the queuing delay in the local  computing queue and task processing delay, which is given by  dt  n,l = ot  nηm,c (Bt  n + ζ (xt  n))  fn  , ∀n ∈ Nm,  (1)  where fn is the CPU frequency of the n-th industrial IoT  device, and ηm,c denotes the computation intensity of the  compressed DNN for the m-th service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Here, Bt  n is the  backlogged computation tasks (in bits) in the local computing  queue, which is updated via  Bt+1  n  = min  ��  Bt  n + ot  nζ  �  xt  n  �  − fnτ  ηm,c  �+  , Bmax  n  �  ,  (2)  where [x]+ = max{x, 0}, Bmax  n  is the capacity of the local  computing queue, and τ is the duration of a time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Tasks  will be dropped if the local computing queue is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Let  Ψt  b,n = max  �  Bt  n + ot  nζ  �  xt  n  �  − fnτ  ηm,c  − Bmax  n  , 0  �  (3)  denote the amount of the dropped tasks in the local computing  queue of device n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Here, Ψt  n,b > 0 indicates that an event of  local computing queue overflow occurs at the n-th device,  and the corresponding penalty will be incurred to avoid queue  overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  2) Offloading to AP: When a task is offloaded to the AP,  it will be processed by an uncompressed DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The service  delay consists of task offloading delay, queuing delay in the  edge computing queue, and task processing delay, which are  analyzed respectively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Task offloading delay: For the n-th industrial IoT device,  the offloading delay is given by  dt  n,o = (1 − ot  n) ζ (xt  n)  Rtn  ,  (4)  where transmission rate between the n-th industrial IoT  device and the AP, Rt  n, is given by  Rt  n = W  N log2  �  1 + PT G(Ht  n)  Nfσ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (5)  Here, W, PT , G(Ht  n), and Nf represent the system band-  width, transmit power, channel gain, and noise figure, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' σ2 = NoW/N denotes the background noise  where No is thermal noise spectrum density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Channel gain  G(Ht  n) varies in terms of channel state Ht  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Based on  extensive real-time measurements, channel state Ht  n can  be modeled with a finite set of channel states H [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The  evolution of channel states is characterized by a discrete-  time and ergodic Markov chain model, whose transition  matrix is P ∈ R|H|×|H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Task processing delay: The tasks from all industrial IoT  devices subscribed to the m-th service are placed in the  edge computing queue for the m-th service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The amount  of aggregated tasks is given by �  n∈Nm (1 − ot  n) ζ (xt  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The computing resource is dynamically allocated among  multiple services at the AP according to service task  arrivals, which can be realized via containerization tech-  niques, such as Dockers and Kubernetes [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Let ct ∈  RM×1 denote the computing resource allocation decision  vector in time slot t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Each element 0 ≤ ct  m ≤ 1 denotes  the portion of computing resource allocated to the m-th  service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence, the processing delay is given by  dt  n,p = ηm,u (1 − ot  n) ζ (xt  n)  ctmfb  , ∀n ∈ Nm,  (6)  where fb is the CPU frequency of the computing server  at the AP, and ηm,u denotes the computation intensity  of processing the m-th service task by the uncompressed  DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Note that ηm,u > ηm,c, since the uncompressed  DNN consumes more computing resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Queuing delay: The queuing delay consists of two com-  ponents: (i) the time taken to process backlogged tasks  in the edge computing queue, which is given by  dt  n,q = Qt  mηm,u  ctmfb  , ∀n ∈ Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (7)  Here, Qt  m denotes the edge computing queue backlog for  the m-th service in time slot t, which is updated according  to  Qt+1  m  = min  ��  Qt  m + at  m − ct  mfbτ  ηm,u  �+  , Qmax  m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (8)  Here, at  m = �  n∈Nm (1 − ot  n) ζ (xt  n) and Qmax  m  denotes  the capacity of the m-th edge computing queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Similar  to that in local computing queues, tasks will also be  dropped if the edge computing queue is full, and the  amount of dropped tasks for the m-th edge computing  queue is given by  Ψt  q,m = max  �  Qt  m + at  m − ct  mfbτ  ηm,u  − Qmax  m  , 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (9)  Here, Ψt  q,m > 0 indicates that an event of edge computing  queue overflow occurs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' and (ii) average waiting time  among all newly arrived tasks until the task of industrial  IoT device n is processed, which is given by  dt  n,w =  ηm,u  �  i̸=n,i∈Nm (1 − ot  i) ζ (xt  i)  2ctmfb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (10)  Here, �  i̸=n,i∈Nm (1 − ot  i) ζ (xt  n) denotes the amount of  aggregated tasks except the task of industrial IoT de-  vice n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Taking both local execution and offloading into account, the  service delay in time slot t is given by  D  �  Xt, ot, ct�  =  �  n∈N  �  dt  n,l + dt  n,o + dt  n,p + dt  n,q + dt  n,w  �  + wp  � �  n∈N  1{Ψt  b,n>0} +  �  m∈M  1{Ψt  q,m>0}  �  ,  (11)  where 1{x} = 1 and wp > 0 are the indicator function and  the positive unit penalty cost for queue overflow, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The first term represents the experienced delay to complete all  tasks in time slot t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The second term represents the penalty for  potential overflow events in local and edge computing queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  5  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Inference Accuracy Model  The inference accuracy depends on the sampling rate of a  task and the type of DNN that executes a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Firstly, we  characterize the relationship between the inference accuracy  and the sampling rate, which is specified by accuracy function  g(θk), ∀θk ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Specifically, we implement a DNN inference  algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', AlexNet [19], and apply the AlexNet to diag-  nose facility fault type based on the collected bearing vibration  signal from the dataset [8], and then measure the accuracy  function values with respect to sampling rates, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Secondly, the relationship between the inference  accuracy and the type of DNN is also characterized via  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Here, hm,c and hm,u represent the inference  accuracy of the compressed DNN and the uncompressed DNN  for the m-th service, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Note that, hm,c < hm,u,  as an uncompressed DNN achieves higher fault diagnosis  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Since the DNN model selection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', task offloading de-  cision) and the sampling rate selection are independent,  inference accuracy is the product of the accuracy value  with respect to the selected sampling rate and the accu-  racy value with respect to the selected DNN type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  g  ��  k∈K xt  n,kθk  �  (ot  nhm,c + (1 − ot  n) hm,u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence, the av-  erage inference accuracy for the m-th service in time slot t  can be given by  Am  �  Xt, ot�  =  �  n∈Nm  1  |Nm|g  ��  k∈K  xt  n,kθk  � �  ot  nhm,c +  �  1 − ot  n  �  hm,u  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (12)  Note that the model can be readily extended to cases when  other inference methods are adopted, since the accuracy values  with respect to sampling rates and DNN types are obtained via  practical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Problem Formulation  DNN inference services require not only minimizing service  delay, but also guaranteeing their long-term accuracy require-  ments, which can be modeled via a CMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Its action, state,  reward, and state transition matrix are defined as follows:  Action: The action includes the sampling rate selection,  task offloading, and edge computing resource alloca-  tion decisions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', ˆat = {Xt, ot, ct}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Note that the  components of the action should satisfy following con-  straints: (1) xt  n,k ∈ {0, 1} constrains the sampling rate  selection decision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' (2) ot  n ∈ {0, 1} requires the binary  task offloading decision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' and (3) �  m∈M ct  m ≤ 1 and  0 ≤ ct  m ≤ 1 constrain a continuous computing resource  allocation decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  State: The state includes local computing queues backlog  of industrial IoT devices Bt  n, edge computing queues  backlog Qt  m, channel conditions of industrial IoT devices  Ht  n, and the raw data size of the generated tasks at  industrial IoT devices ξt  n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  ˆst ={{Bt  n}n∈N , {Qt  m}m∈M, {Ht  n}n∈N , {ξt  n}n∈N }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (13)  The queue backlogs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', {Bt  n}n∈N and {Qt  m}m∈M,  adopt a unit in bits, which result in large state space,  especially for a large number of industrial IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Reward: The reward is designed to minimize the service  delay in (22) in time slot t, which is defined as ˆrt =  −D (Xt, ot, ct) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  State transition probability: State transition probability is  given by  Pr  �  ˆst+1|ˆst, ˆat�  =  �  n∈N  Pr  �  Bt+1  n  |Bt  n, xt  n,k, ot  n  � �  m∈M  Pr  �  Qt+1  m |Qt  m, Xt, ot� �  n∈N  Pr  �  Ht+1  n  |Ht  n  � �  n∈N  Pr  �  ξt+1  n  |ξt  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (14)  The equality holds due to the independence of different  state components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The first two components are gov-  erned by the evolution of local computing queues and  edge computing queues in (2) and (8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The  third component is evolved according to the discrete-  time Markov chain of channel conditions, and the last  component is governed by the memoryless task arrival  pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Note that each of those state components only  depends on its previous state components, which means  the state transition is Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Our goal is to find a stationary policy π ∈ Π that dynam-  ically configures sampling rates selection Xt, task offload-  ing ot, and edge computing resource allocation ct according  to state ˆst, to minimize the service delay while guaranteeing  long-term inference accuracy requirements {Ath  m}m∈M, which  is formulated as the following problem:  P0 : min  π∈Π  lim  T →∞  1  T  T  �  t=1  Eπ  �  D  �  Xt, ot, ct��  (15a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  lim  T →∞  1  T  T  �  t=1  Am  �  Xt, ot�  ≥ Ath  m, ∀m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' (15b)  Here, P0 is a CMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Directly solving the above CMDP via  dynamic programming solutions [15] is challenging due to  the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Firstly, state transition probability is  unknown due to the lack of statistic information on the channel  condition variation and task arrival patterns of all industrial  IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Secondly, even the state transition probability  is known, large action space and state space that grow with  respect to the number of industrial IoT devices incur an ex-  tremely high computational complexity, which makes dynamic  programming solutions intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence, we propose a deep  RL-based algorithm to solve the CMDP, which can be applied  in large-scale networks without requiring statistic information  of network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' DEEP RL-BASED SAMPLING RATE ADAPTION AND  RESOURCE ALLOCATION ALGORITHM  As mentioned before, a CDMP problem cannot be directly  solved via traditional RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' We first leverage the  Lyapunov optimization technique to deal with the long-term  constraints and transform the problem into an MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Then,  6  we develop a deep RL-based algorithm to solve the MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  To further reduce the training complexity, an optimization  subroutine is embedded to directly obtain the optimal edge  computation resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Lyapunov-Based Problem Transformation  The major challenge in solving problem P0 is to han-  dle the long-term constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' We leverage the Lyapunov  technique [20], [21] to address this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The core  idea is to construct accuracy deficit queues to characterize  the satisfaction status of the long-term accuracy constraints,  thereby guiding the learning agent to meet the long-term  accuracy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The problem transformation procedure  is presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Firstly, we construct inference accuracy deficit queues for  all services, whose dynamics evolves as follows:  Zt+1  m  =  �  Ath  m − Am  �  Xt, ot�  + Zt  m  �+ , ∀m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (16)  Here, Zt  m indicates the deviation of the achieved instantaneous  accuracy from the long-term accuracy requirement, whose  initial state is set to Z0  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Then, a Lyapunov function  is introduced to characterize the satisfaction status of the  long-term accuracy constraint, which is defined as L (Zt  m) =  (Zt  m)2 /2 [20]–[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' A smaller value of L (Zt  m) indicates  better satisfaction of the long-term accuracy constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Secondly, the Lyapunov function should be consistently  pushed to a low value in order to guarantee the long-term  accuracy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence, we introduce a one-shot Lya-  punov drift to capture the variation of the Lyapunov function  across two subsequent time slots [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Given Zt  m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' the one-shot  Lyapunov drift is defined as ∆ (Zt  m) = L  �  Zt+1  m  �  − L (Zt  m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  which is upper bounded by  ∆  �  Zt  m  �  = 1  2  ��  Zt+1  m  �2 −  �  Zt  m  �2�  ≤ 1  2  ��  Zt  m + Ath  m − Am  �  Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' ot��2 −  �  Zt  m  �2�  = 1  2  �  Ath  m − Am  �  Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' ot��2 + Zt  m  �  Ath  m − Am  �  Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' ot��  ≤ Cm + Zt  m  �  Ath  m − Am  �  Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' ot��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (17)  where Cm =  �  Ath  m − Amin  m  �2 /2 is a constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' and Amin  m  is the  lowest inference accuracy that can be achieved for service m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The first inequality is due to the substitution of (16), and the  second inequality is because Am (Xt, ot) ≥ Amin  m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Thirdly, based on the Lyapunov optimization theory, the  original CMDP of minimizing the service delay while guar-  anteeing the long-term accuracy requirements boils down to  minimizing a drift-plus-cost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  �  m∈M  ∆  �  Zt  m  �  + V · D  �  Xt, ot, ct�  ≤  �  m∈M  Cm +  �  m∈M  Zt  m  �  Ath  m − Am  �  Xt, ot��  + V · D  �  Xt, ot, ct�  ,  (18)  where the inequality is due to the upper bound in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Here  V is a positive parameter to adjust the tradeoff between the  service delay minimization and the satisfaction status of the  long-term accuracy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The underlying rationale is  that, if the long-term accuracy constraint is violated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  Zt  m > 0, stratifying the long-term constraints by improving the  instantaneous inference accuracy becomes more urgent than  reducing the service delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  In this way, the CMDP is transformed into an MDP with  the objective of minimizing the drift-plus-cost in each time  slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Equivalent MDP  In the equivalent MDP, the action, state, reward, and state  transition matrix are modified as follows due to the incorpo-  ration of accuracy deficit queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Action: The action is the same as that in the CMDP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  at = ˆat = {Xt, ot, ct}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  State: Compared with the state of the CMDP, the accuracy  deficit queue backlog of services {Zt  m}m∈M should be  incorporated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  st = {ˆst, {Zt  m}m∈M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (19)  Reward: The reward is modified to minimize the drift-  plus-cost in (18) in time slot t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  rt = −V · D  �  Xt, ot, ct�  −  �  m∈M  Zt  m  �  Ath  m − Am  �  Xt, ot��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (20)  Note that the constant term �  m∈M Cm in (18) is ignored  in the reward for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  State transition probability: Since accuracy deficit queue  backlogs are incorporated in the state, the state transition  probability evolves according to  Pr  �  st+1|st, at�  = Pr  �  ˆst+1|ˆst, ˆat� �  m∈M  Pr  �  Zt+1  m |Zt  m, Xt, ot�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (21)  where the second term is the evolution of the accuracy  deficit queue backlog according to (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Note that the  overall state transition is still Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Then, problem P0 is transformed into the following MDP  problem:  P1 : min  π∈Π  lim  T →∞  1  T  T  �  t=1  Eπ  � �  m∈M  Zt  m  �  Ath  m − Am  �  Xt, ot��  +V · D  �  Xt, ot, ct��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (22)  Similar to CMDP, solving an MDP via dynamic programming  solutions also suffers from the curse of dimensionality due  to large state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence, we propose a deep RL-based  algorithm to solve the MDP, which is detailed in Section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Optimization Subroutine for Edge Computing Resource  Allocation  Although P1 can be directly solved by RL algorithms, an  inherent property on edge computing resource allocation can  7  be leveraged, in order to reduce the training complexity of  RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Through analysis on (22), the edge computing  resource allocation is independent of the inference accuracy  performance, and hence it only impacts the one-shot service  delay performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In time slot t, once task offloading and  sampling rate selection decisions are made, the optimal com-  puting resource allocation decision can be obtained via solving  the following optimization problem:  P2 : min  ct  D  �  Xt, ot, ct�  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  �  m∈M  ct  m ≤ 1  (23a)  0 ≤ ct  m ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (23b)  A further analysis of (11) indicates that only the task  processing delay and queuing delay at the AP are im-  pacted by the edge computing resource allocation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  �  n∈N  �  dt  n,p + dt  n,q + dt  n,w  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In addition, the aggregated de-  lay from the perspective of all devices is equivalent to the  aggregated delay from the perspective of all services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence,  the objective function in P2 can be rewritten as �  m∈M dt  m,  where  dt  m =  �  n∈Nm  �ηm,u (1 − ot  n) ζ (xt  n)  ctmfb  + Qt  mηm,u  ctmfb  +  ηm,u  �  i̸=n,i∈Nm (1 − ot  i) ζ (xt  i)  2ctmfb  �  (24)  denotes the experienced delay of the m-th service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' By ana-  lyzing the convexity property of the problem, we have the  following theorem to obtain the optimal edge computation  resource allocation in each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The optimal edge computing resource allocation  for problem P2 is given by  ct,⋆  m =  �  Λtm  �  m∈M  �  Λtm  , ∀m ∈ M,  (25)  where  Λt  m =  �  n∈Nm  �  ηm,u  �  1 − ot  n  �  ζ  �  xt  n  �  + Qt  mηm,u  +ηm,u  2  �  i̸=n,i∈Nm  �  1 − ot  i  �  ζ  �  xt  i  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (26)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Proof is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  This optimization subroutine for the edge computing re-  source allocation is embedded in the following proposed deep  RL-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In this way, the training complexity can  be reduced, because it is no longer necessary to train the neural  networks to obtain optimal edge computing resource allocation  policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Deep RL-based Algorithm  To solve problem P1, we propose a deep RL-based al-  gorithm, which is extended from the celebrated DDPG al-  gorithm [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The main difference between DDPG and the  Algorithm 1: Deep RL-based algorithm for sampling  rate adaption and resource allocation  1 Initialization: Initialize all neural networks and the  experience replay memory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  2 for each episode do  3  Reset the environment and obtain initial state s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  4  for time slot t ∈ T do  5  Determine the sampling rate selection and task  offloading actions {Xt, ot} by the actor  network according to current state st;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  6  Determine edge computing resource allocation  action ct by (25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  7  Send joint action at = {Xt, ot, ct} to all  industrial IoT devices by the AP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  8  Execute the joint action at industrial IoT  devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  9  Observe reward rt and new state st+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  10  Store transition {st, at, rt, st+1} in the  epxerience replay memory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  11  Sample a random minibatch transitions from  the epxerience replay memory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  12  Train the critic and actor network by (27) and  (28), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  13  Update target networks by (29);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  14  end  15 end  proposed algorithm is that the above optimization subroutine  for computing resource allocation is embedded to reduce the  training complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The proposed algorithm can be deployed  at the AP, which collects the entire network state information  and enforces the policy to all connected industrial IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  In the algorithm, the learning agent has two parts: (a) an  actor network, which is to determine the action based on the  current state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' and (b) a critic network, which is to evaluate  the determined action based on the reward feedback from the  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Let µ(s|φµ) and Q(s, a|φQ) denote the actor  network and the critic network, respectively, whose neural  network weights are φµ and φQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' As shown in Algorithm 1, the  deep RL-based algorithm operates in a time-slotted manner,  which consists of the following three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The first step is to obtain experience by interacting with  the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Based on current network state st, the  actor network generates the sampling rate selection and task  offloading actions with an additive policy exploration noise  that follows Gaussian distribution N  �  0, σ2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The optimization  subroutine generates the edge computation resource allocation  action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Then, the joint action is executed at all industrial  IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The corresponding reward rt and the next state  st+1 are observed from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The state transition  {st, at, rt, st+1} is stored in the experience replay memory for  actor and critic network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The second step is to train the actor and critic network  based on the stored experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' To avoid the divergence issue  caused by DNN, a minibatch of transitions are randomly sam-  pled from the experience replay memory to break experience  8  Table II  SIMULATION PARAMETERS [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Parameter  Value  Thermal noise spectrum density (No)  −174 dBm/Hz [25]  Communication bandwidth (W)  [5, 25] MHz  Transmit power (PT )  20 dBm [24]  Average task arrival rate (λ)  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='6, 1] request/sec  Noise figure (Nf)  5 dB  Intensity of compressed DNN (η1,c, η2,c)  (80, 160) cycles/bit  Intensity of uncompressed DNN (η1,u, η2,u)  (200, 400) cycles/bit  Device and edge server CPU frequency (fn, fb)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='1, 2) GHz  Number of Type I/II devices (N1, N2)  5, 5  Time slot duration (τ)  1 second  Balance parameter (V)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='05  Unit penalty for queue overflow (wp)  1  Accuracy of compressed DNN (h1,c, h2,c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='8  Accuracy of uncompressed DNN (h1,u, h2,u)  1, 1  Local/edge queue capacity (Bmax, Qmax)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='84, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='2) megabits  Table III  PARAMETERS OF THE PROPOSED RL-BASED ALGORITHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Parameter  Value  Parameter  Value  Actor learning rate  10−4  Critic learning rate  10−3  Actor hidden units  (64, 32)  Critic hidden units  (64, 32)  Hidden activation  ReLU  Actor output activation  Tanh  Optimizer  Adam  Policy noise (σ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='2  Target update (δ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='005  Discount factor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='85  Minibatch size  64  Replay memory size  100,000  Training episodes  1, 000  Time slots per episode  200  correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The critic network is trained by minimizing the  loss function  Loss  �  φQ�  = 1  Nb  Nb  �  i=1  �  yi − Q(si, ai|φQ)  �2 ,  (27)  where yi = ri + γQ′(si+1, µ′(si+1|φµ′)|φQ′), and Nb is the  minibatch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Here, µ′(s|φµ′) and Q′(s, a|φQ′) represent  actor and critic target networks with weights φµ′ and φQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The actor network is trained via the policy gradient  ∇φµ ≈ 1  Nb  Nb  �  i=1  ∇aQ(si, a|φQ)|s=si,a=µ(si)∇θµµ(si|φµ)|si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (28)  The third step is to update target networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In order to ensure  network training stability, the actor and critic target networks  are softly updated by  φQ′ = δφQ + (1 − δ)φQ′, φµ′ = δφµ + (1 − δ)φµ′,  (29)  where 0 < δ ≪ 1 denotes the target network update ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' SIMULATION RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Simulation Setup  We consider a smart factory scenario in our simulation,  in which industrial IoT devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', vibration sensors, are  randomly scattered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The industrial IoT devices installed on  industrial facilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', robot arms) sense their operating  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The sensing data is processed locally or offloaded  to an AP in the smart factory for processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The transmit  power of an industrial IoT device is set to 20 dBm [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The  channel condition is modeled with three states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', “Good  0  200  400  600  800  1000  Training Episodes  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='45  Service Delay (s)  (a) Average delay performance  0  200  400  600  800  1000  Training Episodes  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='95  Average Accuracy  Type I Service  Type II Service  (b) Accuracy Performance of two services  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Performance of the proposed algorithm in the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (G)”, “Normal (N)”, and “Bad (B)”, and the corresponding  transition matrix is given by [17]  P =  �  �  PGG  PGN  0  PNG  PNN  PNB  0  PNB  PBB  �  � =  �  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='3  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (30)  Two types of DNN inference services are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Type  I service: a facility fault diagnosis service to identify the fault  type based on the collected bearing vibration signal from the  dataset [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Since the duration of a time slot in the simulation  is set to be one second, the task data size is the data volume of  a one-second signal, which is a product of the raw sampling  rate and the quantization bits of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In the dataset,  the bearing vibration signal is collected at 48 KHz sampling  rate and 16 bit quantization, and hence the corresponding task  data size is 768 kilobits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The long-term accuracy requirement  of the service is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Type II service: a service extended  from the Type I service to diagnose facility fault based on  a low-grade bearing vibration dataset while requiring higher  inference accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The low-grade dataset collects vibra-  tion signal at a lower sampling rate of 32 KHz, and hence  the task data size is 512 kilobits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' For both services, the task  arrival rate of each industrial IoT device at each time slot  follows a uniform distribution U(λ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='5, λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='5), where λ  is the average task arrival rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' We consider four candidate  sampling rates for industrial IoT devices, which are 25%, 50%,  75% and 100% of the raw sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The corresponding  accuracy with respect to the sampling rates are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='59, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='884,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='950 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='987, respectively, based on extensive experiments  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='0  Task Arrival Rate (req/s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='50  Service Delay (s)  Static Configuration  Delay Myopic  Proposed  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Service delay performance with respect to task arrival rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
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+page_content='910  Type II Service  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Inference accuracy performance with respect to task arrival rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  on the bearing vibration dataset [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Balance parameter V is  set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='05 based on extensive simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Other important  simulation parameters are listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The parameters  of the proposed algorithm are given in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The proposed  algorithm is compared to the following benchmarks:  Delay myopic: Each industrial IoT device dynamically  makes sampling rate selection and task offloading deci-  sions by maximizing the one-step reward in (20) accord-  ing to the network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Static configuration: Each industrial IoT device takes  a static configuration on the sampling rate selection and  task offloading decisions, which can guarantee services’  accuracy requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Performance Evaluation  1) Convergence of the proposed algorithm: The service  delay performance in the training stage is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  We can clearly see that the average service delay gradually de-  creases as the increase of training episodes, which validates the  convergence of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In addition, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 3(b)  shows the accuracy performance for both services with respect  to training episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The accuracy performance is not good at  the beginning of the training stage, but after 1,000 episodes  of training, the accuracy performance converges to the pre-  determined requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  2) Impact of task arrival rate: Once well-trained offline,  we evaluate the performance of the proposed algorithm in  the online inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 4, we compare the  average service delay performance of the proposed algorithm  with benchmark schemes in terms of task arrival rates for  5  10  15  20  25  Bandwidth (MHz)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
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+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='65  Service Delay (s)  Static Configuration  Delay Myopic  Proposed  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Service delay performance with respect to communication bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='20  CPU Frequency of Edge Server (GHz)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='2  Service Delay (s)  Proposed  Proposed-fixed  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Service delay in terms of CPU frequency of the edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  W = 20 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Each simulation point is plotted with a 95%  confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Several observations can be obtained from  the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Firstly, the service delay increases with the task  arrival rate due to constrained communication and computing  resources in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Secondly, the proposed algorithm  significantly outperforms benchmark schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The reason is  that the proposed RL-based algorithm can capture network dy-  namics, such as the task arrival pattern and channel condition  variation, via interacting with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The learned  knowledge is utilized to make online decisions that target at the  long-term performance, while benchmark schemes only focus  on the short-term performance and do not adapt to network  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Specifically, the proposed algorithm can reduce  the average service delay by 19% and 25%, respectively, as  compared with delay myopic and static configuration schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 5, boxplot accuracy distribution of two  services is presented with respect to different task arrival rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The long-term accuracy requirements for two services are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='8  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' It can be seen that the proposed algo-  rithm guarantees the long-term accuracy requirements of both  services with a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Specifically, the maximum  error probability is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  3) Impact of communication bandwidth: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 6 shows the  impact of communication bandwidth on the average service  delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Firstly, we can see that the average service delay  decreases as the growth of bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The reason is that  the transmission delay is reduced when the communication  resource becomes sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In addition, the proposed al-  10  gorithm achieves good performance when the bandwidth is  scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' When system bandwidth is only 5 MHz, the proposed  algorithm achieves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='20× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='42× delay reduction com-  pared with delay myopic and static configuration schemes,  respectively, which is larger than that when the system band-  width is 25 MHz (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='15× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='31×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The reason is that the  proposed algorithm efficiently utilizes the on-board computing  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Simulation results show that the proposed algorithm  decides 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='5% computation tasks to be executed locally with  5 MHz bandwidth, while the delay myopic benchmark only  decides 17%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Due to the efficient resource orchestration among  industrial IoT devices and the AP, the proposed algorithm can  effectively reduce average service delay for both services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  4) Impact of optimization subroutine: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' 7,  we evaluate the performance of the proposed algorithm  with the fixed computing resource allocation (referred to as  proposed-fixed), in which the edge computing resource is  allocated based on the average computing demand of two  services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Compared with the proposed-fixed solution, the pro-  posed algorithm achieves significant performance gain when  the edge computing resource is constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Specifically, the  performance gain in reducing the service delay decreases  from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='98× at 1 GHz CPU frequency to only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='02× at  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='2 GHz CPU frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The reason is that efficient resource  allocation is more important in resource-constrained scenarios,  as compared to resource-rich scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The results validate  the effectiveness of the optimization subroutine for edge  computing resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In addition to the performance  gain, another merit of the optimization subroutine is to reduce  the training complexity of RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' CONCLUSION  In this paper, we have studied the sampling rate adaption  and resource allocation problem for collaborative DNN infer-  ence in industrial IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' A deep RL-based algorithm  has been developed to determine the channel variation and  the task arrival pattern which are then exploited to provide  accuracy-guaranteed DNN inference services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The proposed  algorithm can optimize service delay performance on the fly,  without requiring statistic information of network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  The Lyapunov-based transformation technique can be applied  to other CMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' For the future work, we will investigate the  impact of device mobility on the inference performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Proof of Theorem 1  Firstly, the problem is proved to be a convex optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' For brevity of notations, we omit t in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' With  the definition of Λm in (26), the objective function can be  rewritten as �  m∈M Λm/(cmfb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' The second-order derivative  of the objective function shows 2Λm/  �  fbc3  m  �  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' In addition,  the inequality constraint is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Hence, the problem is a  convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Secondly, a Lagrange function for the problem without  considering the inequality constraints is constructed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=',  L (c, a) =  �  m∈M  Λm  cmfb  + a  � �  m∈M  cm − 1  �  ,  (31)  where a denotes the Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Based on Karush-  Kuhn-Tucker conditions [26], we have  ∂L (c, a)  ∂cm  = − Λm  fbc2m  + a = 0, ∀m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  (32)  By solving the above equation, we can obtain c⋆  m  =  �  Λm/afb, ∀m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=' Substituting the above result into the  complementary slackness condition �  m∈M c⋆  m − 1 = 0, the  optimal value of a is given by a⋆ =  ��  m∈M  √Λm  �2/fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  From the above equation, a⋆ takes a positive value, and  hence {c⋆  m}m∈M are positive values, which means constraint  (23b), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content=', ct  m ≥ 0, ∀m ∈ M, is automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
+page_content='  Substituting a⋆ into the complementary slackness condition  proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQfUvcz/content/2301.00130v1.pdf'}
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+arXiv:2301.02705v1  [math.PR]  6 Jan 2023
+COVARIANCE LOSS, SZEMEREDI REGULARITY,
+AND DIFFERENTIAL PRIVACY
+MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN
+Abstract. We show how randomized rounding based on Grothendieck’s identity can be
+used to prove a nearly tight bound on the covariance loss–the amount of covariance that
+is lost by taking conditional expectation. This result yields a new type of weak Szemeredi
+regularity lemma for positive semidefinite matrices and kernels. Moreover, it can be used to
+construct differentially private synthetic data.
+1. Introduction
+Consider a random vector X taking values in Rd and a σ-algebra F. Let Y denote the
+conditional expectation: Y = E[X|F]. In dimension d = 1, where X and Y are random
+variables, the law of total variance states that
+Var(X) − Var(Y ) = E X2 − E Y 2 = E(X − Y )2 ≥ 0.
+(1.1)
+Thus, taking conditional expectation underestimates the variance. This observation extends
+to higher dimensions. Namely, let ΣX = E(X −E X)(X −E X)T denote the covariance matrix
+of X, and similarly for ΣY . Then
+ΣX − ΣY = E XXT − E Y Y T = E(X − Y )(X − Y )T ⪰ 0,
+(1.2)
+where ⪰ denotes the Loewner order, in which A ⪰ B if A − B is positive semidefinite. Just
+like in the one-dimensional case, we see that taking conditional expectation underestimates
+the covariance.
+In [2], we asked the basic question: “How big is the covariance loss ΣX −ΣY ?” The answer
+will obviously depend on the choice of the sigma-algebra F, prompting the next question:
+“What sigma-algebra F of given complexity minimizes the covariance loss?”
+It was shown in [2] that there exists a partition of the sample space into at most k sets
+such that for the sigma-algebra F generated by this partition, the covariance loss (1.2) is
+upper bounded by C
+�
+log log k/ log k, where C is an absolute constant. In [6], Jain, Sah,
+and Sawhney were able to improve this bound to C/√log k.
+It follows from Proposition
+3.14 in [2] that this bound is optimal up to the value of the constant C. The proofs of the
+bounds in both [2] and [6] are somewhat technical. In this paper we give a new approach
+to this problem, which is conceptually related to Szemeredi regularity. This approach yields
+a tighter bound in terms of the constant on the covariance loss (Corollary 5.1), as well as a
+much shorter and more elementary proof.
+The celebrated Szemeredi regularity lemma states that that for every large enough graph,
+the set of nodes can be divided into subsets of about the same size so that the edges between
+different subsets behave almost randomly [10]. Expressed in the language of linear algebra,
+it says that the adjacency matrix of a graph can be approximated by a sum of cut matrices1.
+The Szemeredi regularity lemma, both in the combinatorial and linear-algebraic forms, has
+1A cut matrix is a rank-one matrix which is constant on a block and zero elsewhere [5].
+1
+
+2
+MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN
+many deep algorithmic and combinatorial applications, see e.g. [5, 8, 7, 1]. We prove a new
+version of (weak) Szemeredi regularity lemma, which states that any positive semidefinite
+kernel K(x, y) whose diagonal is uniformly bounded can be decomposed into a sum of k
+step-functions plus an error term whose L2-norm is O(1/√log k).
+The outline of the paper is as follows.
+We will first bound the covariance loss by the
+“covariance increment” (Proposition 2.1), find a nice representation of the covariance incre-
+ment (Proposition 2.3), and combine it with randomized rounding based on Grothendieck’s
+identity to bound the covariance increment (Theorem 3.1). Thus we can not only bound
+the covariance loss (Corollary 5.1) but also derive a new type of weak Szemeredi regularity
+for matrices (Theorem 5.3) and kernels (Theorem 5.6). We conclude by illustrating how our
+results can be used in connection with differential privacy and synthetic data to improve the
+accuracy bounds obtained in [2].
+1.1. Notation. The subgaussian norm of a random variable X is denoted by ∥X∥ψ2 (see
+e.g. [11]), E(X) is the expectation of X and ∥X∥Lp = (E|X|p)1/p. For d × d matrices A, B,
+define the inner product ⟨A, B⟩ = Tr(ABT ) and the Frobenius norm ∥A∥F =
+�
+⟨A, A⟩.
+2. The covariance increment
+We start out by bounding the covariance loss E XXT−E Y Y T by the covariance increment.
+In the sequel, X is a random vector taking values in Rd, F is a σ-algebra and Y = E[X|F].
+Moreover, (X′, F′, Y ′) is an independent copy of (X, F, Y ) so that the sample space is a
+product space for which X, F, Y are based on the first component of the product space and
+X′, F′, Y ′ are based on the second.
+With a slight abuse of notation, the product σ-algebra F × F denotes the σ-algebra gen-
+erated by F and F′. Moreover, the inner product ⟨Y, Y ′⟩ = E(⟨X, X′⟩| F × F).
+Proposition 2.1 (Covariance loss and covariance increment).
+���E XXT − E Y Y T���
+2
+F ≤
+���E XXT���
+2
+F −
+���E Y Y T���
+2
+F .
+(2.1)
+The proof of Proposition 2.1 will be based on the following lemma:
+Lemma 2.2. Let (U, V ) be a pair of random vectors taking values in Rd, and let (U ′, V ′) be
+an independent copy. Then
+⟨E UU T, E V V T⟩ = E⟨U, V ′⟩2.
+In particular, setting V = U, we have
+���E UU T���
+2
+F = E⟨U, U ′⟩2.
+Proof. The first identity readily follows if we first use the identical distribution, and then
+independence:
+⟨E UU T, E V V T⟩ = ⟨E UU T, E V ′(V ′)T⟩ = E⟨UU T, V ′(V ′)T⟩ = E⟨U, V ′⟩2.
+The lemma is proved.
+□
+
+COVARIANCE LOSS, SZEMEREDI REGULARITY,
+AND DIFFERENTIAL PRIVACY
+3
+Proof of Proposition 2.1. Expanding the square of the Frobenius norm, we can express the
+left hand side of (2.1) as
+��E XXT��2
+F − 2⟨E XXT, E Y Y T⟩ + ∥E Y Y T∥2
+F . After simplification,
+we see that inequality (2.1) is equivalent to
+���E Y Y T���
+2
+F ≤ ⟨E XXT, E Y Y T⟩.
+Using Lemma 2.2 we can rewrite this as
+E⟨Y, Y ′⟩2 ≤ E⟨X, Y ′⟩2,
+(2.2)
+To check (2.2), recall that Y = E[X|F] and apply the conditional Jensen’s inequality.
+□
+The covariance increment has a nice representation, which will come handy in our further
+analysis.
+Proposition 2.3.
+���E XXT���
+2
+F −
+���E Y Y T���
+2
+F = E
+�
+⟨X, X′⟩ − ⟨Y, Y ′⟩
+�2
+.
+(2.3)
+Proof. By Lemma 2.2, the left hand side of (2.3) equals E⟨X, X′⟩2 − E⟨Y, Y ′⟩2. Note that
+⟨Y, Y ′⟩ = E(⟨X, X′⟩| F × F). To finish the proof, apply the law of total variance (1.1) for
+⟨X, X′⟩ instead of X and ⟨Y, Y ′⟩ instead of Y .
+□
+3. Bounding the covariance increment
+Theorem 3.1. Let X be a random vector taking values Rd such that ∥X∥2 ≤ 1 a.s. Then,
+for any r ∈ N, there exists a partition of the sample space into at most 2r parts such that for
+the σ-algebra F generated by this partition, the conditional expectation Y = E[X|F] satisfies
+E
+�
+⟨X, X′⟩ − ⟨Y, Y ′⟩
+�2
+≤ π2
+r .
+The proof of Theorem 3.1 will utilize Grothendieck’s identity (see e.g. [11, Lemma 3.6.6]):
+Lemma 3.2 (Grothendieck’s identity). Let x, x′ be a pair of unit vectors in Rd, and let
+g ∼ N(0, Id). Then
+⟨x, x′⟩ = sin
+�π
+2 E sign⟨x, g⟩ sign⟨x′, g⟩
+�
+.
+Proof of Theorem 3.1. Step 1. Let us first make a stronger assumption, namely that∥X∥2 = 1
+a.s., and prove a weaker conclusion, namely that there exists a random variable Z measurable
+with respect to the product σ-algebra F × F and such that
+��⟨X, X′⟩ − Z
+��
+L2 ≤
+π
+2√r.
+(3.1)
+To this end, consider independent random vectors g1, . . . , gk ∈ N(0, Id), and for x, x′ ∈ Rd,
+define the random variable
+Fg(x, x′) = sin
+
+π
+2 · 1
+r
+r
+�
+k=1
+sign⟨x, gk⟩ sign⟨x′, gk⟩
+
+ .
+
+4
+MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN
+Denoting ξk = sign⟨x, gk⟩ sign⟨x′, gk⟩, noting that the function sin(·) is 1-Lipschitz and ap-
+plying Lemma 3.2, we obtain
+��⟨x, x′⟩ − Fg(x, x′)
+�� ≤ π
+2 · 1
+r
+������
+r
+�
+k=1
+(ξk − E ξk)
+������
+.
+(3.2)
+By independence, this yields
+Eg
+�
+⟨x, x′⟩ − Fg(x, x′)
+�2 ≤
+� π
+2r
+�2
+· r Var(ξ1) ≤ π2
+4r .
+Substitute x = X, x′ = X′ and take expectation with respect to X and X′. By Fubini
+theorem, there exists a realization of the random vectors g1, . . . , gk such that
+E
+�
+⟨X, X′⟩ − Fg(X, X′)
+�2 ≤ π2
+4r ,
+Fix such a realization. Let F = σ(V (X)) be the σ-algebra generated by the random vector
+V (X) = (sign⟨x, gk⟩)r
+k=1. Since V (X) takes at most 2r values, F satisfies the requirement of
+the theorem. Moreover, the random vector
+Z = Fg(X, X′) = sin
+� π
+2r⟨V (X), V (X′)⟩
+�
+is measurable with respect to the product sigma-algebra F × F. Thus, we proved (3.1).
+Step 2: replacing Z with ⟨Y, Y ′⟩. The conditional expectation E
+�
+·| F × F
+�
+is an orthogonal
+projection in L2 onto the subspace of random variables that are F × F-measurable. Since
+the random variable Z constructed in the previous step is F × F-measurable, and ⟨Y, Y ′⟩ =
+E
+�
+⟨X, X′⟩| F × F
+�
+, it follows from (3.1) that
+��⟨X, X′⟩ − ⟨Y, Y ′⟩
+��
+L2 ≤
+��⟨X, X′⟩ − Z
+��
+L2 ≤
+π
+2√r.
+Step 3: removing the unit norm requirement. We proved the theorem under the additional
+assumption that X is a unit random vector.
+Now let X be an arbitrary random vector
+satisfying ∥X∥2 ≤ 1.
+Let N ∈ N, and consider the random vector W =
+�
+1 −∥X∥2
+2 · θ,
+where θ is a random vector that is uniformly distributed in the unit basis {e1, . . . , eN} of RN.
+Then the direct sum X ⊕ W = (X1, . . . , Xd, W1, . . . , WN) is a unit random vector in Rd+N.
+Applying the previous step for X ⊕ W, we find that
+π
+2√r ≥
+��⟨X ⊕ W, X′ ⊕ W ′⟩ − ⟨Y ⊕ η, Y ′ ⊕ η′⟩
+��
+L2
+(where η = E(W|F))
+=
+��⟨X, X′⟩ − ⟨Y, Y ′⟩ + ⟨W, W ′⟩ − ⟨η, η′⟩
+��
+L2
+≥
+��⟨X, X′⟩ − ⟨Y, Y ′⟩
+��
+L2 −
+��⟨W, W ′⟩ − ⟨η, η′⟩
+��
+L2 .
+(3.3)
+Since ⟨η, η′⟩ = E(⟨W, W ′⟩| F × F), the law of total variance (1.1) and definition of W yields
+��⟨W, W ′⟩ − ⟨η, η′⟩
+��
+L2 ≤
+��⟨W, W ′⟩
+��
+L2 ≤
+��⟨θ, θ′⟩
+��
+L2 =
+1
+√
+N
+,
+(3.4)
+where the last identity follows since ⟨θ, θ′⟩ is Bernoulli with parameter 1/N. Taking N to be
+large enough, we conclude from (3.3) that
+��⟨X, X′⟩ − ⟨Y, Y ′⟩
+��
+L2 ≤ π
+√r.
+
+COVARIANCE LOSS, SZEMEREDI REGULARITY,
+AND DIFFERENTIAL PRIVACY
+5
+Theorem 3.1 is proved.
+□
+Remark 3.3. In the conclusion of Theorem 3.1, we can replace the constant π2 by π2/4 + ǫ
+for any fixed ǫ > 0.
+4. Subgaussian error
+We can extend the bound in Theorem 3.1 to the subgaussian norm, which is the Orlicz
+norm with respect to the Young function ψ2(x) = ex2 − 1. Thus, a random variable X is
+subgaussian if
+∥X∥ψ2 = inf{t > 0 : E ψ2(X/t) ≤ 1} < ∞,
+(4.1)
+and this quantity is called the subgaussian norm of X, see [11, Sections 2.5, 2.7.1].
+Theorem 4.1. Let X be a random vector taking values Rd such that ∥X∥2 ≤ 1 a.s. Then,
+for any r ∈ N, there exists a partition of the sample space into at most 2r parts such that for
+the σ-algebra F generated by this partition, the conditional expectation Y = E[X|F] satisfies
+��⟨X, X′⟩ − ⟨Y, Y ′⟩
+��
+ψ2 ≤ C
+√r.
+(4.2)
+We can prove this result by modifying the proof of Theorem 3.1. Let us explain how to do
+this.
+In Step 1, instead of using additivity of variance, we can use Hoeffding’s inequality to
+control the sum of independent Bernoulli random variables ξk. Denoting ζk = ξk − E ξk, we
+have
+���r
+k=1 ζk
+��
+ψ2 ≲ √r, see [11, Proposition 2.6.1]. Using this in (3.2), we get
+��⟨x, x′⟩ − Fg(x, x′)
+��
+ψ2 ≲ 1/√r.
+By definition of the subgaussian norm, this means that
+Eg ψ2
+�
+(⟨x, x′⟩ − Fg(x, x′))c√r
+�
+≤ 1,
+where c > 0 is some absolute constant. Substituting here x = X and x′ = X′ and applying
+the Fubini inequality as in Step 1, we obtain the following version of (3.1):
+��⟨X, X′⟩ − Z
+��
+ψ2 ≲ 1
+√r.
+In Step 2, although the conditional expectation is not a metric projection in the subgaussian
+norm, it is an approximate metric projection:
+Lemma 4.2. Let X be a random variable and F be a σ-algebra. Then, for any random
+variable Z that is F-measurable, and any p ≥ 1, we have
+��X − E(X|F)
+��
+ψ2 ≤ 2∥X − Z∥ψ2 .
+Proof. Subtracting and adding Z and using triangle inequality, we get
+��X − E(X|F)
+��
+ψ2 ≤∥X − Z∥ψ2 +
+��E(X|F) − Z
+��
+ψ2 .
+Since Z is F-measurable, we have
+��E(X|F) − Z
+��
+ψ2 =
+��E(X − Z|F)
+��
+ψ2 ≤∥X − Z∥ψ2 ,
+where the last step follows from definition of subgaussian norm (4.1) and conditional Jensen’s
+inequality. Combine the two bounds to complete the proof.
+□
+
+6
+MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN
+Using this lemma for ⟨X, X′⟩ instead of X, we obtain in Step 2 that
+��⟨X, X′⟩ − ⟨Y, Y ′⟩
+��
+ψ2 ≤ 2
+��⟨X, X′⟩ − Z
+��
+ψ2 ≲ 1
+√r.
+In Step 3, we argue in a similar way about the subgaussian norm. The bound (3.4) becomes
+��⟨W, W ′⟩ − ⟨η, η′⟩
+��
+ψ2 ≤ 2
+��⟨W, W ′⟩
+��
+ψ2 ≤
+��⟨θ, θ′⟩
+��
+ψ2 ≲
+1
+√log N ,
+Here in the first step we use Lemma 4.2 for X = ⟨W, W ′⟩ and Z = 0, and the last step
+is a straightforward bound on the subgaussian norm of a Bernoulli random variable with
+parameter 1/N. This completes the proof.
+5. Implications
+5.1. Covariance loss. Combining Proposition 2.1, Proposition 2.3 and Theorem 3.1, we
+obtain:
+Corollary 5.1 (Covariance loss). Let X be a random vector taking values Rd such that
+∥X∥2 ≤ 1 a.s. Then, for any r ∈ N, there exists a partition of the sample space into at most
+2r parts such that for the σ-algebra F generated by this partition, the conditional expectation
+Y = E[X|F] satisfies
+���E XXT − E Y Y T���
+F ≤ π
+√r.
+In particular, for any k with log2 k ∈ N, there exists a partition of the sample space into
+at most k parts such that for the σ-algebra F generated by this partition, we have
+���E XXT − E Y Y T���
+F ≤
+π
+�
+log2 k
+.
+(5.1)
+Remark 5.2 (Optimality). Proposition 3.14 in [2] implies that there exists a random vector X
+taking values in the unit ball of Rd such
+���E XXT − E Y Y T���
+F ≥
+√
+log2(e)
+80√
+log2 k for any σ-algebra
+F generated by a partition into at most k parts. Thus the bound in (5.1) is sharp up to an
+absolute constant.
+5.2. Weak Szemeredi regularity.
+Theorem 5.3 (Weak Szemeredi regularity). Let A be an n × n positive semidefinite matrix
+such that Aii ≤ 1 for all i. Then, for any r ∈ N, there exists a partition [n] = I1 ∪ . . . ∪ Ik
+with k ≤ 2r, and a matrix B that is constant on each block Ii × Ij and such that
+1
+n∥A − B∥F ≤ π
+√r.
+(5.2)
+Moreover, B can be computed by averaging the entries of A in each block.
+Proof. By assumption, A can be represented as the Gram matrix of some vectors x1, . . . , xn
+satisfying ∥xi∥2 ≤ 1 for all i. Thus A = [⟨xi, xj⟩]n
+i,j=1. Apply Theorem 3.1 for the random
+vector X that is uniformly distributed on {x1, . . . , xn} to complete the proof.
+□
+Remark 5.4. Rewriting the approximation error in Theorem 5.3 as
+1
+n2
+n
+�
+i,j=1
+(Aij − Bij)2 ≤ π2
+r ,
+we can interpret it as a bound on the mean squared error of the entries.
+
+COVARIANCE LOSS, SZEMEREDI REGULARITY,
+AND DIFFERENTIAL PRIVACY
+7
+Remark 5.5. Applying Theorem 4.1 instead of Theorem 3.1, we can replace the Frobenius
+norm in (5.2) by stronger matrix norms, such as the ℓp norm of the entries.
+Theorem 5.6 (Weak Szemeredi regularity, analytic form). Let (Ω, µ) be a probability measure
+space. Let K : Ω × Ω → R be a measurable positive definite kernel such that K(t, t) ≤ 1 for
+all t ∈ Ω. Then for any r ∈ N, there exists a partition Ω = I1 ∪ . . . ∪ Ik with k ≤ 2r, and a
+function L : Ω × Ω → R that is constant on each block Ii × Ij such that
+∥K − L∥L2(µ×µ) ≤ π
+√r.
+(5.3)
+Moreover, L can be computed by averaging K on each block.
+Proof. When Ω is finite, using Theorem 3.1 (see also Remark 3.3), one can easily establish
+∥K − L∥L2(µ×µ) ≤
+π
+2√r + ǫ for any fixed ǫ > 0.
+The general case follows by a standard
+approximation argument, which we leave to the reader.
+□
+5.3. Comparison to the existing work on weak Szemeredi regularity. Our result is
+more restrictive but stronger that the classical weak regularity lemma by Frieze and Kan-
+nan [5]. There the approximation error R = K − L is measured in the cut norm
+∥R∥□ = sup
+S,T⊂Ω
+�����
+�
+S×T
+R(x, y) dµ(x)dµ(y)
+����� .
+The cut norm is equivalent to the operator norm L1 → L∞ (see e.g. [8]) and clearly satisfies
+∥R∥□ ≤ ∥R∥L2(µ×µ). So the Hilbert-Schmidt norm, which is the focus of the current paper,
+is stronger than the cut norm. As a result, our bound (5.3) automatically extends to the
+cut norm. A similar bound for the cut norm was established in the original work of Frieze
+and Kannan [5] in wider generality: it holds only for kernels but for an arbitrary measurable
+function K(x, y) that is pointwise bounded by 1, see [8, Lemma 3.1]. In contrast to this,
+positive semidefiniteness is required for any nontrivial bound on the error in the stronger
+Hilbert-Schmidt norm, such as the one in Theorem 5.6. We will see this in the next section.
+A matrix decomposition in the spirit of Theorem 5.3, i.e.
+with error bounded in the
+Frobenius norm, appears in [3, Theorem 7] by Deshpande, Kannan, and Srivastava. Their
+theorem, which does not require the matrix to be positive semidefinite, is only nontrivial for
+low-rank matrices A, namely for matrices whose rank is at most logarithmic2 in the dimension
+n. Our results do not have a rank restriction.
+6. Optimality
+The decay rate 1/r in the conclusion of Theorem 3.1 is optimal, because the decay rate
+1/√r in the conclusion of Corollary 5.1 is optimal.
+All previous results in the literature on weak Szemeredi regularity do not need the assump-
+tion that the matrix A is positive semidefinite, see e.g. [5, 8, 1]. Can it be removed from our
+Theorem 5.3? The following result says that it cannot be removed.
+2Theorem 7 in [3] approximates A by a sum of roughly t = rank(A) log4 n cut-matrices; they can be further
+broken down into 2t smaller cut-matrices with disjoint support. This ultimate decomposition is nontrivial
+only if 2t < n2.
+
+8
+MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN
+Proposition 6.1 (Positive semidefiniteness is essential). Let A be an n×n Hadamard matrix,
+and let [n] = I1 ∪ . . . ∪ Ik be any partition. Then, for any matrix B that is constant on each
+block Ii × Ij, we have
+1
+n∥A − B∥F ≥ 1
+2
+�
+1 − k
+n.
+Thus, the error does not vanish unless the number of parts k is extremely large, namely
+k = n − o(n).
+The proof will use the following elementary bound:
+Lemma 6.2. Let x1, . . . , xm are orthonormal vectors in Rn. Then for each k = 1, . . . , m we
+have
+���xk − 1
+m
+m
+�
+t=1
+xt
+���
+2
+≥ 1 − 1
+m.
+Proof. The norm of the vector xk − 1
+m
+�m
+t=1 xt is bounded below by the inner product of that
+vector and xk, which equals 1 − 1/m.
+□
+Proof of Proposition 6.1. The minimum of ∥A − B∥F over all matrices B in the proposition
+can only get smaller if we minimize over all matrices that are constant on smaller blocks,
+namely on the sets {ℓ} × Ij, where ℓ = 1, . . . , n and j = 1, . . . , k.
+The latter minimum
+is attained3 for the matrix B that is obtained by averaging the entries of A in each block
+{ℓ} × Ij. Equivalently, B is obtained by averaging the columns of A in each set Ij. Formally,
+the columns of B are
+Bℓ =
+1
+��Ij
+��
+�
+t∈Ij
+At
+for ℓ ∈ Ij,
+where Bℓ is the ℓ column of B and At is the tth column of A. Note that there at least n − k
+indices belong to the non-singleton blocks Ij, i.e.
+T :=
+�
+j:|Ij|>1
+Ij
+satisfies
+|T| ≥ n − k.
+(Indeed, since T c is the union of the singleton blocks, its cardinality equals the number of
+such blocks Ij, which is bounded by the total number of blocks k.)
+Pick any index ℓ ∈ T, so ℓ ∈ Ij where
+��Ij
+�� ≥ 2. Since A is an Hadamard matrix, its
+columns At are orthogonal, and their Euclidean norms equal √n. Thus, applying Lemma 6.2
+and rescaling, we have
+∥Aℓ − Bℓ∥2 ≥
+�
+1 − 1
+��Ij
+��
+�√n ≥
+√n
+2 .
+It follows that
+∥A − B∥2
+F =
+n
+�
+ℓ=1
+∥Aℓ − Bℓ∥2
+2 ≥
+�
+ℓ∈T
+∥Aℓ − Bℓ∥2
+2 ≥|T|
+�√n
+2
+�2
+≥ n(n − k)
+4
+.
+□
+3To see this, argue like in Step 2 of the proof of Theorem 3.1.
+
+COVARIANCE LOSS, SZEMEREDI REGULARITY,
+AND DIFFERENTIAL PRIVACY
+9
+7. Application to differential privacy and synthetic data
+In this section we briefly describe how the findings in this paper can be applied to improve
+the results in [2] on generating private synthetic data. The partitioning result behind the
+covariance loss bound in Corollary 5.1 and the Szemeredi regularity bound in Theorem 5.3
+provide a natural technique towards data privacy related to k-anonymity and differential
+privacy. While the Szemeredi regularity lemma has been proposed in [4, 9] as a mechanism
+for data anonymization, the results in those papers are merely empirical and come without
+any theoretical guarantees.
+The findings in this paper can be readily used to improve upon the data privacy and utility
+guarantees for k-anonymity and differential privacy in [2]. For instance, using Corollary 5.1
+in lieu of Theorem 1.2 in [2], but otherwise following the same procedure, we arrive at the
+following theorem which gives an improved accuracy guarantee compared to Theorem 5.14
+in [2]. We leave the details to the reader and refer to [2] for the precise problem statement
+and a detailed description of the underlying concepts.
+Theorem 7.1. Let K be a convex set in Rp that lies in the unit Euclidean ball Bp
+2. Let α > 0.
+If n, p ∈ N, ε > 0 satisfy n > C(α)p
+ε, then there is an ε-differentially private algorithm with
+input x1, . . . , xn ∈ K and output u1, . . . , um(α) ∈ K such that
+E
+����
+1
+n
+n
+�
+i=1
+x⊗d
+i
+−
+1
+m(α)
+m
+�
+i=1
+u⊗d
+i
+����
+2
+2
+≲ 16dα,
+for all d ∈ N. The run time of the algorithm is C(α)(np + q), where q is the complexity to
+find a best approximation element of K to a given vector in Rp.
+Acknowledgement
+M.B. acknowledges support from NSF DMS-2140592. T.S. acknowledges support from NIH
+R01HL16351, NSF DMS-2027248, and NSF DMS-2208356. R.V. acknowledges support from
+NSF DMS-1954233, NSF DMS-2027299, U.S. Army 76649-CS, and NSF+Simons Research
+Collaborations on the Mathematical and Scientific Foundations of Deep Learning.
+References
+[1] Greg Bodwin and Santosh Vempala. A unified view of graph regularity via matrix decompositions. Ran-
+dom Structures & Algorithms, 61(1):62–83, 2022.
+[2] March Boedihardjo, Thomas Strohmer, and Roman Vershyin. Covariance’s Loss is Privacy’s Gain: Com-
+putationally Efficient, Private and Accurate Synthetic Data. Foundations of Computational Mathematics,
+to appear.
+[3] Amit Deshpande, Ravindran Kannan, and Nikhil Srivastava. Zero-one rounding of singular vectors. In
+International Colloquium on Automata, Languages, and Programming, pages 278–289. Springer, 2012.
+[4] Daniele Foffano, Luca Rossi, and Andrea Torsello. You can’t see me: Anonymizing graphs using the
+Szemeredi regularity lemma. Frontiers in big Data, 2:7, 2019.
+[5] Alan Frieze and Ravi Kannan. Quick approximation to matrices and applications. Combinatorica,
+19(2):175–220, 1999.
+[6] Vishesh Jain, Ashwin Sah, and Mehtaab Sawhney. Optimal minimization of the covariance loss. arXiv
+preprint arXiv:2205.01773, 2022.
+[7] J´anos Koml´os, Ali Shokoufandeh, Mikl´os Simonovits, and Endre Szemer´edi. The regularity lemma and
+its applications in graph theory. Summer school on theoretical aspects of computer science, pages 84–112,
+2000.
+[8] L´aszl´o Lov´asz and Bal´azs Szegedy. Szemer´edi’s lemma for the analyst. GAFA Geometric And Functional
+Analysis, 17(1):252–270, 2007.
+
+10
+MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN
+[9] Giorgia Minello, Luca Rossi, and Andrea Torsello. k-anonymity on graphs using the Szemer´edi regularity
+lemma. IEEE Transactions on Network Science and Engineering, 8(2):1283–1292, 2020.
+[10] Endre Szemer´edi. Regular partitions of graphs. In J.-C. Bermond, J.-C. Fournier, M. Las Vergnas, and
+D. Sotteau, editors, Proc. Colloque Inter. CNRS, page 399–401, 1978.
+[11] Roman Vershynin. High-dimensional probability. An introduction with applications in data science. Cam-
+bridge University Press, 2018.
+Department of Mathematics, ETH Z¨urich
+Email address: march.boedihardjo@ifor.math.ethz.ch
+Department of Mathematics, University of California, Davis and Center of Data Science
+and Artificial Intelligence Research, UC Davis
+Email address: strohmer@math.ucdavis.edu
+Department of Mathematics, University of California, Irvine
+Email address: rvershyn@uci.edu
+
diff --git a/SdE0T4oBgHgl3EQf1wL7/content/tmp_files/load_file.txt b/SdE0T4oBgHgl3EQf1wL7/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c9d82a25c54ddf99f6b2ea5683aedbdd87a475a9
--- /dev/null
+++ b/SdE0T4oBgHgl3EQf1wL7/content/tmp_files/load_file.txt
@@ -0,0 +1,439 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf,len=438
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='02705v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='PR]  6 Jan 2023  COVARIANCE LOSS, SZEMEREDI REGULARITY,  AND DIFFERENTIAL PRIVACY  MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' We show how randomized rounding based on Grothendieck’s identity can be  used to prove a nearly tight bound on the covariance loss–the amount of covariance that  is lost by taking conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' This result yields a new type of weak Szemeredi  regularity lemma for positive semidefinite matrices and kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Moreover, it can be used to  construct differentially private synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Introduction  Consider a random vector X taking values in Rd and a σ-algebra F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let Y denote the  conditional expectation: Y = E[X|F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' In dimension d = 1, where X and Y are random  variables, the law of total variance states that  Var(X) − Var(Y ) = E X2 − E Y 2 = E(X − Y )2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1)  Thus, taking conditional expectation underestimates the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' This observation extends  to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Namely, let ΣX = E(X −E X)(X −E X)T denote the covariance matrix  of X, and similarly for ΣY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then  ΣX − ΣY = E XXT − E Y Y T = E(X − Y )(X − Y )T ⪰ 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2)  where ⪰ denotes the Loewner order, in which A ⪰ B if A − B is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Just  like in the one-dimensional case, we see that taking conditional expectation underestimates  the covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  In [2], we asked the basic question: “How big is the covariance loss ΣX −ΣY ?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The answer  will obviously depend on the choice of the sigma-algebra F, prompting the next question:  “What sigma-algebra F of given complexity minimizes the covariance loss?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  It was shown in [2] that there exists a partition of the sample space into at most k sets  such that for the sigma-algebra F generated by this partition, the covariance loss (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2) is  upper bounded by C  �  log log k/ log k, where C is an absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' In [6], Jain, Sah,  and Sawhney were able to improve this bound to C/√log k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  It follows from Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='14 in [2] that this bound is optimal up to the value of the constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The proofs of the  bounds in both [2] and [6] are somewhat technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' In this paper we give a new approach  to this problem, which is conceptually related to Szemeredi regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' This approach yields  a tighter bound in terms of the constant on the covariance loss (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1), as well as a  much shorter and more elementary proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  The celebrated Szemeredi regularity lemma states that that for every large enough graph,  the set of nodes can be divided into subsets of about the same size so that the edges between  different subsets behave almost randomly [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Expressed in the language of linear algebra,  it says that the adjacency matrix of a graph can be approximated by a sum of cut matrices1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  The Szemeredi regularity lemma, both in the combinatorial and linear-algebraic forms, has  1A cut matrix is a rank-one matrix which is constant on a block and zero elsewhere [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  1  2  MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN  many deep algorithmic and combinatorial applications, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' [5, 8, 7, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' We prove a new  version of (weak) Szemeredi regularity lemma, which states that any positive semidefinite  kernel K(x, y) whose diagonal is uniformly bounded can be decomposed into a sum of k  step-functions plus an error term whose L2-norm is O(1/√log k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  The outline of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  We will first bound the covariance loss by the  “covariance increment” (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1), find a nice representation of the covariance incre-  ment (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3), and combine it with randomized rounding based on Grothendieck’s  identity to bound the covariance increment (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Thus we can not only bound  the covariance loss (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1) but also derive a new type of weak Szemeredi regularity  for matrices (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3) and kernels (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' We conclude by illustrating how our  results can be used in connection with differential privacy and synthetic data to improve the  accuracy bounds obtained in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The subgaussian norm of a random variable X is denoted by ∥X∥ψ2 (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' [11]), E(X) is the expectation of X and ∥X∥Lp = (E|X|p)1/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' For d × d matrices A, B,  define the inner product ⟨A, B⟩ = Tr(ABT ) and the Frobenius norm ∥A∥F =  �  ⟨A, A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The covariance increment  We start out by bounding the covariance loss E XXT−E Y Y T by the covariance increment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  In the sequel, X is a random vector taking values in Rd, F is a σ-algebra and Y = E[X|F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Moreover, (X′, F′, Y ′) is an independent copy of (X, F, Y ) so that the sample space is a  product space for which X, F, Y are based on the first component of the product space and  X′, F′, Y ′ are based on the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  With a slight abuse of notation, the product σ-algebra F × F denotes the σ-algebra gen-  erated by F and F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Moreover, the inner product ⟨Y, Y ′⟩ = E(⟨X, X′⟩| F × F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 (Covariance loss and covariance increment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  ���E XXT − E Y Y T���  2  F ≤  ���E XXT���  2  F −  ���E Y Y T���  2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1)  The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 will be based on the following lemma:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let (U, V ) be a pair of random vectors taking values in Rd, and let (U ′, V ′) be  an independent copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then  ⟨E UU T, E V V T⟩ = E⟨U, V ′⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  In particular, setting V = U, we have  ���E UU T���  2  F = E⟨U, U ′⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The first identity readily follows if we first use the identical distribution, and then  independence:  ⟨E UU T, E V V T⟩ = ⟨E UU T, E V ′(V ′)T⟩ = E⟨UU T, V ′(V ′)T⟩ = E⟨U, V ′⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  The lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  □  COVARIANCE LOSS, SZEMEREDI REGULARITY,  AND DIFFERENTIAL PRIVACY  3  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Expanding the square of the Frobenius norm, we can express the  left hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1) as  ��E XXT��2  F − 2⟨E XXT, E Y Y T⟩ + ∥E Y Y T∥2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' After simplification,  we see that inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1) is equivalent to  ���E Y Y T���  2  F ≤ ⟨E XXT, E Y Y T⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2 we can rewrite this as  E⟨Y, Y ′⟩2 ≤ E⟨X, Y ′⟩2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2)  To check (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2), recall that Y = E[X|F] and apply the conditional Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  □  The covariance increment has a nice representation, which will come handy in our further  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  ���E XXT���  2  F −  ���E Y Y T���  2  F = E  �  ⟨X, X′⟩ − ⟨Y, Y ′⟩  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2, the left hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3) equals E⟨X, X′⟩2 − E⟨Y, Y ′⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Note that  ⟨Y, Y ′⟩ = E(⟨X, X′⟩| F × F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' To finish the proof, apply the law of total variance (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1) for  ⟨X, X′⟩ instead of X and ⟨Y, Y ′⟩ instead of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Bounding the covariance increment  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let X be a random vector taking values Rd such that ∥X∥2 ≤ 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then,  for any r ∈ N, there exists a partition of the sample space into at most 2r parts such that for  the σ-algebra F generated by this partition, the conditional expectation Y = E[X|F] satisfies  E  �  ⟨X, X′⟩ − ⟨Y, Y ′⟩  �2  ≤ π2  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 will utilize Grothendieck’s identity (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' [11, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='6]):  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2 (Grothendieck’s identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let x, x′ be a pair of unit vectors in Rd, and let  g ∼ N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then  ⟨x, x′⟩ = sin  �π  2 E sign⟨x, g⟩ sign⟨x′, g⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let us first make a stronger assumption, namely that∥X∥2 = 1  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=', and prove a weaker conclusion, namely that there exists a random variable Z measurable  with respect to the product σ-algebra F × F and such that  ��⟨X, X′⟩ − Z  ��  L2 ≤  π  2√r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1)  To this end, consider independent random vectors g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , gk ∈ N(0, Id), and for x, x′ ∈ Rd,  define the random variable  Fg(x, x′) = sin  \uf8ee  \uf8f0π  2 · 1  r  r  �  k=1  sign⟨x, gk⟩ sign⟨x′, gk⟩  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  4  MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN  Denoting ξk = sign⟨x, gk⟩ sign⟨x′, gk⟩, noting that the function sin(·) is 1-Lipschitz and ap-  plying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2, we obtain  ��⟨x, x′⟩ − Fg(x, x′)  �� ≤ π  2 · 1  r  ������  r  �  k=1  (ξk − E ξk)  ������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2)  By independence, this yields  Eg  �  ⟨x, x′⟩ − Fg(x, x′)  �2 ≤  � π  2r  �2  r Var(ξ1) ≤ π2  4r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Substitute x = X, x′ = X′ and take expectation with respect to X and X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' By Fubini  theorem, there exists a realization of the random vectors g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , gk such that  E  �  ⟨X, X′⟩ − Fg(X, X′)  �2 ≤ π2  4r ,  Fix such a realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let F = σ(V (X)) be the σ-algebra generated by the random vector  V (X) = (sign⟨x, gk⟩)r  k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Since V (X) takes at most 2r values, F satisfies the requirement of  the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Moreover, the random vector  Z = Fg(X, X′) = sin  � π  2r⟨V (X), V (X′)⟩  �  is measurable with respect to the product sigma-algebra F × F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Thus, we proved (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Step 2: replacing Z with ⟨Y, Y ′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The conditional expectation E  �  | F × F  �  is an orthogonal  projection in L2 onto the subspace of random variables that are F × F-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Since  the random variable Z constructed in the previous step is F × F-measurable, and ⟨Y, Y ′⟩ =  E  �  ⟨X, X′⟩| F × F  �  , it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1) that  ��⟨X, X′⟩ − ⟨Y, Y ′⟩  ��  L2 ≤  ��⟨X, X′⟩ − Z  ��  L2 ≤  π  2√r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Step 3: removing the unit norm requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' We proved the theorem under the additional  assumption that X is a unit random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Now let X be an arbitrary random vector  satisfying ∥X∥2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Let N ∈ N, and consider the random vector W =  �  1 −∥X∥2  2 · θ,  where θ is a random vector that is uniformly distributed in the unit basis {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , eN} of RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Then the direct sum X ⊕ W = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , Xd, W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , WN) is a unit random vector in Rd+N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Applying the previous step for X ⊕ W, we find that  π  2√r ≥  ��⟨X ⊕ W, X′ ⊕ W ′⟩ − ⟨Y ⊕ η, Y ′ ⊕ η′⟩  ��  L2  (where η = E(W|F))  =  ��⟨X, X′⟩ − ⟨Y, Y ′⟩ + ⟨W, W ′⟩ − ⟨η, η′⟩  ��  L2  ≥  ��⟨X, X′⟩ − ⟨Y, Y ′⟩  ��  L2 −  ��⟨W, W ′⟩ − ⟨η, η′⟩  ��  L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3)  Since ⟨η, η′⟩ = E(⟨W, W ′⟩| F × F), the law of total variance (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1) and definition of W yields  ��⟨W, W ′⟩ − ⟨η, η′⟩  ��  L2 ≤  ��⟨W, W ′⟩  ��  L2 ≤  ��⟨θ, θ′⟩  ��  L2 =  1  √  N  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='4)  where the last identity follows since ⟨θ, θ′⟩ is Bernoulli with parameter 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Taking N to be  large enough, we conclude from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3) that  ��⟨X, X′⟩ − ⟨Y, Y ′⟩  ��  L2 ≤ π  √r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  COVARIANCE LOSS, SZEMEREDI REGULARITY,  AND DIFFERENTIAL PRIVACY  5  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' In the conclusion of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1, we can replace the constant π2 by π2/4 + ǫ  for any fixed ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Subgaussian error  We can extend the bound in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 to the subgaussian norm, which is the Orlicz  norm with respect to the Young function ψ2(x) = ex2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Thus, a random variable X is  subgaussian if  ∥X∥ψ2 = inf{t > 0 : E ψ2(X/t) ≤ 1} < ∞,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1)  and this quantity is called the subgaussian norm of X, see [11, Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let X be a random vector taking values Rd such that ∥X∥2 ≤ 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then,  for any r ∈ N, there exists a partition of the sample space into at most 2r parts such that for  the σ-algebra F generated by this partition, the conditional expectation Y = E[X|F] satisfies  ��⟨X, X′⟩ − ⟨Y, Y ′⟩  ��  ψ2 ≤ C  √r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2)  We can prove this result by modifying the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let us explain how to do  this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  In Step 1, instead of using additivity of variance, we can use Hoeffding’s inequality to  control the sum of independent Bernoulli random variables ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Denoting ζk = ξk − E ξk, we  have  ���r  k=1 ζk  ��  ψ2 ≲ √r, see [11, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Using this in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2), we get  ��⟨x, x′⟩ − Fg(x, x′)  ��  ψ2 ≲ 1/√r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  By definition of the subgaussian norm, this means that  Eg ψ2  �  (⟨x, x′⟩ − Fg(x, x′))c√r  �  ≤ 1,  where c > 0 is some absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Substituting here x = X and x′ = X′ and applying  the Fubini inequality as in Step 1, we obtain the following version of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1):  ��⟨X, X′⟩ − Z  ��  ψ2 ≲ 1  √r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  In Step 2, although the conditional expectation is not a metric projection in the subgaussian  norm, it is an approximate metric projection:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let X be a random variable and F be a σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then, for any random  variable Z that is F-measurable, and any p ≥ 1, we have  ��X − E(X|F)  ��  ψ2 ≤ 2∥X − Z∥ψ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Subtracting and adding Z and using triangle inequality, we get  ��X − E(X|F)  ��  ψ2 ≤∥X − Z∥ψ2 +  ��E(X|F) − Z  ��  ψ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Since Z is F-measurable, we have  ��E(X|F) − Z  ��  ψ2 =  ��E(X − Z|F)  ��  ψ2 ≤∥X − Z∥ψ2 ,  where the last step follows from definition of subgaussian norm (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1) and conditional Jensen’s  inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Combine the two bounds to complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  □  6  MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN  Using this lemma for ⟨X, X′⟩ instead of X, we obtain in Step 2 that  ��⟨X, X′⟩ − ⟨Y, Y ′⟩  ��  ψ2 ≤ 2  ��⟨X, X′⟩ − Z  ��  ψ2 ≲ 1  √r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  In Step 3, we argue in a similar way about the subgaussian norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='4) becomes  ��⟨W, W ′⟩ − ⟨η, η′⟩  ��  ψ2 ≤ 2  ��⟨W, W ′⟩  ��  ψ2 ≤  ��⟨θ, θ′⟩  ��  ψ2 ≲  1  √log N ,  Here in the first step we use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2 for X = ⟨W, W ′⟩ and Z = 0, and the last step  is a straightforward bound on the subgaussian norm of a Bernoulli random variable with  parameter 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Implications  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Covariance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Combining Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1, we  obtain:  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 (Covariance loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let X be a random vector taking values Rd such that  ∥X∥2 ≤ 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then, for any r ∈ N, there exists a partition of the sample space into at most  2r parts such that for the σ-algebra F generated by this partition, the conditional expectation  Y = E[X|F] satisfies  ���E XXT − E Y Y T���  F ≤ π  √r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  In particular, for any k with log2 k ∈ N, there exists a partition of the sample space into  at most k parts such that for the σ-algebra F generated by this partition, we have  ���E XXT − E Y Y T���  F ≤  π  �  log2 k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1)  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2 (Optimality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='14 in [2] implies that there exists a random vector X  taking values in the unit ball of Rd such  ���E XXT − E Y Y T���  F ≥  √  log2(e)  80√  log2 k for any σ-algebra  F generated by a partition into at most k parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Thus the bound in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1) is sharp up to an  absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Weak Szemeredi regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3 (Weak Szemeredi regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let A be an n × n positive semidefinite matrix  such that Aii ≤ 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then, for any r ∈ N, there exists a partition [n] = I1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' ∪ Ik  with k ≤ 2r, and a matrix B that is constant on each block Ii × Ij and such that  1  n∥A − B∥F ≤ π  √r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2)  Moreover, B can be computed by averaging the entries of A in each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' By assumption, A can be represented as the Gram matrix of some vectors x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , xn  satisfying ∥xi∥2 ≤ 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Thus A = [⟨xi, xj⟩]n  i,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 for the random  vector X that is uniformly distributed on {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , xn} to complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  □  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Rewriting the approximation error in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3 as  1  n2  n  �  i,j=1  (Aij − Bij)2 ≤ π2  r ,  we can interpret it as a bound on the mean squared error of the entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  COVARIANCE LOSS, SZEMEREDI REGULARITY,  AND DIFFERENTIAL PRIVACY  7  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 instead of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1, we can replace the Frobenius  norm in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2) by stronger matrix norms, such as the ℓp norm of the entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='6 (Weak Szemeredi regularity, analytic form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let (Ω, µ) be a probability measure  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let K : Ω × Ω → R be a measurable positive definite kernel such that K(t, t) ≤ 1 for  all t ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then for any r ∈ N, there exists a partition Ω = I1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' ∪ Ik with k ≤ 2r, and a  function L : Ω × Ω → R that is constant on each block Ii × Ij such that  ∥K − L∥L2(µ×µ) ≤ π  √r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3)  Moreover, L can be computed by averaging K on each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' When Ω is finite, using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 (see also Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3), one can easily establish  ∥K − L∥L2(µ×µ) ≤  π  2√r + ǫ for any fixed ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  The general case follows by a standard  approximation argument, which we leave to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Comparison to the existing work on weak Szemeredi regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Our result is  more restrictive but stronger that the classical weak regularity lemma by Frieze and Kan-  nan [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' There the approximation error R = K − L is measured in the cut norm  ∥R∥□ = sup  S,T⊂Ω  �����  �  S×T  R(x, y) dµ(x)dµ(y)  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  The cut norm is equivalent to the operator norm L1 → L∞ (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' [8]) and clearly satisfies  ∥R∥□ ≤ ∥R∥L2(µ×µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' So the Hilbert-Schmidt norm, which is the focus of the current paper,  is stronger than the cut norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' As a result, our bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3) automatically extends to the  cut norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' A similar bound for the cut norm was established in the original work of Frieze  and Kannan [5] in wider generality: it holds only for kernels but for an arbitrary measurable  function K(x, y) that is pointwise bounded by 1, see [8, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' In contrast to this,  positive semidefiniteness is required for any nontrivial bound on the error in the stronger  Hilbert-Schmidt norm, such as the one in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' We will see this in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  A matrix decomposition in the spirit of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  with error bounded in the  Frobenius norm, appears in [3, Theorem 7] by Deshpande, Kannan, and Srivastava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Their  theorem, which does not require the matrix to be positive semidefinite, is only nontrivial for  low-rank matrices A, namely for matrices whose rank is at most logarithmic2 in the dimension  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Our results do not have a rank restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Optimality  The decay rate 1/r in the conclusion of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 is optimal, because the decay rate  1/√r in the conclusion of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  All previous results in the literature on weak Szemeredi regularity do not need the assump-  tion that the matrix A is positive semidefinite, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' [5, 8, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Can it be removed from our  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The following result says that it cannot be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  2Theorem 7 in [3] approximates A by a sum of roughly t = rank(A) log4 n cut-matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' they can be further  broken down into 2t smaller cut-matrices with disjoint support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' This ultimate decomposition is nontrivial  only if 2t < n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  8  MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 (Positive semidefiniteness is essential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let A be an n×n Hadamard matrix,  and let [n] = I1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' ∪ Ik be any partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then, for any matrix B that is constant on each  block Ii × Ij, we have  1  n∥A − B∥F ≥ 1  2  �  1 − k  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Thus, the error does not vanish unless the number of parts k is extremely large, namely  k = n − o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  The proof will use the following elementary bound:  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , xm are orthonormal vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Then for each k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , m we  have  ���xk − 1  m  m  �  t=1  xt  ���  2  ≥ 1 − 1  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The norm of the vector xk − 1  m  �m  t=1 xt is bounded below by the inner product of that  vector and xk, which equals 1 − 1/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  □  Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The minimum of ∥A − B∥F over all matrices B in the proposition  can only get smaller if we minimize over all matrices that are constant on smaller blocks,  namely on the sets {ℓ} × Ij, where ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , n and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  The latter minimum  is attained3 for the matrix B that is obtained by averaging the entries of A in each block  {ℓ} × Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Equivalently, B is obtained by averaging the columns of A in each set Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Formally,  the columns of B are  Bℓ =  1  ��Ij  ��  �  t∈Ij  At  for ℓ ∈ Ij,  where Bℓ is the ℓ column of B and At is the tth column of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Note that there at least n − k  indices belong to the non-singleton blocks Ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  T :=  �  j:|Ij|>1  Ij  satisfies  |T| ≥ n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  (Indeed, since T c is the union of the singleton blocks, its cardinality equals the number of  such blocks Ij, which is bounded by the total number of blocks k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=')  Pick any index ℓ ∈ T, so ℓ ∈ Ij where  ��Ij  �� ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Since A is an Hadamard matrix, its  columns At are orthogonal, and their Euclidean norms equal √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Thus, applying Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2  and rescaling, we have  ∥Aℓ − Bℓ∥2 ≥  �  1 − 1  ��Ij  ��  �√n ≥  √n  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  It follows that  ∥A − B∥2  F =  n  �  ℓ=1  ∥Aℓ − Bℓ∥2  2 ≥  �  ℓ∈T  ∥Aℓ − Bℓ∥2  2 ≥|T|  �√n  2  �2  ≥ n(n − k)  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  □  3To see this, argue like in Step 2 of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  COVARIANCE LOSS, SZEMEREDI REGULARITY,  AND DIFFERENTIAL PRIVACY  9  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Application to differential privacy and synthetic data  In this section we briefly describe how the findings in this paper can be applied to improve  the results in [2] on generating private synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The partitioning result behind the  covariance loss bound in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1 and the Szemeredi regularity bound in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='3  provide a natural technique towards data privacy related to k-anonymity and differential  privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' While the Szemeredi regularity lemma has been proposed in [4, 9] as a mechanism  for data anonymization, the results in those papers are merely empirical and come without  any theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  The findings in this paper can be readily used to improve upon the data privacy and utility  guarantees for k-anonymity and differential privacy in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' For instance, using Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1  in lieu of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='2 in [2], but otherwise following the same procedure, we arrive at the  following theorem which gives an improved accuracy guarantee compared to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='14  in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' We leave the details to the reader and refer to [2] for the precise problem statement  and a detailed description of the underlying concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let K be a convex set in Rp that lies in the unit Euclidean ball Bp  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Let α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  If n, p ∈ N, ε > 0 satisfy n > C(α)p  ε, then there is an ε-differentially private algorithm with  input x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , xn ∈ K and output u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' , um(α) ∈ K such that  E  ����  1  n  n  �  i=1  x⊗d  i  −  1  m(α)  m  �  i=1  u⊗d  i  ����  2  2  ≲ 16dα,  for all d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The run time of the algorithm is C(α)(np + q), where q is the complexity to  find a best approximation element of K to a given vector in Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Acknowledgement  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' acknowledges support from NSF DMS-2140592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' acknowledges support from NIH  R01HL16351, NSF DMS-2027248, and NSF DMS-2208356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' acknowledges support from  NSF DMS-1954233, NSF DMS-2027299, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Army 76649-CS, and NSF+Simons Research  Collaborations on the Mathematical and Scientific Foundations of Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  References  [1] Greg Bodwin and Santosh Vempala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' A unified view of graph regularity via matrix decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Ran-  dom Structures & Algorithms, 61(1):62–83, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  [2] March Boedihardjo, Thomas Strohmer, and Roman Vershyin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Covariance’s Loss is Privacy’s Gain: Com-  putationally Efficient, Private and Accurate Synthetic Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Foundations of Computational Mathematics,  to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  [3] Amit Deshpande, Ravindran Kannan, and Nikhil Srivastava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Zero-one rounding of singular vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' In  International Colloquium on Automata, Languages, and Programming, pages 278–289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Springer, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  [4] Daniele Foffano, Luca Rossi, and Andrea Torsello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' You can’t see me: Anonymizing graphs using the  Szemeredi regularity lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Frontiers in big Data, 2:7, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  [5] Alan Frieze and Ravi Kannan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Quick approximation to matrices and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Combinatorica,  19(2):175–220, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  [6] Vishesh Jain, Ashwin Sah, and Mehtaab Sawhney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Optimal minimization of the covariance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' arXiv  preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='01773, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  [7] J´anos Koml´os, Ali Shokoufandeh, Mikl´os Simonovits, and Endre Szemer´edi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' The regularity lemma and  its applications in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Summer school on theoretical aspects of computer science, pages 84–112,  2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  [8] L´aszl´o Lov´asz and Bal´azs Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Szemer´edi’s lemma for the analyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' GAFA Geometric And Functional  Analysis, 17(1):252–270, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  10  MARCH BOEDIHARDJO, THOMAS STROHMER, AND ROMAN VERSHYNIN  [9] Giorgia Minello, Luca Rossi, and Andrea Torsello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' k-anonymity on graphs using the Szemer´edi regularity  lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' IEEE Transactions on Network Science and Engineering, 8(2):1283–1292, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  [10] Endre Szemer´edi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Regular partitions of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' In J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Bermond, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Fournier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Las Vergnas, and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Sotteau, editors, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Colloque Inter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' CNRS, page 399–401, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  [11] Roman Vershynin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' High-dimensional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' An introduction with applications in data science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content=' Cam-  bridge University Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='  Department of Mathematics, ETH Z¨urich  Email address: march.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='boedihardjo@ifor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='ch  Department of Mathematics, University of California, Davis and Center of Data Science  and Artificial Intelligence Research, UC Davis  Email address: strohmer@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='edu  Department of Mathematics, University of California, Irvine  Email address: rvershyn@uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE0T4oBgHgl3EQf1wL7/content/2301.02705v1.pdf'}
diff --git a/SdE4T4oBgHgl3EQf_w7x/content/tmp_files/2301.05374v1.pdf.txt b/SdE4T4oBgHgl3EQf_w7x/content/tmp_files/2301.05374v1.pdf.txt
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+ 
+1 
+Effect of Annealing Temperature on Minimum Domain Size of 
+Ferroelectric Hafnia 
+ 
+Seokjung Yun#, Hoon Kim#, Myungsoo Seo, Min-Ho Kang, Taeho Kim, Seongwoo Cho, 
+Min Hyuk Park, Sanghun Jeon, Yang-Kyu Choi, and Seungbum Hong* 
+ 
+aDepartment of Materials Science and Engineering, KAIST, Daejeon, 34141, Republic of 
+Korea 
+bKAIST Institute for NanoCentury (KINC), KAIST, Daejeon, 34141, Republic of Korea 
+cDepartment of Materials Science and Engineering, KAIST, Daejeon, 34141, Republic of 
+Korea 
+dDepartment of Materials Science and Engineering, Seoul National University, Seoul, 
+Republic of Korea 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+2 
+Abstract 
+Here, we optimized the annealing temperature of HZO/TiN thin film heterostructure via 
+multiscale analysis of remnant polarization, crystallographic phase, minimum ferroelectric 
+domain size, and average grain size. We found that the remnant polarization was closely related 
+to the relative amount of the orthorhombic phase whereas the minimum domain size was to the 
+relative amount of the monoclinic phase. The minimum domain size was obtained at the 
+annealing temperature of 500 ℃ while the optimum remnant polarization and capacitance at 
+the annealing temperature of 600℃. We conclude that the minimum domain size is more 
+important than the sheer magnitude of remnant polarization considering the retention and 
+fatigue of switchable polarization in nanoscale ferroelectric devices. Our results are expected 
+to contribute to the development of ultra-low-power logic transistors and next-generation non-
+volatile memory devices. 
+ 
+Keywords: Ferroelectric hafnia, domain behavior, piezoresponse force microscopy, rapid 
+thermal annealing, multi-scale analysis, grain size, domain size 
+ 
+ 
+
+ 
+3 
+Introduction 
+Hf0.5Zr0.5O2 (HZO), which has ferroelectric properties through Zr doping with the conventional 
+gate dielectric material HfO2, has attracted much interest due to its potential application to next 
+generation non-volatile memory devices and logic circuits. In addition, HZO is more suitable 
+for the fabrication of FETs owing to its higher scalability, good thermal stability, and 
+compatibility with Si processes than the previously introduced Pb(Zr,Ti)O3 (PZT), SrBi2Ta2O9 
+(SBT) and BaTiO3 (BTO) ferroelectric materials. 
+Hafnia can have ferroelectric properties through various dopants such as Zr [10], Si [11], La 
+[12], Y [13], Al [14], Gd [15], and Sr [16], as well as by post heat treatment and strain 
+engineering between electrodes and oxide films. Through the process introduced above, hafnia 
+with ferroelectric properties shows the formation of the non-centrosymmetric orthorhombic 
+phase with a space group of Pca21, unlike conventional HfO2 [17]. Because of this non-
+centrosymmetric orthorhombic phase, Park et al. reported ferroelectricity and non-
+ferroelectricity depending on the Zr concentration of HZO [18]. In addition, it was reported 
+that the other dopants such as Y, Al, Gd, Sr, and La induce ferroelectric properties at a particular 
+dopant concentration and heat treatment temperature [18]. The ferroelectric properties are lost 
+due to the appearance of M-phase replacing O-phase above a specific temperature [19]. 
+Furthermore, the strain between the electrode and the corresponding ferroelectric thin film 
+affects the ferroelectric properties [20]. In the semiconductor process, as various temperatures, 
+dopant concentrations, and electrodes can be used, it is crucial to find the optimum process 
+condition to fabricate ferroelectric HZO thin with a high yield. 
+Many research groups have reported on the macroscopic characteristics of ferroelectrics, but 
+only a few have analyzed the origin of ferroelectricity from the perspective of the interplay 
+between domain configuration, defect distribution, polarization and screening charges 
+
+ 
+4 
+[17,18,21]. To understand the polarization switching mechanism and retention or fatigue 
+behavior of ferroelectric HfO2 materials at the nanoscale, piezoresponse force microscopy 
+(PFM) is one of the most powerful tools as it can visualize both out-of-plane and in-plane 
+polarization vectors [22,23]. However, the challenge lies in the fact that the piezoelectric 
+coefficient, which is the marker for polarization vector in PFM, is less than 10 pm/V. Stolichnov 
+et al. reported deff of HZO to be 5 pm/V, and Liam et al. reported effective piezoelectric 
+coupling coefficient (deff) of < 2 pm/V in Si-doped hafnia [24,25]. The low piezoelectric 
+coefficient results in low signal to noise ratio in PFM images, which makes the analysis difficult 
+due to the various artifacts contributing to the PFM signal [23].  
+Here, we designed the voltage profile for PFM imaging of HZO thin films, which were 
+fabricated with various rapid thermal annealing temperatures. We successfully analyzed the 
+domain configuration of HZO thin films with different voltage pulse amplitude and duration to 
+find the optimal process condition for minimizing the written domain size. 
+ 
+ 
+
+ 
+5 
+Experiment 
+Fabrication of Hf0.5Zr0.5O2 thin films 
+8.5 nm thick Hf0.5Zr0.5O2 (HZO) films were deposited on 200 nm thick TiN electrode as 
+the bottom electrode/1.5 nm SiO2 interfacial layer, using atomic layer deposition (ALD). 
+Tetrakis(ethylmethylamido)hafnium(IV) (TEMAHf), 
+Tetrakis(ethylmethylamido)zirconium(IV) (TEMAZr), and O3 were used as Hf precursor, Zr 
+precursor, and oxygen reactant, respectively. A 200 nm TiN electrode was deposited on top of 
+the device via physical vapor deposition (PVD). The top electrode of TiN was wet etched 
+using an NH4OH:H2O2:H2O solution at a concentration ratio of 1:2:5 and the pattern size was 
+30~500 ㎛. The ferroelectric HZO film was crystallized by post-metal rapid thermal 
+annealing (RTA) in the temperature range from 400 to 900℃.  
+We confirmed the thickness of 9.8 nm thick HZO through a cross-sectional TEM image, 
+which was deposited by ALD on the TiN bottom electrode followed by the deposition of the 
+TiN top electrode. The atomic percent ratio of the HZO film was confirmed by X-ray 
+photoelectron spectroscopy (XPS) (Sigma Probe, Thermo VG Scientific) depth profiling, 
+which was around 1:1 in terms of Hf: Zr ratio. The crystal structure of the HZO films was 
+analyzed using grazing incidence X-ray diffraction (GIXRD) (Rigaku, SmartLab) after the 
+post-metal annealing. The polarization-electric field (P-E) hysteresis loops and the capacitance-
+voltage (C-V) characteristics were obtained by Precision LC II. 
+ 
+Piezoresponse force microscopy (PFM) imaging of Hf0.5Zr0.5O2 thin films 
+To understand the nano-scale polarization and domain properties of HZO, PFM imaging was 
+conducted on the HZO surface after removing the top electrode. We used NX-10 (Parks system) 
+and Cypher ES (Asylum Research, Oxford Instruments) to measure the grain size and conduct 
+
+ 
+6 
+piezoresponse force microscopy (PFM) analysis, respectively. To measure the grain size of 
+HZO thin films with RTA temperature between 500 and 800 ℃, we conducted non-contact 
+mode AFM imaging using an Al-coated AFM tip (AC160TS, Olympus). Scan size and speed 
+were 500 nm × 500 nm and 0.5 Hz, respectively. In the case of PFM analysis, we conducted 
+dual ac resonance tracking (DART) PFM using a Pt/Ir-coated AFM tip (PPP-EFM, Nanoworld) 
+[26,27]. 
+The ac modulation voltage and loading force were 1V and 40 nN, respectively. The drive 
+frequencies were near the contact resonance between 320 and 350 kHz. To measure the domain 
+size, we conducted the nanoscale bit formation process. First, we performed background poling 
+as the first step, where we applied -7 V and +7 V respectively for up and down switching over 
+area of 3 µm × 3 µm. For dot poling, we applied +7 V and +8 V on the up-domain region and 
+-7 V and -8 V on the down-domain region as a function of pulse time, respectively. After the 
+poling process, we scanned and confirmed that the nanoscale bits were successfully formed. 
+The imaging experiments were repeated more than 10 times. 
+ 
+Results and Discussion 
+Macro-scale characteristic of HZO thin films with various RTA temperatures 
+The thickness of the HZO thin film is 98 Å, as confirmed by transmission electron 
+microscopy (TEM) in Fig. 1 (a). Fig. 1 (b) and (c) show the grazing incidence X-ray diffraction 
+(GI-XRD) spectrum and the relative ratio of monoclinic, orthorhombic, and tetragonal phases, 
+respectively. The crystalline peaks started to appear from 500 ℃ up to 900 ℃. Park et al. 
+reported that the 2 theta peaks between 28 and 31º could be assigned as m(-111), o(111), t(011), 
+and m(111), respectively [26]. As such, one can deconvolute the overlapped peaks into m(-111), 
+o(111), t(011), and m(111) that are located at 28.54º, 30.4º, 30.8º, and 31.64º [28]. The relative 
+
+ 
+7 
+ratio of each phase obtained by Gaussian peaks from each RTA temperature is shown in Fig. 1 
+(c). The m-phase increases with increasing RTA temperature, the o-phase increases up to 700 ℃ 
+and then slightly decreases at 800 ℃. Our results are in good agreement with the prior XRD 
+results [28]. We conclude that the optimum temperature range for o-phase crystallization is 
+between 600 and 700 ℃. 
+In ferroelectric HZO MFM structure, the electrode is one of the critical factors for high 
+ferroelectricity and polarization switching behavior. Among the various electrodes, TiN has 
+been reported as one of the most suitable electrodes to have high ferroelectricity in HZO 
+capacitors. 
+P-V and C-V characteristics were measured using Precision LC II equipment. Among the 
+samples fabricated by various RTA temperatures, the well-defined ferroelectric properties with 
+a large 2Pr of >30 μC/cm2 were observed in the case of the HZO films with RTA at 600℃, as 
+shown Fig. 2 (a), from P-E hysteresis loops. In the case of RTA at 900℃, the capacitor failed 
+to display a well-defined hysteresis loop and showed a doughnut shape, a signature of a leaky 
+ferroelectric film. 
+Figure 2 (b) shows the butterfly-like feature of the capacitance-voltage (C-V) characteristic 
+of the HZO thin films at various RTA temperatures. P-E hysteresis and C-V measurements of 
+the HZO samples with RTA between 500 and 800℃, show well-defined ferroelectric 
+characteristics, of which results are in good agreement with prior report [29]. 
+Figure 2 (c) shows the wake-up effect, which is a typical characteristic of HZO thin films 
+where Pr starts to increase from 102 cycles to 104 cycles of polarization switching. Based on 
+macroscale characterization via XRD, P-V, and C-V analysis, we found that the optimum RTA 
+temperature was 600℃. 
+
+ 
+8 
+ 
+Nanoscale characterization of HZO thin films as a function of RTA temperature 
+We conducted AFM experiment to complement the optimization of annealing temperature, 
+which was based on the macroscale analysis. In the case of HZO thin films, unlike conventional 
+ferroelectric materials such as PbTiO3 (PTO), PZT, BTO, and P(VDF-TrFE) (Poly(vinylidene 
+fluoride-trifluoroethylene), the polarization poling process of ferroelectric domain is 
+challenging [30-34]. 
+The reasons behind the challenge of poling process are as follows. First, the piezoelectric 
+coefficient of HZO thin film is ~10 pm/V [35], which is very low compared to that of the 
+conventional piezoelectric material such as PZT, PTO, and BiFeO3 (BFO) [36-38]. This 
+means that PFM signal may be influenced by other artifacts such as electrostatic force induced 
+vibration [23]. Furthermore, HZO thin film is likely to have defects such as non-ferroelectric 
+layer, oxygen vacancy, and electron trap inside the film during deposition and heat treatment 
+[39]. Since HZO thin films have low d33 compared to other ferroelectric materials with potential 
+imprint in the hysteresis loop, the PFM amplitude and phase of HZO thin films may suffer from 
+low signal-to-noise ratio [40]. To remove the internal bias between the electrode and HZO thin 
+films, we applied an additional bias (1~2 V) to the tip to neutralize the internal bias. After 
+removing the internal bias, we could observe clear PFM amplitude and phase signals with 
+domain boundaries as shown in Fig. S1. 
+We measured the domain size using the method introduced by Woo et al. [41]. Fig. 3(a) 
+shows the sequences of nanoscale bit formation. We used lithography function where we could 
+form up- and down-domains by applying predefined bias over a region of 3 µm × 3 µm as 
+shown in Fig. 3(a). More specifically, a negative bias was applied to the AFM tip to form up-
+
+ 
+9 
+domains over a rectangle region of 1.5 µm × 3 µm, and a positive bias was applied to form 
+down-domains over a rectangle region of 1.5 µm × 3 µm next to the up-domain region. 
+Subsequently, we formed a matrix of dot array by changing the amplitude and duration of 
+voltage pulse applied to the AFM tip. After the matrix formation, we acquired the topography 
+(left), PFM amplitude (middle), and phase (right) images as shown in Fig. 3(b). The upward 
+(downward) polarizations show bright (dark) contrast without inducing any change in the 
+topography. The matrix formation was repeated on the HZO thin films with RTA conducted 
+between 500 and 800 ℃. 
+Fig. 4 shows the domain dot size (i.e. domain diameter) of RTA 600℃ sample as a function 
+of pulse time (500 μs, 5 ms, 50 ms, 500 ms, and 1 s) for different bias voltages of +7, +8, -7 
+and -8 V, which was measured from PFM phase and amplitude images by taking the full-width 
+at half maximum (FWHM) value of each line profile [41]. The reason why the coercive voltage 
+is much larger in PFM hysteresis loops than that in P-E hysteresis loops can be explained by 
+the inhomogeneous electric field generated by the PFM tip and critical size of domain nucleus 
+[41]. The lateral domain size from amplitude and phase images was linearly proportional to 
+logarithmic value of the pulse width. 
+Fig. 5(a)-(d) shows the grain size of HZO thin films as a function of RTA temperature. We 
+found that the grain size increases as a function of RTA temperature, as shown in Fig. 5 (e). 
+The orthorhombic phase (O-phase) is theoretically unstable, which is responsible for the 
+unexpected ferroelectric property of HZO thin films. As such, grain size and surface energy are 
+important factors in understanding the appearance of the O-phase in HZO thin films [42,43]. 
+As the grain size increases, the portion of M-phase increases while the portion of O-phase 
+decreases as shown in Fig. 1. We speculate that the low surface energy of M-phase played an 
+important role in increasing the grain size at the expense of the O-phase as the RTA temperature 
+
+ 
+10 
+increased. 
+Fig. 6(a) shows the correlation between the minimum domain size and the RTA temperature. 
+We observed that the minimum domain size increases with RTA temperature. If the M-phase is 
+mostly distributed at the surface, then the surface layer can act as a dead layer, which will 
+decrease the effective electric field applied over the O-phase. At the same time, it will act like 
+a lens through which the electric field will distribute more sparsely over the O-phase layer. This 
+hypothesis can explain the correlation between the minimum domain size, grain size and the 
+RTA temperature. 
+We found that the minimum domain size was 36.6 nm at RTA temperature of 500℃ as 
+shown in Fig. 6 (a). This is the smallest domain of HZO observed by PFM reported up to date. 
+The domain size increased by 39.4 nm when the RTA temperature increased by 100℃. This 
+information helps the engineers to design the RTA process for ferroelectric hafnia memory 
+devices, as it will influence the processing window and the resulting yield of the device. 
+Fig. 6 (b) shows the correlation between domain size, Pr, M-phase, and RTA temperature. 
+In the case of Pr and M-phase, the optimal condition lies between 600 and 700℃, whereas, in 
+the case of the minimum domain size, the optimum condition can be found below 600 ℃ as 
+described in Table 1. Kim et al. reported that the memory window and reliability of the device 
+deteriorated when Ps increased in the Fe-FETs using ferroelectric hafnia [44]. Based on our 
+results and Kim et al.’s work, we conclude that the RTA temperature of 600 ℃ or lower is more 
+suitable for high-performance, high-endurance, and highly integrated Fe-FETs using 
+ferroelectric hafnia in the future. 
+ 
+Conclusions 
+
+ 
+11 
+We investigated the minimum domain size of ferroelectric hafnia thin films as a function of 
+RTA temperature using dual ac resonance tracking (DART) PFM. We found that the minimum 
+domain size scales linearly with the RTA temperature. The maximum capacitance and remnant 
+polarization were found at 600 and 700 ℃, whereas the minimum domain size was found at 
+500 ℃. The above findings were analyzed in terms of the ratio of monoclinic phase to 
+orthorhombic phase and its role in domain size and grain size. Our findings provide direct 
+evidence of high-density ferroelectric memory devices based on nanoscale domain switching. 
+ 
+Acknowledgement 
+This work was supported by the Samsung Electronics Semiconductor Research Project funded 
+by Samsung Electronics, and the National Research Foundation of Korea (NRF) grant funded 
+by the Korea government (MSIT) (No. 2020R1A2C201207811). 
+ 
+
+ 
+12 
+ 
+ 
+Figure 1. (a) transmission electron microscopy (TEM) images of HfZrO M-F-M structures, 
+(b) GI-XRD, (c) relative ratio of HZO thin films with various RTA temperature 
+ 
+ 
+ 
+ 
+
+(a)
+TiN (Top electrode)
+HZO(98 A)
+ TiN (Bottom electrode
+HAADF
+(b)
+(c)
+0.7
+0-(111)/t-(111)
+0.6
+0-(002)
+0-(220) 0-(311)
+Intensity [a.u.]
+Relative ratio
+0.5
+RTA 900°
+0.4
+RTA 800°℃
+Break.
+Amorp
+S
+RTA 700°C
+0.3
+dowr
+RTA 600°C
+0.2
+RTA 500°C
++ m-phase
+0.1
+RTA400℃
+o-phase
+As deposition
+t-phase
+0.0
+10
+20
+30
+40
+50
+60
+70
+80
+as-
+400 500 600 700 800 900 1000
+dep.
+20[degree]
+Anneal Temp. [℃] 
+13 
+ 
+Figure 2. (a) Polarization–Electric field (P-E) curve, (b) Capacitance–Voltage (C-V) curve, 
+and (c) cycle properties of HZO films with various RTA temperature. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+30
+3.5
+20
+RTA 900°C
+Capacitance (nF)
+Dlarization [μC/cm’]
+20
+prll uogezy
+5-0.0.0.
+10
+10
+ol
+3.0
+0
+0
+0:0:0:0:0-0-0
+8
+0:0:0:0:0-0
+-10
+ASDEPO
+RTA400℃
+2.5
+RTA500℃
+-10
+20
+RTA600℃
+Pol
+8:
+RTA700℃
+RTA800℃
+30
+2.0
+-20
+-5
+-3
+-2-10
+2
+3
+4
+5
+-3
+.2
+-1
+0
+1
+2
+3
+10-1
+100
+101
+102
+103
+104
+105
+Electric Field [MVlcm]
+Voltage (V)
+Cycle 
+14 
+ 
+ 
+Figure 3. (a) Process sequences of nano-scale bit formation, (b) After poling process, the 
+topography(left), amplitude(middle), and phase(right) images 
+ 
+ 
+ 
+ 
+ 
+
+7 V-8 V
+nm
+pm
+10.0
+125
+270°
+.00
+600 nm
+10.00
+600
+nm
+600
+n 
+15 
+ 
+Figure 4. PFM (a) phase, (b) FHWM domain size from phase image, (c) amplitude, and (d) 
+FHWM domain size from amplitude image of HZO at RTA 600 ℃, 
+ 
+ 
+ 
+ 
+
++7 V switching
++8 V switching
+-7 V switching
+-8 V switching
+250
+200
+Domain dot size 
+150
+100
+50
+0
+0.5
+5
+5
+500
+1000
+0.5
+5
+5
+0
+1000
+5
+5
+5
+500
+1000
+0.5
+5
+5
+500
+1000
+5
+0
+Pulse time (ms)
++7 V switching
++8 V switching
+-7 V switching
+-8 V switching
+250
+(wu)
+200
+Domain dot size (
+150
+100
+50
+0
+0.5
+5
+500
+1000
+5.
+5
+009
+1000
+0.5
+5
+5
+500
+1000
+0.5
+5
+5
+500
+1000
+0
+5
+Pulse time (ms) 
+16 
+ 
+ 
+Figure 5. Grain size of HZO thin film from (a) RTA 500 ℃, (b) RTA 600 ℃, (c) RTA 
+700 ℃, (d) RTA 800 ℃ temperatures, and (e) correlation between domain size and RTA 
+temperatures 
+ 
+ 
+
+Grain size (nm)
+Grain size (nm)
+30
+AVG.
+23.0
+AVG.
+24.8
+STD.
+10.4
+STD.
+12.7
+(wu)
+25
+Grain size (nm)
+Grain size (nm)
+Grainsize
+Grain size (nm)
+Grain size (nm)
+100
+20
+AVG.
+25.8
+AVG
+26.9
+STD
+13.3
+STD
+15.7
+500
+600
+700
+800 
+17 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 6.  (a) Correlation between domain size and RTA temperatures and (b) correlation 
+between domain size, Pr, M-phase, and RTA temperatures. 
+ 
+ 
+
+Amplitude
+Phase
+dot size (nm)
+200
+24
+0.58
+250
+Domain size
+口
+口
+0.57
+O-phase
+23
+200
+150
+0.56
+wu
+222
+0.55
+150
+ize
+(μC/cm
+lase
+Domain (
+100
+0.54
+omain
+口
+100
+0.53
+50
+0.52
+Min.
+50
+20
+0.51
+0
+19
+0.50
+500 600 700 800 500 600 700 800
+500
+600
+700
+800
+RTA Temperature (℃)200
+24
+0.5
+Domain size
+-口-Pr
+-△-M-phase
+口-
+一口
+23
+0.4
+Domain size (nm)
+p
+150
+M-phase fraction
+Amorphous
+Break-
+down
+22
+0.3
+口
+100
+21
+0.2
+Domainsize
+50
+20
+0.1
+M-phase
+A
+0
+19
+10.0
+As
+400
+500
+600
+700
+800
+900
+Dep.
+RTA Temperature (℃) 
+18 
+ 
+ 
+ 
+Table 1. Summary of O-phase, Pr, Min. grain size, and domain size from experimental results. 
+ 
+ 
+ 
+ 
+ 
+
+RTA temperature
+500℃
+D。 009
+700 ℃
+800 ℃
+O-phase (%)
+52
+54
+55
+52
+M-phase (%)
+4
+8
+12
+25
+Remnant polarization (μC/cm²)
+21.9
+23.3
+23.3
+21.3
+Average grain size (nm)
+23.0
+24.8
+25.8
+26.9
+Minimum domain size (nm)
+36.6
+43.2
+87.5
+120.6 
+19 
+References 
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+doi:10.1039/c7nr02121f. 
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+Ferroelectric Field‐Effect Transistor Memory with a Spherical Recess Channel. physica status solidi 
+(RRL) – Rapid Research Letters 2021, 15, doi:10.1002/pssr.202100018. 
+ 
+ 
+ 
+
+ 
+22 
+Supporting Information for 
+ 
+ 
+Effect of Annealing Temperature on Minimum Domain Size of 
+Ferroelectric Hafnia 
+ 
+Seokjung Yun#, Hoon Kim#, Myungsoo Seo, Min-Ho Kang, Taeho Kim, Seongwoo Cho, 
+Min Hyuk Park, Sanghun Jeon, Yang-Kyu Choi, and Seungbum Hong* 
+ 
+aDepartment of Materials Science and Engineering, KAIST, Daejeon, 34141, Republic of Korea 
+bKAIST Institute for NanoCentury (KINC), KAIST, Daejeon, 34141, Republic of Korea 
+cDepartment of Materials Science and Engineering, KAIST, Daejeon, 34141, Republic of Korea 
+dDepartment of Materials Science and Engineering, Seoul National University, Seoul, Republic of Korea 
+ 
+ 
+
+ 
+23 
+ 
+ 
+ 
+ 
+ 
+Figure S1. The schematic of (a) the domain from HZO thin film with internal bias, (b) the 
+domain signal distorted due to internal bias, (c) the domain from HZO thin film applied bias 
+with removed internal bias, and (c) the domain signal after removed internal bias 
+
+Negative bias -1 --2V 
+24 
+ 
+ 
+Figure S2. (a) topography, (b) amplitude, and (c) phase images from the HZO thin film after 
+poling process with removed internal bias 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+nm
+pm
+10.0
+145
+270
+M
+0.00
+80.0
+90.0
+um
+10.00
+90.0 
+25 
+ 
+ 
+ 
+ 
+Figure S3. (a) phase image, (b) domain size of phase image, (c) amplitude image, and (d) 
+domain size of amplitude images from the HZO thin film that annealed at 500 °C. 
+
++7 V switching
++8 V switching
+-7 V switching
+-8 V switching
++
+350
+270
+Domain dot size (nm)
+300
+250
+Phase
+200
+150
+100
+50
+600 nm
+0
+0.5
+500
+0001
+0.5
+5
+500
+000
+5
+5
+500
+1000
+0.5
+5
+500
+1000
++7 V switching
++8 V switching
+-7 V switching
+-8 V switching
+pm
+450
+350
+Domain dot size (nm)
+300
+Amplitude
+250
+240
+200
+150
+100
+50
+600 nm
+0
+0.5
+5
+5
+500
+000
+0.5
+5
+500
+1000
+0.5
+5
+500
+1000
+0.5
+5
+500
+1000 
+26 
+ 
+ 
+ 
+ 
+Figure S4. (a) phase image, (b) domain size of phase image, (c) amplitude image, and (d) 
+domain size of amplitude images from the HZO thin film that annealed at 700 °C. 
+
++7 V switching
++8 V switching
+-7 V switching
+-8 V switching
+350
+270
+Domain dot size (nm)
+300
+250
+Phase
+200
+150
+100
+50
+600
+nm
+0
+0.5
+5
+50
+500
+1000
+0.5
+500
+000
+0.5
+5
+5
+500
+1000
+0.5
+50
+500
+1000
++7 V switching
++8 V switching
+-7 V switching
+-8 V switching
+pm
+170
+350
+Domain dot size (nm)
+300
+Amplitude
+250
+70.0
+200
+150
+100
+50
+600 nm
+0
+0.5
+5
+500
+1000
+5
+5
+500
+1000
+0.5
+5
+5
+500
+1000
+0.5
+5
+500
+1000
+0 
+27 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure S5. (a) phase image, (b) domain size of phase image, (c) amplitude image, and (d) 
+domain size of amplitude images from the HZO thin film that annealed at 800 °C. 
+
++7 V switching
++8 V switching
+-7 V switching
+-8 V switching
+270
+350
+Domain dot size (nm)
+300
+250
+Phase
+200
+150
+100
+50
+600 nm
+0
+0.5
+5
+5
+500
+0001
+0.5
+5
+5
+500
+1000
+0.5
+5
+500
+1000
+0.5
+5
+500
+1000
+1
++7 V switching
++8 V switching
+-7 V switching
+-8 V switching
+pm
+110
+350
+Domain dot size (nm)
+Amplitude
+300
+250
+200
+60.0
+150
+100
+50
+600 nm
+10.00
+0
+0.5
+5
+500
+1000
+0.5
+5
+500
+1000
+0.5
+5
+500
+1000
+0.5
+5
+500
+0001 
+28 
+Graphical Summary 
+ 
+ 
+
+KAIST
+270
+&SAMSUNG
+90.0
+1
+-90.0
+2μmKAIST
+270
+&SAMSUNG
+90.0
+1
+-90.0
+2μmpm
+KNST& SAMSUNG
+310
+160
+10.0
+2umpm
+KNST& SAMSUNG
+310
+160
+10.0
+2um
\ No newline at end of file
diff --git a/SdE4T4oBgHgl3EQf_w7x/content/tmp_files/load_file.txt b/SdE4T4oBgHgl3EQf_w7x/content/tmp_files/load_file.txt
new file mode 100644
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+page_content=' Seoul National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Republic of Korea  2 Abstract Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' we optimized the annealing temperature of HZO/TiN thin film heterostructure via multiscale analysis of remnant polarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' crystallographic phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' minimum ferroelectric domain size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' and average grain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We found that the remnant polarization was closely related to the relative amount of the orthorhombic phase whereas the minimum domain size was to the relative amount of the monoclinic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The minimum domain size was obtained at the annealing temperature of 500 ℃ while the optimum remnant polarization and capacitance at the annealing temperature of 600℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We conclude that the minimum domain size is more important than the sheer magnitude of remnant polarization considering the retention and fatigue of switchable polarization in nanoscale ferroelectric devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Our results are expected to contribute to the development of ultra-low-power logic transistors and next-generation non- volatile memory devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  Keywords: Ferroelectric hafnia, domain behavior, piezoresponse force microscopy, rapid thermal annealing, multi-scale analysis, grain size, domain size  3 Introduction Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5O2 (HZO), which has ferroelectric properties through Zr doping with the conventional gate dielectric material HfO2, has attracted much interest due to its potential application to next generation non-volatile memory devices and logic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' In addition, HZO is more suitable for the fabrication of FETs owing to its higher scalability, good thermal stability, and compatibility with Si processes than the previously introduced Pb(Zr,Ti)O3 (PZT), SrBi2Ta2O9 (SBT) and BaTiO3 (BTO) ferroelectric materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Hafnia can have ferroelectric properties through various dopants such as Zr [10], Si [11], La [12], Y [13], Al [14], Gd [15], and Sr [16], as well as by post heat treatment and strain engineering between electrodes and oxide films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Through the process introduced above, hafnia with ferroelectric properties shows the formation of the non-centrosymmetric orthorhombic phase with a space group of Pca21, unlike conventional HfO2 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Because of this non- centrosymmetric orthorhombic phase, Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' reported ferroelectricity and non- ferroelectricity depending on the Zr concentration of HZO [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' In addition, it was reported that the other dopants such as Y, Al, Gd, Sr, and La induce ferroelectric properties at a particular dopant concentration and heat treatment temperature [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The ferroelectric properties are lost due to the appearance of M-phase replacing O-phase above a specific temperature [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  Furthermore, the strain between the electrode and the corresponding ferroelectric thin film affects the ferroelectric properties [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' In the semiconductor process, as various temperatures, dopant concentrations, and electrodes can be used, it is crucial to find the optimum process condition to fabricate ferroelectric HZO thin with a high yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Many research groups have reported on the macroscopic characteristics of ferroelectrics, but only a few have analyzed the origin of ferroelectricity from the perspective of the interplay between domain configuration, defect distribution, polarization and screening charges  4 [17,18,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' To understand the polarization switching mechanism and retention or fatigue behavior of ferroelectric HfO2 materials at the nanoscale, piezoresponse force microscopy (PFM) is one of the most powerful tools as it can visualize both out-of-plane and in-plane polarization vectors [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' However, the challenge lies in the fact that the piezoelectric coefficient, which is the marker for polarization vector in PFM, is less than 10 pm/V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Stolichnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' reported deff of HZO to be 5 pm/V, and Liam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' reported effective piezoelectric coupling coefficient (deff) of < 2 pm/V in Si-doped hafnia [24,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The low piezoelectric coefficient results in low signal to noise ratio in PFM images, which makes the analysis difficult due to the various artifacts contributing to the PFM signal [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Here, we designed the voltage profile for PFM imaging of HZO thin films, which were fabricated with various rapid thermal annealing temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We successfully analyzed the domain configuration of HZO thin films with different voltage pulse amplitude and duration to find the optimal process condition for minimizing the written domain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  5 Experiment Fabrication of Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5O2 thin films 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5 nm thick Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5O2 (HZO) films were deposited on 200 nm thick TiN electrode as the bottom electrode/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5 nm SiO2 interfacial layer, using atomic layer deposition (ALD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Tetrakis(ethylmethylamido)hafnium(IV) (TEMAHf), Tetrakis(ethylmethylamido)zirconium(IV) (TEMAZr), and O3 were used as Hf precursor, Zr precursor, and oxygen reactant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' A 200 nm TiN electrode was deposited on top of the device via physical vapor deposition (PVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The top electrode of TiN was wet etched using an NH4OH:H2O2:H2O solution at a concentration ratio of 1:2:5 and the pattern size was 30~500 ㎛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The ferroelectric HZO film was crystallized by post-metal rapid thermal annealing (RTA) in the temperature range from 400 to 900℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We confirmed the thickness of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='8 nm thick HZO through a cross-sectional TEM image, which was deposited by ALD on the TiN bottom electrode followed by the deposition of the TiN top electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The atomic percent ratio of the HZO film was confirmed by X-ray photoelectron spectroscopy (XPS) (Sigma Probe, Thermo VG Scientific) depth profiling, which was around 1:1 in terms of Hf: Zr ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The crystal structure of the HZO films was analyzed using grazing incidence X-ray diffraction (GIXRD) (Rigaku, SmartLab) after the post-metal annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The polarization-electric field (P-E) hysteresis loops and the capacitance- voltage (C-V) characteristics were obtained by Precision LC II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  Piezoresponse force microscopy (PFM) imaging of Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5O2 thin films To understand the nano-scale polarization and domain properties of HZO, PFM imaging was conducted on the HZO surface after removing the top electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We used NX-10 (Parks system) and Cypher ES (Asylum Research, Oxford Instruments) to measure the grain size and conduct  6 piezoresponse force microscopy (PFM) analysis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' To measure the grain size of HZO thin films with RTA temperature between 500 and 800 ℃, we conducted non-contact mode AFM imaging using an Al-coated AFM tip (AC160TS, Olympus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Scan size and speed were 500 nm × 500 nm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5 Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' In the case of PFM analysis, we conducted dual ac resonance tracking (DART) PFM using a Pt/Ir-coated AFM tip (PPP-EFM, Nanoworld) [26,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The ac modulation voltage and loading force were 1V and 40 nN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The drive frequencies were near the contact resonance between 320 and 350 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' To measure the domain size, we conducted the nanoscale bit formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' First, we performed background poling as the first step, where we applied -7 V and +7 V respectively for up and down switching over area of 3 µm × 3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' For dot poling, we applied +7 V and +8 V on the up-domain region and -7 V and -8 V on the down-domain region as a function of pulse time, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' After the poling process, we scanned and confirmed that the nanoscale bits were successfully formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The imaging experiments were repeated more than 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  Results and Discussion Macro-scale characteristic of HZO thin films with various RTA temperatures The thickness of the HZO thin film is 98 Å, as confirmed by transmission electron microscopy (TEM) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 1 (b) and (c) show the grazing incidence X-ray diffraction (GI-XRD) spectrum and the relative ratio of monoclinic, orthorhombic, and tetragonal phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The crystalline peaks started to appear from 500 ℃ up to 900 ℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' reported that the 2 theta peaks between 28 and 31º could be assigned as m(-111), o(111), t(011), and m(111), respectively [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' As such, one can deconvolute the overlapped peaks into m(-111), o(111), t(011), and m(111) that are located at 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='54º, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='4º, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='8º, and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='64º [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The relative  7 ratio of each phase obtained by Gaussian peaks from each RTA temperature is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The m-phase increases with increasing RTA temperature, the o-phase increases up to 700 ℃ and then slightly decreases at 800 ℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Our results are in good agreement with the prior XRD results [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We conclude that the optimum temperature range for o-phase crystallization is between 600 and 700 ℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' In ferroelectric HZO MFM structure, the electrode is one of the critical factors for high ferroelectricity and polarization switching behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Among the various electrodes, TiN has been reported as one of the most suitable electrodes to have high ferroelectricity in HZO capacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' P-V and C-V characteristics were measured using Precision LC II equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Among the samples fabricated by various RTA temperatures, the well-defined ferroelectric properties with a large 2Pr of >30 μC/cm2 were observed in the case of the HZO films with RTA at 600℃, as shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 2 (a), from P-E hysteresis loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' In the case of RTA at 900℃, the capacitor failed to display a well-defined hysteresis loop and showed a doughnut shape, a signature of a leaky ferroelectric film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Figure 2 (b) shows the butterfly-like feature of the capacitance-voltage (C-V) characteristic of the HZO thin films at various RTA temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' P-E hysteresis and C-V measurements of the HZO samples with RTA between 500 and 800℃, show well-defined ferroelectric characteristics, of which results are in good agreement with prior report [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  Figure 2 (c) shows the wake-up effect, which is a typical characteristic of HZO thin films where Pr starts to increase from 102 cycles to 104 cycles of polarization switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Based on macroscale characterization via XRD, P-V, and C-V analysis, we found that the optimum RTA temperature was 600℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  8  Nanoscale characterization of HZO thin films as a function of RTA temperature We conducted AFM experiment to complement the optimization of annealing temperature, which was based on the macroscale analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' In the case of HZO thin films, unlike conventional ferroelectric materials such as PbTiO3 (PTO), PZT, BTO, and P(VDF-TrFE) (Poly(vinylidene fluoride-trifluoroethylene), the polarization poling process of ferroelectric domain is challenging [30-34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The reasons behind the challenge of poling process are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' First, the piezoelectric coefficient of HZO thin film is ~10 pm/V [35], which is very low compared to that of the conventional piezoelectric material such as PZT, PTO, and BiFeO3 (BFO) [36-38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' This means that PFM signal may be influenced by other artifacts such as electrostatic force induced vibration [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Furthermore, HZO thin film is likely to have defects such as non-ferroelectric layer, oxygen vacancy, and electron trap inside the film during deposition and heat treatment [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Since HZO thin films have low d33 compared to other ferroelectric materials with potential imprint in the hysteresis loop, the PFM amplitude and phase of HZO thin films may suffer from low signal-to-noise ratio [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' To remove the internal bias between the electrode and HZO thin films, we applied an additional bias (1~2 V) to the tip to neutralize the internal bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  After removing the internal bias, we could observe clear PFM amplitude and phase signals with domain boundaries as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We measured the domain size using the method introduced by Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 3(a) shows the sequences of nanoscale bit formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We used lithography function where we could form up- and down-domains by applying predefined bias over a region of 3 µm × 3 µm as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' More specifically, a negative bias was applied to the AFM tip to form up-  9 domains over a rectangle region of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5 µm × 3 µm, and a positive bias was applied to form down-domains over a rectangle region of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5 µm × 3 µm next to the up-domain region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Subsequently, we formed a matrix of dot array by changing the amplitude and duration of voltage pulse applied to the AFM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' After the matrix formation, we acquired the topography (left), PFM amplitude (middle), and phase (right) images as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The upward (downward) polarizations show bright (dark) contrast without inducing any change in the topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The matrix formation was repeated on the HZO thin films with RTA conducted between 500 and 800 ℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 4 shows the domain dot size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' domain diameter) of RTA 600℃ sample as a function of pulse time (500 μs, 5 ms, 50 ms, 500 ms, and 1 s) for different bias voltages of +7, +8, -7 and -8 V, which was measured from PFM phase and amplitude images by taking the full-width at half maximum (FWHM) value of each line profile [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The reason why the coercive voltage is much larger in PFM hysteresis loops than that in P-E hysteresis loops can be explained by the inhomogeneous electric field generated by the PFM tip and critical size of domain nucleus [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The lateral domain size from amplitude and phase images was linearly proportional to logarithmic value of the pulse width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 5(a)-(d) shows the grain size of HZO thin films as a function of RTA temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  We found that the grain size increases as a function of RTA temperature, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 5 (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The orthorhombic phase (O-phase) is theoretically unstable, which is responsible for the unexpected ferroelectric property of HZO thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' As such, grain size and surface energy are important factors in understanding the appearance of the O-phase in HZO thin films [42,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' As the grain size increases, the portion of M-phase increases while the portion of O-phase decreases as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We speculate that the low surface energy of M-phase played an important role in increasing the grain size at the expense of the O-phase as the RTA temperature  10 increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 6(a) shows the correlation between the minimum domain size and the RTA temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We observed that the minimum domain size increases with RTA temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' If the M-phase is mostly distributed at the surface, then the surface layer can act as a dead layer, which will decrease the effective electric field applied over the O-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' At the same time, it will act like a lens through which the electric field will distribute more sparsely over the O-phase layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' This hypothesis can explain the correlation between the minimum domain size, grain size and the RTA temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We found that the minimum domain size was 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='6 nm at RTA temperature of 500℃ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 6 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' This is the smallest domain of HZO observed by PFM reported up to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The domain size increased by 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='4 nm when the RTA temperature increased by 100℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' This information helps the engineers to design the RTA process for ferroelectric hafnia memory devices, as it will influence the processing window and the resulting yield of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 6 (b) shows the correlation between domain size, Pr, M-phase, and RTA temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' In the case of Pr and M-phase, the optimal condition lies between 600 and 700℃, whereas, in the case of the minimum domain size, the optimum condition can be found below 600 ℃ as described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' reported that the memory window and reliability of the device deteriorated when Ps increased in the Fe-FETs using ferroelectric hafnia [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  Based on our results and Kim et al.’s work, we conclude that the RTA temperature of 600 ℃ or lower is more suitable for high-performance, high-endurance, and highly integrated Fe-FETs using ferroelectric hafnia in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  Conclusions  11 We investigated the minimum domain size of ferroelectric hafnia thin films as a function of RTA temperature using dual ac resonance tracking (DART) PFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' We found that the minimum domain size scales linearly with the RTA temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The maximum capacitance and remnant polarization were found at 600 and 700 ℃, whereas the minimum domain size was found at 500 ℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' The above findings were analyzed in terms of the ratio of monoclinic phase to orthorhombic phase and its role in domain size and grain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Our findings provide direct evidence of high-density ferroelectric memory devices based on nanoscale domain switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  Acknowledgement This work was supported by the Samsung Electronics Semiconductor Research Project funded by Samsung Electronics, and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 2020R1A2C201207811).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  12  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' (a) transmission electron microscopy (TEM) images of HfZrO M-F-M structures, (b) GI-XRD, (c) relative ratio of HZO thin films with various RTA temperature  (a) TiN (Top electrode) HZO(98 A) TiN (Bottom electrode HAADF (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='7 0-(111)/t-(111) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='6 0-(002) 0-(220) 0-(311) Intensity [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='] Relative ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5 RTA 900° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='4 RTA 800°℃ Break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Amorp S RTA 700°C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='3 dowr RTA 600°C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='2 RTA 500°C + m-phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='1 RTA400℃ o-phase As deposition t-phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0 10 20 30 40 50 60 70 80 as- 400 500 600 700 800 900 1000 dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 20[degree] Anneal Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' [℃] 13  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' (a) Polarization–Electric field (P-E) curve, (b) Capacitance–Voltage (C-V) curve, and (c) cycle properties of HZO films with various RTA temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  20  RTA 900°C  Capacitance (nF)  Dlarization [μC/cm’]  20  prll uogezy  5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  10  10  ol  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0  0  0  0:0:0:0:0 0 0  8  0:0:0:0:0 0 10  ASDEPO  RTA400℃  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  RTA500℃ 10  20  RTA600℃  Pol  8:  RTA700℃  RTA800℃  30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0 20 5 3 2 10  2  3  4  5 3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='2 1  0  1  2  3  10 1  100  101  102  103  104  105  Electric Field [MVlcm]  Voltage (V)  Cycle  14  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' (a) Process sequences of nano-scale bit formation, (b) After poling process, the topography(left), amplitude(middle), and phase(right) images  7 V 8 V  nm  pm  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0  125  270°  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='00  600 nm  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='00  600  nm  600  n  15  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' PFM (a) phase, (b) FHWM domain size from phase image, (c) amplitude, and (d) FHWM domain size from amplitude image of HZO at RTA 600 ℃,  +7 V switching  +8 V switching 7 V switching 8 V switching  250  200  Domain dot size  150  100  50  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  5  5  500  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  5  5  0  1000  5  5  5  500  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  5  5  500  1000  5  0  Pulse time (ms)  +7 V switching  +8 V switching 7 V switching 8 V switching  250  (wu)  200  Domain dot size (  150  100  50  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  5  500  1000  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  5  009  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  5  5  500  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  5  5  500  1000  0  5  Pulse time (ms)  16  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Grain size of HZO thin film from (a) RTA 500 ℃, (b) RTA 600 ℃, (c) RTA 700 ℃, (d) RTA 800 ℃ temperatures, and (e) correlation between domain size and RTA temperatures  Grain size (nm)  Grain size (nm)  30  AVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0  AVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='8  STD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='4  STD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='7  (wu)  25  Grain size (nm)  Grain size (nm)  Grainsize  Grain size (nm)  Grain size (nm)  100  20  AVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='8  AVG  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='9  STD  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='3  STD  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='7  500  600  700  800  17  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' (a) Correlation between domain size and RTA temperatures and (b) correlation between domain size, Pr, M-phase, and RTA temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  Amplitude Phase dot size (nm) 200 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='58 250 Domain size 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='57 O-phase 23 200 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='56 wu 222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='55 150 ize (μC/cm lase Domain ( 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='54 omain 口 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='53 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='52 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 50 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='51 0 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='50 500 600 700 800 500 600 700 800 500 600 700 800 RTA Temperature (℃)200 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5 Domain size -口-Pr -△-M-phase 口- 一口 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='4 Domain size (nm) p 150 M-phase fraction Amorphous Break- down 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='3 口 100 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='2 Domainsize 50 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='1 M-phase A 0 19 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0 As 400 500 600 700 800 900 Dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' RTA Temperature (℃) 18  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Summary of O-phase, Pr, Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' grain size, and domain size from experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  RTA temperature 500℃ D。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 009 700 ℃ 800 ℃ O-phase (%) 52 54 55 52 M-phase (%) 4 8 12 25 Remnant polarization (μC/cm²) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='3 Average grain size (nm) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='8 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='9 Minimum domain size (nm) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='6 19 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Valasek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Piezo-Electric and Allied Phenomena in Rochelle Salt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Physical Review 1921, 17, 475-481, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='1103/PhysRev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Scott, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Ferroelectric Memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Springer 2000, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='1007/978-3-662-04307-3, doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='1007/978-3-662-04307-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content=' Principles and Applications of Ferroelectrics and Related Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Oxford University Press 2001, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='1093/acprof:oso/9780198507789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0001, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='1093/acprof:oso/9780198507789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Fridkin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Domains and ferroelectricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Crystallography Reports 2017, 62, 169-173, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Integration technology for ferroelectric memory devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Microelectronics Reliability 2003, 43, 385-398, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='1016/s0026-2714(02)00285-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content=' Takahashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Sakai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Self-Aligned-Gate Metal/Ferroelectric/Insulator/Semiconductor Field-Effect Transistors with Long Memory Retention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Japanese Journal of Applied Physics 2005, 44, L800-L802, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content=' Mischenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content=' Fengler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content=' (a) topography, (b) amplitude, and (c) phase images from the HZO thin film after poling process with removed internal bias  nm  pm  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content=' (a) phase image, (b) domain size of phase image, (c) amplitude image, and (d) domain size of amplitude images from the HZO thin film that annealed at 500 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  +7 V switching  +8 V switching 7 V switching 8 V switching  +  350  270  Domain dot size (nm)  300  250  Phase  200  150  100  50  600 nm  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content='  +7 V switching  +8 V switching 7 V switching 8 V switching  350  270  Domain dot size (nm)  300  250  Phase  200  150  100  50  600  nm  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content='5  50  500  1000  +7 V switching  +8 V switching 7 V switching 8 V switching  pm  170  350  Domain dot size (nm)  300  Amplitude  250  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0  200  150  100  50  600 nm  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  5  500  1000  5  5  500  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  5  5  500  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  5  500  1000  0  27  Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content=' (a) phase image, (b) domain size of phase image, (c) amplitude image, and (d) domain size of amplitude images from the HZO thin film that annealed at 800 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='  +7 V switching  +8 V switching 7 V switching 8 V switching  270  350  Domain dot size (nm)  300  250  Phase  200  150  100  50  600 nm  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='5  5  5  500  0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+page_content='5  5  500  1000  1  +7 V switching  +8 V switching 7 V switching 8 V switching  pm  110  350  Domain dot size (nm)  Amplitude  300  250  200  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
+page_content='0  150  100  50  600 nm  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE4T4oBgHgl3EQf_w7x/content/2301.05374v1.pdf'}
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+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL
+STAGE-RESTART METHODS ∗
+ALEX C. FISH†, DANIEL R. REYNOLDS†, AND STEVEN B. ROBERTS‡
+Abstract. Implicit-Explicit (IMEX) methods are flexible numerical time integration methods
+which solve an initial-value problem (IVP) that is partitioned into stiff and nonstiff processes with the
+goal of lower computational costs than a purely implicit or explicit approach. A complementary form
+of flexible IVP solvers are multirate infinitesimal methods for problems partitioned into fast- and
+slow-changing dynamics, that solve a multirate IVP by evolving a sequence of “fast” IVPs using any
+suitably accurate algorithm. This article introduces a new class of high-order implicit-explicit multi-
+rate methods that are designed for multirate IVPs in which the slow-changing dynamics are further
+partitioned in an IMEX fashion.
+This new class, which we call implicit-explicit multirate stage-
+restart (IMEX-MRI-SR), both improves upon the previous implicit-explicit multirate generalized-
+structure additive Runge Kutta (IMEX-MRI-GARK) methods, and extends multirate exponential
+Runge Kutta (MERK) methods into the IMEX context. We leverage GARK theory to derive con-
+ditions guaranteeing orders of accuracy up to four. We provide second-, third-, and fourth-order
+accurate example methods and perform numerical simulations demonstrating convergence rates and
+computational performance in both fixed-step and adaptive-step settings.
+Key words. multirate time integration, initial-value problems, implicit-explicit methods
+AMS subject classifications. 65L20, 65L05, 65L06
+1. Introduction. Flexible time integration methods for solving systems of initial-
+value problems (IVPs) have seen growing interest in recent years, largely due to their
+ability to provide highly accurate approximations of the IVP solution with increased
+computational efficiency. These integrators strive to reduce computational costs by
+partitioning the IVP into different components, and then treating each using different
+step sizes or numerical methods. Some of the primary families of flexible methods in-
+clude implicit-explicit (IMEX) partitioning [2, 9, 11, 23], linear-nonlinear partitioning
+[8, 13, 14, 15, 16], and multirate partitioning [6, 22, 24, 26].
+IMEX time integration methods solve IVPs in which the right-hand side function
+∗Submitted to the editors DATE.
+Funding: The work of Alex Fish and Daniel Reynolds was supported by the U.S. Department
+of Energy, Office of Science, Office of Advanced Scientific Computing Research, Scientific Discovery
+through Advanced Computing (SciDAC) Program through the FASTMath Institute, under DOE
+award DE-SC0021354.
+The work of Steven Roberts was performed under the auspices of the U.S. Department of Energy by
+Lawrence Livermore National Laboratory under Contract DEAC52-07NA27344 and was supported
+by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Re-
+search. LLNL-JRNL-843436.
+This document was prepared as an account of work sponsored by an agency of the United States
+government. Neither the United States government nor Lawrence Livermore National Security, LLC,
+nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or
+responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product,
+or process disclosed, or represents that its use would not infringe privately owned rights. Reference
+herein to any specific commercial product, process, or service by trade name, trademark, manufac-
+turer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or
+favoring by the United States government or Lawrence Livermore National Security, LLC. The views
+and opinions of authors expressed herein do not necessarily state or reflect those of the United States
+government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or
+product endorsement purposes.
+†Department of Mathematics, Southern Methodist University, Dallas, TX, USA (afish@smu.edu,
+reynolds@smu.edu).
+‡Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore,
+CA, USA (roberts115@llnl.gov).
+1
+arXiv:2301.00865v1  [math.NA]  2 Jan 2023
+
+2
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+f(t, y(t)) is additively split into stiff {I} and nonstiff {E} processes,
+y′(t) = f(t, y) := f {I}(t, y) + f {E}(t, y),
+t ≥ t0,
+y(t0) = y0.
+(1.1)
+IMEX methods then couple two different numerical methods to treat these compo-
+nents: f {I} typically uses a stiff but computationally expensive solver, whereas f {E}
+may use a cheaper but nonstiff solver. For example, additive Runge–Kutta (ARK)
+methods typically combine an A-stable diagonally-implicit Runge–Kutta (DIRK) method
+with an explicit Runge–Kutta (RK) method.
+Similarly, multirate methods solve IVPs in which the right-hand side is additively
+partitioned into rapidly and slowly evolving dynamics, {F} and {S},
+y′(t) = f(t, y) := f {F }(t, y) + f {S}(t, y),
+t ≥ t0,
+y(t0) = y0.
+(1.2)
+Multirate methods then apply numerical methods with different step sizes for each
+component to save on computation time while retaining a desired level of accuracy.
+In this work we combine the above approaches to consider a three-way additively
+partitioned IVP, wherein the slow partition f {S} from (1.2) is split in an IMEX fashion,
+y′(t) = f {F }(t, y) + f {I}(t, y) + f {E}(t, y),
+t ≥ t0,
+y(t0) = y0.
+(1.3)
+In particular, we add a new class of methods to the ever-growing family of multirate
+infinitesimal (MRI) methods. These approximate the solution to (1.2) or (1.3) through
+solving a sequence of “fast” IVPs,
+v′
+i(θ) = f {F }(θ, vi) + gi(θ),
+θ ∈ [θ0,i, θf,i],
+v(θ0,i) = v0,i.
+(1.4)
+The forcing functions gi(θ) incorporate information from the slow dynamics defined
+by f {I} and f {E} in a manner defined by the method. MRI methods assume that
+these fast IVPs (1.4) are solved exactly, but in practice these are approximated using
+an additional “inner” numerical method with a smaller step size than the multirate
+method. This inner method can itself further decompose the problem through IMEX,
+linear-nonlinear, or multirate approaches.
+Our proposed class of methods is called Implicit-Explicit Multirate Infinitesimal
+Stage-Restart (IMEX-MRI-SR) methods. Each stage of an IMEX-MRI-SR method
+consists of evolving a fast IVP followed by an implicit solve. This allows derivation
+of IMEX-MRI-SR methods by extending a base ARK method. We discuss the role of
+base methods further in Section 2.1, particularly focusing on their role in satisfying
+order conditions.
+IMEX-MRI-SR methods are defined by nΩ coefficient matrices Ω{k} ∈ Rs{S}×s{S},
+k = 0, ..., nΩ − 1, a coefficient matrix Γ ∈ Rs{S}×s{S}, and an abscissae vector c{S} ∈
+Rs{S}.
+Embedded versions of these methods include additional coefficient vectors
+ˆω{k} ∈ Rs{S} and ˆγ ∈ Rs{S}. The algorithm for evolving a solution to an IVP of the
+form (1.3) is defined as follows.
+Definition 1.1 (IMEX-MRI-SR methods for additively partitioned systems).
+An IMEX-MRI-SR method evolves the solution to the problem (1.3) from tn to tn +H
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+3
+according to the following algorithm.
+Let: Y {S}
+1
+:= yn
+(1.5a)
+For: i = 2, ..., s{S}
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Let:
+vi(0) := yn,
+Solve:
+v′
+i(θ) = f {F }(tn + θ, vi(θ)) + gi(θ), for θ ∈ [0, c{S}
+i
+H]
+where gi(θ) =
+1
+c{S}
+i
+i−1
+�
+j=1
+ωi,j
+�
+θ
+c{S}
+i
+H
+� �
+f {E}
+j
++ f {I}
+j
+�
+Solve:
+Y {S}
+i
+= vi
+�
+c{S}
+i
+H
+�
++ H
+i
+�
+j=1
+γi,jf {I}
+j
+,
+(1.5b)
+Let: yn+1 := Y {S}
+s{S},
+(1.5c)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Let:
+ˆv(0) := yn,
+Solve:
+ˆv(θ) = f {F }(tn + θ, ˆv(θ)) + ˆg(θ), for θ ∈ [0, H]
+where ˆg(θ) =
+s{S}−1
+�
+j=1
+ˆωj
+� θ
+H
+� �
+f {E}
+j
++ f {I}
+j
+�
+Solve:
+ˆyn+1 = ˆv (H) + H
+s{S}−1
+�
+j=1
+ˆγjf {I}
+j
++ Hˆγs{S}f {I}(tn + H, ˆyn+1),
+(1.5d)
+where f {E}
+j
+:= f {E} �
+tn + c{S}
+j
+H, Y {S}
+j
+�
+and f {I}
+j
+:= f {I} �
+tn + c{S}
+j
+H, Y {S}
+j
+�
+. Here
+yn+1 is the time-evolved approximation to y(tn+H), and ˆyn+1 is an embedded solution
+used for temporal error esimation. If temporal error estimation is not needed, then
+step (1.5d) may be omitted.
+Definition 1.2 (Slow tendency coefficients). The coefficients ωi,j in (1.5b) are
+defined as in [3]
+(1.6)
+ωi,j(τ) :=
+nΩ−1
+�
+k=0
+ω{k}
+i,j τ k,
+ωi,j :=
+� 1
+0
+ωi,j(τ)dτ =
+nΩ−1
+�
+k=0
+ω{k}
+i,j
+k + 1,
+and we refer to Ω{k}, Ω, and Γ as the s{S} × s{S} matrices containing the coeffi-
+cients {ω{k}
+i,j }, {ωi,j}, and {γi,j}, respectively. As in [3] we assume the first row of
+the coefficient matrices are identically zero, and that Ω{k} and Ω are strictly lower-
+triangular. The embedding functions ˆωj(τ) are defined similarly to (1.6), with vectors
+of coefficients ˆω{k}.
+The rest of this paper is structured as follows. In the next subsection, we present
+related methods to IMEX-MRI-SR, discussing both their similarities and their limita-
+tions that are improved upon by the proposed methods. In Section 2 we prove order
+conditions for IMEX-MRI-SR methods up to order four, and in Section 3 we examine
+their linear stability. In Section 4 we provide embedded IMEX-MRI-SR methods of
+orders 2 through 4. In Section 5 we show that a previous class of multirate methods
+may be reformulated as IMEX-MRI-SR methods, and we use our convergence theory
+to show previously unproven features of those methods. We then present numerical
+results in Section 6 to validate our convergence theory, and to compare the efficiency
+of IMEX-MRI-SR methods against existing methods for problems of the form (1.3).
+
+4
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+Finally, in Section 7 we conclude this article with a summary of our contributions,
+and an outlook toward future work.
+1.1. Related methods. To our knowledge, there exist three MRI algorithms
+that allow both IMEX partitioning of the slow dynamics and infinitesimal treatment
+of the fast dynamics. These are the first-order accurate “Lie–Trotter” [4, 21] and the
+second-order accurate “Strang–Marchuk” [17, 25] operator splitting methods, and
+the recent fourth-order IMEX-MRI-GARK [3] method, an IMEX variation of MRI-
+GARK [22] methods. Both Lie–Trotter and Strang–Marchuk operate by sequentially
+applying distinct solvers to each component, only communicating with one another
+through the initial conditions applied within each sub-solve. Variations of both classes
+of methods for the IMEX multirate splitting (1.3) are shown in [3]. However, due to
+their weak “initial condition” coupling, Lie–Trotter and Strang–Marchuk are limited
+to at most first and second order accuracy in time, regardless of the order of accuracy
+of each component solver.
+IMEX-MRI-GARK methods are organized similarly to IMEX-MRI-SR, in that
+they advance the solution by alternating between evolving fast IVPs and solving im-
+plicit algebraic equations involving f {I}, and they use the result from each stage to
+provide a contribution to gi(θ) for later stages. However, unlike IMEX-MRI-SR meth-
+ods, IMEX-MRI-GARK methods evolve each fast IVP over an interval [ci−1H, ciH],
+with an initial condition given as the result of the previous stage.
+In each stage,
+either a fast evolution or implicit solve may occur, with this choice dictated by the
+abscissae: when ∆ci := ci − ci−1 > 0 a fast evolution occurs, but when ∆ci = 0 an
+algebraic system must be solved. In all existing implicit MRI-GARK and IMEX-MRI-
+GARK methods, authors have derived schemes by beginning with a base DIRK or
+ARK method, and then introduced additional internal stages to ensure an alternating
+pattern of ∆ci ̸= 0 followed by ∆ci+1 = 0, thereby ensuring an appropriate structure
+[3, 22]. However, these methods inherently require abscissae vectors c{S} that are
+non-decreasing. This is generally an uncommon feature in RK methods, especially
+for orders of accuracy higher than two [2, 9, 10, 11, 19, 20, 23], so there are relatively
+few base DIRK and ARK methods available for deriving new MRI-GARK and IMEX-
+MRI-GARK methods. Additionally, the “padding” process for adding internal stages
+to ensure ∆ci = 0 is not obvious, and frequently results in an overly complicated trial-
+and-error process to decide where to insert stages. As a result of these challenges, no
+authors have successfully created IMEX-MRI-GARK methods with embeddings for
+temporal error estimation.
+A second class of methods that are closely related to IMEX-MRI-SR are multi-
+rate exponential Runge–Kutta (MERK) methods [13]. While these do not support
+implicitness at the slow time scale (i.e., f {I} = 0), and they assume that the fast
+partition is linear (i.e., f {F }(t, y) = Ly), their structure matches (1.5). Each internal
+stage is computed through evolving a fast IVP over an interval [0, ciH], using a forcing
+function that is determined through the values of f {E} at previous slow stages.
+2. IMEX-MRI-SR Order Conditions. Similar to IMEX-MRI-GARK meth-
+ods, we derive order conditions for IMEX-MRI-SR methods by first representing the
+algorithm in GARK form. Due to the 3-component partitioning (1.3), we must iden-
+tify GARK coefficients A{σ,ν}, b{ν}, and c{ν} for σ ∈ {S, F} and ν ∈ {I, E, F}. We
+refer to A{F,ν}, ν ∈ {I, E, F} as the fast GARK tables, and to A{S,ν}, ν ∈ {I, E, F}
+as the slow GARK tables. We assume the fast IVP (1.5b) is solved with one step of
+a sufficiently accurate Runge–Kutta method having s{F } stages, and defined by the
+Butcher table (A{F }, b{F }, c{F }). The jth sub-stage in computing the solution to the
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+5
+fast IVP v′
+i(θ) is given by
+Vi,j = yn + c{S}
+i
+H
+s{F }
+�
+k=1
+a{F }
+j,k f {F }
+i,k
++ H
+s{F }
+�
+k=1
+a{F }
+j,k
+i−1
+�
+ℓ=1
+ωi,ℓ(c{F }
+k
+)
+�
+f {E}
+ℓ
++ f {I}
+ℓ
+�
+(2.1)
+for i = 1, ..., s{S}, and j = 1, ..., s{F }, where f {F }
+i,k
+:= f {F }(tn + c{F }
+k
+H, Vi,k), f {E}
+ℓ
+:=
+f {E}(tn + c{S}
+ℓ
+H, Y {S}
+ℓ
+) and f {I}
+ℓ
+:= f {I}(tn + c{S}
+ℓ
+H, Y {S}
+ℓ
+). The IMEX-MRI-SR
+method’s slow stages Y {S}
+i
+, i = 1, . . . , s{S}, are then computed by
+Y {S}
+i
+= yn + c{S}
+i
+H
+s{F }
+�
+j=1
+b{F }
+j
+f {F }
+i,j
++ H
+s{F }
+�
+j=1
+b{F }
+j
+i−1
+�
+ℓ=1
+ωi,ℓ
+�
+c{F }
+j
+� �
+f {E}
+ℓ
++ f {I}
+ℓ
+�
++ H
+i
+�
+ℓ=1
+γi,ℓf {I}
+ℓ
+.
+(2.2)
+We leverage GARK theory [23] to construct order conditions as in [3, 22]. Since
+the slow partitions share the same stages, Y {S}
+i
+, these methods have six GARK ma-
+trices, A{F,F }, A{F,E}, A{F,I}, A{S,F }, A{S,E}, and A{S,I} and three GARK vectors
+b{F }, b{E}, and b{I},
+A{F,F }
+A{F,E}
+A{F,I}
+A{S,F }
+A{S,E}
+A{S,I}
+b{F },T
+b{E},T
+b{I},T
+.
+Here, A{F,F } is a square s{SF } × s{SF } matrix (where we define s{SF } = s{S} ·
+s{F }), containing the coefficients relating the fast stages {Vi,j} to each other. It is
+block-diagonal because Vi,j depends only on Vi,k, k = 1, ..., s{F } through the A{F }
+coefficients, and never on Vℓj, ℓ ̸= i. These s{F } × s{F } block-diagonal elements are
+named A{F,F,i}.
+A{F,E} and A{F,I} are tall s{SF } × s{S} matrices relating the fast stages {Vi,j}
+to the explicit and implicit function evaluations of the slow stages {Y {S}
+i
+}, comprised
+of s{S} blocks named A{F,E,i}.
+A{S,F } is a wide s{S} ×s{SF } matrix, containing the coefficients relating the slow
+stages {Y {S}
+i
+} to the fast stages {Vi,j}. We name these s{S} total s{S} × s{F } blocks
+that comprise A{S,F } as A{S,F,i}. Each A{S,F,i} contains at most one row of non-zero
+entries, located in row i, because Y {S}
+i
+depends only on Vi,j, j = 1, ..., s{F }, and never
+on Vℓ,j, ℓ ̸= i.
+A{S,E} and A{S,I} are square s{S}×s{S} matrices relating the slow stages {Y {S}
+i
+}
+to the explicit and implicit function evaluations of those slow stages.
+The vectors b{σ} equal the last rows of A{S,σ}, σ ∈ {F, E, I}, because IMEX-
+MRI-SR methods have the first-same-as-last (FSAL) property, where the last stage is
+used as the solution to the step.
+When an embedding (1.5d) is included, it will correspond to three additional
+GARK vectors, ˆb{F } ˆb{E} and ˆb{I}. Due to the structural similarity of (1.5d) to the
+last stage Y {S}
+s{S}, the contents of these vectors will only differ from b{F }, b{E} and
+b{I} through their dependence on ˆω{k}
+j
+and ˆγj instead of ω{k}
+s{S},j and γs{S},j. Thus
+
+6
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+the order conditions that follow for the primary GARK matrices and vectors can be
+applied to the embedding as well.
+With the above simplifications, the GARK tableau can be expressed in block-
+matrix form as
+A{F,F,1}
+A{F,E,1}
+A{F,I,1}
+...
+...
+...
+A{F,F,s{S}}
+A{F,E,s{S}}
+A{F,I,s{S}}
+A{S,F,1}
+· · ·
+A{S,F,s{S}}
+A{S,E}
+A{S,I}
+b{F },T
+b{E},T
+b{I},T
+.
+A GARK method with this tabular structure has stage update formulas
+(2.3a)
+Vi,j = yn + H
+s{F }
+�
+k=1
+a{F,F,i}
+j,k
+f {F }
+i,j
++ H
+s{F }
+�
+k=1
+a{F,E,i}
+j,k
+f {E}
+j
++ H
+s{F }
+�
+k=1
+a{F,I,i}
+j,k
+f {I}
+j
+,
+(2.3b)
+Y {S}
+i
+= yn + H
+s{F }
+�
+j=1
+a{S,F,i}
+i,j
+f {F }
+ij
++ H
+s{F }
+�
+j=1
+A{S,E}
+i,j
+f {E}
+j
++ H
+s{F }
+�
+j=1
+A{S,I}
+i,j
+f {I}
+j
+.
+By matching coefficients in (2.1) and (2.3a), we identify the fast GARK coefficients,
+a{F,F,i}
+j,k
+= c{S}
+i
+a{F }
+j,k ,
+(2.4a)
+a{F,E,i}
+j,ℓ
+=
+s{F }
+�
+l=1
+a{F }
+j,k ωi,ℓ(c{F }
+l
+) =
+s{F }
+�
+l=1
+nΩ−1
+�
+k=0
+ω{k}
+i,ℓ a{F }
+j,l c{F }×k
+l
+,
+(2.4b)
+a{F,I,i}
+j,ℓ
+= a{F,E,i}
+j,ℓ
+=
+s{F }
+�
+k=1
+b{F }
+j,k ωi,ℓ(c{F }
+k
+),
+(2.4c)
+where the superscript ×k denotes element-wise exponentiation of a vector by k. Con-
+verting these to matrix form, we have the fast GARK tables,
+A{F,F } = C{S} ⊗ A{F } ∈ Rs{SF }×s{SF },
+(2.5a)
+A{F,E} =
+nΩ−1
+�
+k=0
+Ω{k} ⊗ A{F }c{F }×k ∈ Rs{SF }×s{S},
+(2.5b)
+A{F,I} = A{F,E} =
+nΩ−1
+�
+k=0
+Ω{k} ⊗ A{F }c{F }×k ∈ Rs{SF }×s{S},
+(2.5c)
+where ⊗ denotes the Kronecker product and C{σ} = diag(c{σ}).
+We similarly find the slow GARK table coefficients by comparing (2.2) and (2.3b),
+a{S,F,i}
+i,j
+= c{S}
+i
+b{F }
+j
+,
+(2.6a)
+a{S,E}
+i,ℓ
+=
+s{F }
+�
+k=1
+b{F }
+k
+ωi,ℓ(c{F }
+k
+) =
+nΩ−1
+�
+k=0
+ω{k}
+i,ℓ b{F },T c{F }×k,
+(2.6b)
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+7
+a{S,I}
+i,ℓ
+=
+s{F }
+�
+k=1
+b{F }
+k
+ωi,ℓ
+�
+c{F }
+j
+�
++ γi,ℓ =
+nΩ−1
+�
+k=0
+ω{k}
+i,ℓ b{F },T c{F }×k + γi,ℓ.
+(2.6c)
+Due to the infinitesimal nature of the fast method, we assume that it satisfies all
+bushy-tree order conditions,
+b{F },T c{F }×k =
+1
+k + 1,
+k = 0, ..., nΩ − 1.
+Leveraging this, and examining (2.6), the slow GARK tables in matrix-form are
+A{S,F } = C{S} ⊗ b{F },T ∈ Rs{S}×s{SF },
+(2.7a)
+A{S,E} =
+nΩ−1
+�
+k=0
+Ω{k}
+1
+k + 1 = Ω ∈ Rs{S}×s{S},
+(2.7b)
+A{S,I} =
+nΩ−1
+�
+k=0
+Ω{k}
+1
+k + 1 + Γ = Ω + Γ ∈ Rs{S}×s{S}.
+(2.7c)
+We additionally define the following variables, knowing that an IMEX-MRI-SR
+method has the FSAL property with respect to the slow stages Y {S},
+b{F },T = eT
+s{S}A{S,F } = (C{S}es{S} ⊗ b{F },T ) = (es{S} ⊗ b{F })T ,
+(2.8a)
+b{E},T = eT
+s{S}A{S,E} = eT
+s{S}Ω,
+(2.8b)
+b{I},T = eT
+s{S}A{S,I} = eT
+s{S}(Ω + Γ),
+(2.8c)
+where es{S} is an s{S}-length vector of all zeroes except a one in the last position.
+2.1. Base Consistency. If f {F }(t, y) = 0, then an IMEX-MRI-SR method re-
+duces to a simple ARK method defined by slow explicit and implicit base methods,
+(A{E}, b{E}, c{E}) = (A{S,E}, b{E}, A{S,E}1s{S}),
+(A{I}, b{I}, c{I}) = (A{S,I}, b{I}, A{S,I}1s{S}),
+where 1s{S} is a vector of ones with length s{S}. This ARK method has stages
+Y {S}
+i
+= yn + H
+i−1
+�
+j=1
+a{E}
+i,j f {E}(tn + c{E}
+j
+H, Y {S}
+j
+)
++ H
+i
+�
+j=1
+a{I}
+i,j f {I}(tn + c{I}
+j
+H, Y {S}
+j
+),
+with the last stage being equal to the solution for the step. We will refer to this ARK
+method as the slow base method throughout the derivation of the order conditions.
+Remark 2.1 (First order conditions). As long as the slow base method and the
+arbitrary fast method are order one or higher, there are no additional first-order
+coupling conditions for a GARK method. Thus, the effective first-order IMEX-MRI-
+SR condition is that A{S,E} = Ω and A{S,I} = Ω+Γ form an order one ARK method,
+which can be simply achieved if both A{S,E} and A{S,I} have first-order accuracy.
+Remark 2.2 (Deriving IMEX-MRI-SR methods from existing ARK methods).
+Due to the IMEX-MRI-SR structure, a base ARK method should be stiffly-accurate;
+otherwise it must first be converted to stiffly-accurate form by appending the b vectors
+to the bottom and pad a column of zeros to the right of the ARK’s A matrices.
+
+8
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+2.2. Kronecker Product Identities. In the ensuing derivations we leverage
+the following identities:
+(A ⊗ B)T = AT ⊗ BT ,
+(2.9a)
+(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD),
+(2.9b)
+(A ⊗ v)1{r(A)} = (A1{r(A)}) ⊗ v,
+(2.9c)
+(v ⊗ A)1{r(A)} = v ⊗ (A1{r(A)}),
+(2.9d)
+where A, B, C, D are arbitrary matrices with compatible dimemsions, v is an arbitrary
+vector, and 1{r(A)} is a vector of ones with length equal to the number of rows of A.
+Identities (2.9a) and (2.9b) can be found in [12]. Identities (2.9c) and (2.9d) can be
+shown through elementary computation.
+2.3. GARK Internal Consistency.
+Theorem 2.3 (GARK Internal Consistency). An IMEX-MRI-SR method satis-
+fies the GARK internal consistency conditions,
+c{F,F } = c{F,E} = c{F,I},
+(2.10a)
+c{S,F } = c{S,E} = c{S,I},
+(2.10b)
+where c{F,ν} = A{F,ν}1s{SF }, c{S,ν} = A{S,ν}1s{S}, ν ∈ {I, E, F}, if the following
+conditions hold:
+Ω{0}1s{S} = c{S},
+(2.11a)
+Ω{k}1s{S} = 0s{S}, k = 1, ..., nΩ − 1,
+(2.11b)
+Γ1s{S} = 0s{S},
+(2.11c)
+where 0s{S} is a vector of zeros with length s{S}.
+Proof. Computing the fast GARK abscissae from the respective fast GARK ta-
+bles,
+c{F,F } = A{F,F }1s{SF } = (C{S} ⊗ A{F })1s{SF } = c{S} ⊗ c{F },
+c{F,E} = A{F,E}1s{S} =
+nΩ−1
+�
+k=0
+Ω{k} ⊗ A{F }c{F }×k1s{S}
+= (Ω{0}1s{S}) ⊗ c{F } +
+nΩ−1
+�
+k=1
+(Ω{k}1s{S}) ⊗ A{F }c{F }×k,
+c{F,I} = A{F,I}1s{S} = A{F,E}1s{S} = c{F,E},
+thus c{F,F } = c{F,I} = c{F,E} when Ω01s{S} = c{S} and Ω{k}1s{S} = 0s{S}, k =
+1, ..., nΩ − 1. Computing the slow GARK abscissae from the slow GARK tables,
+c{S,F } = A{S,F }1s{SF } = (C{S} ⊗ b{F })1s{SF } = c{S},
+c{S,E} = A{S,E}1s{S} = Ω1s{S} =
+nΩ−1
+�
+k=0
+Ω{k}1s{S}
+1
+k + 1,
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+9
+c{S,E} = A{S,I}1s{S} = (Ω + Γ)1s{S} =
+nΩ−1
+�
+k=0
+Ω{k}1s{S}
+1
+k + 1 + Γ1s{S},
+and thus c{S,F } = c{S,I} = c{S,E} is satisfied using the same conditions as above,
+with the additional constraint that Γ1s{S} = 0s{S}.
+Remark 2.4 (Second order conditions). When the tables that comprise a GARK
+method are each at least second-order accurate and internal consistency holds, there
+are no additional second-order coupling conditions. Thus, the internal consistency
+conditions act as second-order conditions when the slow base method and arbitrary
+fast method are at least second-order accurate.
+2.4. Higher Order Conditions.
+Theorem 2.5 (Third Order Conditions). An internally-consistent IMEX-MRI-
+SR method with third-order accurate slow base method and with fast method of order
+max(3, nΩ + 1) accurate is third-order accurate if the following condition holds:
+(2.12)
+eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+c{S} = 1
+6.
+Proof. From [3], a internally consistent GARK method of this structure with a
+third-order accurate slow base method and an order max(3, nΩ + 1) fast method has
+four third-order coupling conditions,
+b{σ},T A{S,F }c{F } = 1
+6,
+(2.13a)
+b{F },T A{S,σ}c{S} = 1
+6,
+(2.13b)
+for σ ∈ {I, E}. An internally-consistent IMEX-MRI-SR method has
+c{F } = c{F,F } = c{F,E} = c{F,I} = c{S} ⊗ c{F },
+(2.14a)
+c{S} = c{S,F } = c{S,E} = c{S,I} = c{E} = c{I} = c{S}.
+(2.14b)
+Conditions (2.13a) are automatically satisfied for both values of σ:
+1
+6 = b{σ},T A{S,F }c{F }
+= b{σ},T (C{S} ⊗ b{F },T )(c{S} ⊗ c{F }) = 1
+3 · 1
+2
+Conditions (2.13b) reduce to the single condition (2.12) because A{F,E} = A{F,I} for
+an IMEX-MRI-SR method:
+1
+6 = b{F },T A{F,σ}c{S} = (eT
+s{S} ⊗ b{F },T )
+�nΩ−1
+�
+k=0
+Ω{k} ⊗ A{F }c{F }×k
+�
+c{S}
+= eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+c{S}.
+
+10
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+Theorem 2.6 (Fourth Order Conditions). An IMEX-MRI-SR method satisfying
+Theorem 2.5 is fourth-order accurate if the slow base method is fourth-order accurate,
+the arbitrary fast method is order max(4, nΩ+2) accurate, and the following conditions
+hold:
+eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 3)
+�
+c{S} = 1
+8,
+(2.15a)
+eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+C{S}c{S} = 1
+12,
+(2.15b)
+eT
+s{S}ΓC{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+c{S} = 0,
+(2.15c)
+eT
+s{S}ΩC{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+c{S} = 1
+24,
+(2.15d)
+eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+Ωc{S} = 1
+24,
+(2.15e)
+eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+Γc{S} = 0.
+(2.15f)
+Proof. From [3], there are 26 fourth-order coupling conditions for a third-order
+GARK method with this structure; this further reduces to 21 when A{F,E} = A{F,I}.
+These GARK coupling conditions are:
+b{σ},T C{S}A{S,F }c{F } = 1
+8
+(2.16a)
+b{σ},T A{S,ν}A{S,F }c{F } = 1
+24
+(2.16b)
+b{σ},T A{S,F }C{F }c{F } = 1
+12
+(2.16c)
+b{σ},T A{S,F }A{F,F }c{F } = 1
+24
+(2.16d)
+b{F },T C{F }A{F,σ}c{S} = 1
+8
+(2.16e)
+b{F },T A{F,σ}C{S}c{S} = 1
+12
+(2.16f)
+b{σ},T A{S,F }A{F,ν}c{S} = 1
+24
+(2.16g)
+b{F },T A{F,F }A{F,σ}c{S} = 1
+24
+(2.16h)
+b{F },T A{F,σ}A{S,ν}c{S} = 1
+24
+(2.16i)
+b{F },T A{F,σ}A{S,F }c{F } = 1
+24
+(2.16j)
+for σ, ν ∈ {I, E}. c{F } and c{S} are defined as in (2.14a) and (2.14b), respectively.
+We arrive at the conditions (2.15) by checking each of the conditions (2.16) in turn.
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+11
+The first four conditions (2.16a)-(2.16d) are automatically satisfied:
+1
+8 = b{σ},T C{S}A{S,F }c{F }
+= b{σ},T C{S}(C{S} ⊗ b{F },T )(c{S} ⊗ c{F })
+= b{σ},T C{S}C{S}c{S} 1
+2 = 1
+4 · 1
+2,
+1
+24 = b{σ},T A{S,ν}A{S,F }c{F }
+= b{σ},T A{ν}(C{S} ⊗ b{F },T )(c{S} ⊗ c{F })
+= b{σ},T A{ν}C{S}c{S} 1
+2 = 1
+12 · 1
+2,
+1
+12 = b{σ},T A{S,F }C{F }c{F }
+= b{σ},T (C{S} ⊗ b{F },T ) diag(c{S} ⊗ c{F })(c{S} ⊗ c{F })
+= b{σ},T C{S}C{S}c{S} 1
+3 = 1
+4 · 1
+3,
+1
+24 = b{σ},T A{S,F }A{F,F }c{F }
+= b{σ},T (C{S} ⊗ b{F },T )(C{S} ⊗ A{F })(c{S} ⊗ c{F })
+= b{σ},T C{S}C{S}c{S} 1
+6 = 1
+4 · 1
+6.
+Condition (2.16e) is not automatically satisfied and simplifies to (2.15a):
+1
+8 = b{F },T C{F }A{F,σ}c{S}
+= (es{S} ⊗ b{F })T diag(c{S} ⊗ c{F })
+�nΩ−1
+�
+k=0
+Ω{k} ⊗ A{F }c{F }×k
+�
+c{S}
+= eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 3)
+�
+c{S}.
+Condition (2.16f) is not automatically satisfied and simplifies to (2.15b).
+1
+12 = b{F },T A{F,σ}C{S}c{S}
+= (es{S} ⊗ b{F })T
+�nΩ−1
+�
+k=0
+Ω{k} ⊗ A{F }c{F }×k
+�
+C{S}c{S}
+= eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+C{S}c{S}.
+Condition (2.16g) is not automatically satisfied and reduces to (2.15c) and (2.15d)
+when σ = I, E, respectively, and both conditions are enforced simultaneously,
+1
+24 = b{σ},T A{S,F }A{F,ν}c{S}
+= b{σ},T (C{S} ⊗ b{F })
+�nΩ−1
+�
+k=0
+Ω{k} ⊗ A{F }c{F }×k
+�
+c{S}
+
+12
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+= b{σ},T C{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+c{S}.
+Condition (2.16h) is not automatically satisfied but simplifies to (2.12) minus
+(2.15a),
+1
+24 = b{F },T A{F,F }A{F,σ}c{S}
+= (es{S} ⊗ b{F })T (C{S} ⊗ A{F })
+�nΩ−1
+�
+k=0
+Ω{k} ⊗ A{F }c{F }×k
+�
+c{S}
+= eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)(k + 3)
+�
+c{S}.
+Condition (2.16i) is not automatically satisfied and reduces to (2.15e) and (2.15f)
+when ν = I, E, respectively, and both conditions are enforced simultaneously:
+1
+24 = b{F },T A{F,σ}A{S,ν}c{S}
+= (es{S} ⊗ b{F })T
+�nΩ−1
+�
+k=0
+Ω{k} ⊗ A{F }c{F }×k
+�
+A{S,ν}c{S}
+= eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+A{S,ν}c{S}.
+Finally, condition (2.16j) simplifies to the same condition as (2.15b),
+1
+24 = b{F },T A{F,σ}A{S,F }c{F }
+= (es{S} ⊗ b{F })
+�nΩ−1
+�
+k=0
+Ω{k} ⊗ A{F }c{F }×k
+�
+(C{S} ⊗ b{F })(c{S} ⊗ c{F })
+= eT
+s{S}
+�nΩ−1
+�
+k=0
+Ω{k}
+1
+(k + 1)(k + 2)
+�
+C{S}c{S} 1
+2.
+Theorem 2.7 (Minimum nΩ for third-order accuracy). An IMEX-MRI-SR re-
+quires at least nΩ = 2 for third-order accuracy.
+Proof. An IMEX-MRI-SR method with one Ω matrix, Ω{0}, has A{S,E} = Ω = Ω0
+and, from the FSAL property, b{E},T = eT
+s{S}Ω = eT
+s{S}Ω{0}. For an IMEX-MRI-SR
+method of order p, we assume the slow-explicit base method also satisfies all standard
+Runge–Kutta order conditions up to order p. Thus, a third-order IMEX-MRI-SR slow-
+explicit base method with one Ω satisfies the second order condition b{E},T c{S} = 1
+2,
+which simplifies to
+1
+2 =
+s{S}
+�
+j=1
+ω{0}
+s{S},jc{S}
+j
+.
+However, the third-order IMEX-MRI-SR coupling condition (2.12) simplifies to
+1
+3 =
+s{S}
+�
+j=1
+ω{0}
+s{S},jc{S}
+j
+.
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+13
+Since these cannot hold simultaneously, a third-order IMEX-MRI-SR method with one
+Ω matrix is not possible. There are no such mutually exclusive conditions for nΩ > 1,
+and in Section 4 we introduce third and a fourth order methods with nΩ = 2.
+3. Linear Stability. As in [3, 22], we analyze the stability of IMEX-MRI-SR
+methods when applied to the linear, scalar test problem
+y′(t) = λ{F }y + λ{E}y + λ{I}y,
+t ≥ 0,
+y(0) = 1,
+(3.1)
+with λ{F }, λ{E}, λ{I} ∈ C−. For convenience, we additionally define z{F } = Hλ{F },
+z{E} = Hλ{E}, and z{I} = Hλ{I}. When applied to (3.1), the IVP (1.5b) becomes
+v′
+i(θ) = λ{F }vi(θ) +
+1
+c{S}
+i
+i−1
+�
+j=1
+ωi,j
+�
+θ
+c{S}
+i
+H
+�
+(λ{E}Y {S}
+j
++ λ{I}Y {S}
+j
+)
+= λ{F }vi(θ) +
+1
+c{S}
+i
+i−1
+�
+j=1
+nΩ−1
+�
+k=0
+ω{k}
+i,j
+�
+θ
+c{S}
+i
+H
+�k
+(λ{E}Y {S}
+j
++ λ{I}Y {S}
+j
+)
+for i = 2, ..., s{S}, with θ ∈ [0, c{S}
+i
+H] and vi(0) = yn.
+The solution to this at
+θ = c{S}
+i
+H is
+vi(c{S}
+i
+H) = ec{S}
+i
+z{F }yn + (z{E} + z{I})
+i−1
+�
+j=1
+nΩ−1
+�
+k=0
+ω{k}
+i,j
+�� 1
+0
+ec{S}
+i
+z{F }(1−t)tkdt
+�
+Y {S}
+j
+= ϕ0(c{S}
+i
+z{F })yn + (z{E} + z{I})
+i−1
+�
+j=1
+ηi,j(z{F })Y {S}
+j
+.
+Here we define η as in [3] as a function of the fast eigenvalue z{F },
+ηi,j(z{F }) =
+nΩ−1
+�
+k=0
+ω{k}
+i,j ϕk+1(c{S}
+i
+z{F }),
+(3.2)
+where the family of analytical functions {ϕk} are given by [22],
+ϕ0(z) = ez,
+ϕk(z) =
+� 1
+0
+ez(1−t)tk−1dt,
+k ≥ 1.
+The ith IMEX-MRI-SR stage for the stability problem (3.1) then becomes
+Y {S}
+i
+= ϕ0(c{S}
+i
+z{F })yn + (z{E} + z{I})
+i−1
+�
+j=1
+ηi,j(z{F })Y {S}
+j
++ z{I}
+i
+�
+j=1
+γi,jY {S}
+j
+.
+(3.3)
+Concatenating Y = [Y {S}
+1
+· · · Y {S}
+s{S}]T and writing (3.3) in matrix form,
+Y = ϕ0(c{S}z{F })yn + (z{E} + z{I})η(z{F })Y + z{I}ΓY
+=
+�
+I − (z{E} + z{I})η(z{F }) − z{I}Γ
+�−1
+ϕ0(c{S}z{F })yn
+
+14
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+where
+η(z{F }) =
+nΩ−1
+�
+k=0
+diag(ϕk+1(c{S}z{F }))Ω{k}.
+Thus the stability function for an IMEX-MRI-SR method applied to (3.1) is
+R(z{F },z{E}, z{I}) =
+eT
+s{S}
+�
+I − (z{E} + z{I})η(z{F }) − z{I}Γ
+�−1
+ϕ0(c{S}z{F }).
+(3.4)
+We consider a few definitions of joint stability for IMEX-MRI-SR methods. As
+with IMEX-MRI-GARK methods, we consider a region that incorporates all three z,
+Jα,ρ,β,ξ = {z{E} ∈ C− : |R(z{F }, z{E}, z{I})| ≤ 1, ∀z{F } ∈ S{F }
+α,ρ , ∀z{I} ∈ S{I}
+β,ξ },
+(3.5)
+where S{σ}
+α,ρ = {z{σ} ∈ C− : | arg(z{σ}) − π| ≤ α, |z{σ}| ≤ ρ}. We note that this
+definition of joint stability may be overly-restrictive, as it demands the implicit and
+fast parts of the method to be A(β)- and A(α)-stable, respectively, for any joint
+stability region to exist.
+Thus, we also analyze the implicit and explicit stability
+regions of IMEX-MRI-SR methods independently, with stability regions defined as:
+J {I}
+α,ρ = {z{I} ∈ C− : |R(z{F }, 0, z{I})| ≤ 1, ∀z{F } ∈ S{F }
+α,ρ },
+(3.6)
+J {E}
+α,ρ = {z{E} ∈ C− : |R(z{F }, z{E}, 0)| ≤ 1, ∀z{F } ∈ S{F }
+α,ρ }.
+(3.7)
+These are more consistent with standard stability analyses of ARK methods, wherein
+explicit and implicit stability are considered separately.
+4. Example Methods. We introduce three IMEX-MRI-SR methods of orders
+2, 3 and 4, each of which includes an embedding with accuracy one order lower (1, 2
+and 3, resp.) for temporal error estimation. We note that we designed the embedding
+coefficients with efficiency in mind, so that computation of the embedded solutions
+do not require an additional implicit solve at the slow time scale (i.e., ˆγs{S} = 0).
+When presenting the coefficients for each method we use the notation
+Ω{k} =
+�
+�Ω{k}
+ˆω{k}
+�
+� , Γ =
+�
+�Γ
+ˆγ
+�
+� ,
+where Ω{k} and Γ are the matrices of coefficients defining the primary method, and
+ˆω{k} and ˆγ are the embedding row of coefficients.
+4.1. IMEX-MRI-SR2(1). The first method is second-order with a first-order
+embedding. It has 4 stages, nΩ = 1, and requires 3 slow nonlinear solves per step.
+The coefficients can be found in Appending A.
+To create this method, we used the free coefficients of the primary method to
+maximize the size of the stability region defined by (3.5). We used the free coefficients
+of the embedded method to minimize the value of the C-statistic from Prince and
+Dormand [18],
+C(p+1) = ∥ˆτ (p+1) − τ (p+1)∥2
+∥ˆτ (p)∥2
+,
+(4.1)
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+15
+where τ (p) is the vector of order condition residuals at order p for the primary method,
+ˆτ (p) is the vector of order condition residuals at order p for the embedded method, and
+p is the order of the primary method (in this case, p = 2). The C-statistic gives an
+estimate of how much the error stemming from unsatisfied (p + 1)-order conditions of
+the primary method corrupts the error estimate provided by the embedded method.
+This method has large, robust stability regions. Figures 1a and 1b show Jα,102,β,104
+for α ∈ {10◦, 45◦} and varying β, along with the explicit slow base method’s stability
+region. We see that the multirate method has a stability region essentially identical
+to the explicit slow base method when α = 10◦ for any value of z{I}. When α grows
+to 45◦ we can see a decay of the stability region associated with the β = 85◦, while
+the regions associated with smaller β values experience negligible decay, if any.
+Figure 1c shows J {E}
+α,102 for varying α. We see that when z{I} = 0, the stability
+region is again large, with only a slight decay in area when α = 85◦.
+Figure 1c shows J {I}
+α,102 for varying α. When z{E} = 0, the multirate method is
+A-stable for 0◦ ≤ α ≤ 45◦. When α grows further to 65◦, the region decays slightly
+but is still approximately A(80◦)-stable. When α = 85◦, the stability region decays
+to an enclosed bubble similar to the regions in Figure 1c.
+(a) J10◦,102,α,104
+(b) J45◦,102,α,104
+(c) J {E}
+α,102
+(d) J {I}
+α,102
+Fig. 1: Joint Stability Regions for IMEX-MRI-SR2(1)
+
+4
+2
+Explicit
+β=10°
+0
+β=45°
+-2
+β=65°
+β=85°
+4
+-6
+-4
+-2
+0
+Re4
+2
+Explicit
+β=10°
+0
+β=45°
+-2
+β=65°
+β=85°
+.4
+-6
+-4
+-2
+0
+Re4
+2
+Base
+0
+α=10°
+α=45°
+-2
+α=65°
+α=85°
+.4
+-6
+-4
+-2
+0
+Re6
+4
+2
+Base
+m
+0
+α=10°
+-2
+α=45°
+α=65°
+-4
+α=85°
+-6
+-10-8
+-6
+-4
+-2
+0
+2
+Re16
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+4.2. IMEX-MRI-SR3(2). Our second method is third-order with a second-
+order embedding. It has 5 stages, nΩ = 2, and requires 4 nonlinear solves per step.
+The coefficients can be found in Appendix B.
+To create this method, we used the free coefficients for both the method and
+embedding as described in Section 4.1, this time using the C-statistic (4.1) with p = 3.
+Figures 2a and 2b show the relatively large joint stability regions Jα,102,β,104
+for α = {10◦, 45◦}, respectively, along with the explicit slow base method’s stability
+region. Again, at α = 10◦ and lower values of β, the method is essentially as stable as
+the explicit slow base method. The areas of these stability regions decrease for higher
+values of β. When α grows to 45◦ in Figure 2b, the stability regions shrink slightly
+in comparison with α = 10◦ from Figure 2a.
+Figures 2c and 2d show J {E}
+α,102 and J {I}
+α,102 for varying α, respectively. We see that
+when z{I} = 0, the stability region is reasonably large for smaller value of α, but the
+region shrinks as α grows; however, even for α = 85◦, the region retains a good extent
+along the imaginary axis. Similar to the second-order method, when z{E} = 0, the
+stability region is A-stable for most values of α, only losing A-stability for α = 85◦,
+where the region decays to a rather large, yet enclosed, bubble.
+(a) J10◦,102,β,104
+(b) J45◦,102,β,104
+(c) J {E}
+α,102
+(d) J {I}
+α,102
+Fig. 2: Joint Stability Regions for IMEX-MRI-SR3(2)
+
+4
+2
+Explicit
+β=10°
+0
+β=45°
+-2
+β=65°
+β=85°
+-4
+-6
+-4
+-2
+0
+Re4
+2
+Explicit
+β=10°
+0
+β=45°
+-2
+β=65°
+β=85°
+-4
+-6
+-4
+-2
+0
+Re4
+2
+Base
+α=10°
+0
+α=45°
+-2
+α=65°
+α=85°
+-4
+-6
+-4
+-2
+0
+Re30
+20
+10
+Base
+m
+0
+α=10°
+-10
+α=45°
+-20
+α=65°
+α=85°
+-30
+-50-40-30-20-10
+0
+10
+ReIMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+17
+4.3. IMEX-MRI-SR4(3). Our final method is fourth-order with a third-order
+embedding, has 7 stages, nΩ = 2, and requires 5 nonlinear solves per step.
+The
+coefficients can be found in Appendix C.
+Due to the number of order conditions involved when simultaneously solving all
+IMEX-MRI-SR coupling conditions and base ARK order conditions at fourth-order,
+we based this method off of an existing 4(3) ARK method, LIRK4 [2]. Since this was
+not stiffly accurate, then as discussed in Remark 2.2 we padded the A{E} and A{I}
+matrices with the b vector, i.e.,
+(4.2)
+A{S,E} =
+�
+�A{E}
+0{6}
+bT
+0
+�
+� , A{S,I} =
+�
+�A{I}
+0{6}
+bT
+0
+�
+� ,
+where 0{6} ∈ R6 is all zero.
+Since the last row of A{S,E} and A{S,I} both equal bT , the last row of Γ equals
+zero, as follows from the definitions (2.7b) and (2.7c). When A{S,I} has a zero in
+the bottom right entry (and therefore Γ has a zero in the bottom right entry, from
+(2.7c)), there is no nonlinear solve required to compute this last stage and therefore
+the updated time step solution. We believe that this negatively affects stability, as
+we will show in Figure 3.
+As before, we used the free variables of the IMEX-MRI-SR method to optimize
+stability. This method had an empty joint stability region defined by (3.5) for all of
+our attempts to choose or optimize values of the free variables, so we instead optimized
+the size of the stability region defined by (3.7). To optimize the embedded method,
+we minimized the 2-norm of the fourth-order condition residuals, ∥ˆτ (4)∥2, to reduce
+the overall error in the embedded solution. We note that our previous approach of
+minimizing the C-statistic (4.1) was not possible since we do not yet have the fifth-
+order IMEX-MRI-SR coupling conditions.
+Figure 3a shows the stability regions J {E}
+α,1
+for varying α. Unlike the lower-order
+methods, this method’s explicit stability region never fully matches that of the explicit
+base method. Similarly to the other methods, as α grows the stability region shrinks.
+Figure 3b shows the stability regions J {I}
+α,1 for varying α. Again unlike the lower-
+order methods, these regions are not A-stable for α ̸= 0.
+Because J {I}
+α,ρ is never
+A-stable for this method, the stability region (3.5) is always empty. We suspect that
+this is primarily due to the lack of an implicit solve in the last stage of the method.
+In future work we plan to investigate this issue in more detail to better understand
+the factors that contribute to IMEX-MRI-SR joint stability.
+5. MERK Methods as Explicit IMEX-MRI-SR Methods. In [13], Luan
+et al. define MERK methods with orders of accuracy spanning two through five by
+explicitly defining the abscissae c{S}
+i
+and the forcing functions gi(θ). Due to their
+similar structure to IMEX-MRI-SR methods, we may analyze MERK methods using
+our theory from Section 2. Because MERK methods are always explicit, their IMEX-
+MRI-SR Γ matrices will be all zero. We recall that MERK methods are defined under
+an assumption that the fast function is linear, f {F }(t, y) = Ly, but that the slow
+function can be arbitrary. It is thus natural to assume that MERK methods might
+only satisfy a subset of the IMEX-MRI-SR order conditions, potentially failing those
+that handle nonlinearity in f {F }.
+
+18
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+(a) J {E}
+α,102
+(b) J {I}
+α,102
+Fig. 3: Joint Stability Regions for IMEX-MRI-SR4(3)
+5.1. MERK2. Converting the three-stage second-order accurate MERK2 method
+from [13] into IMEX-MRI-SR form, we have
+Ω{0} =
+�
+����
+0
+0
+0
+c{S}
+2
+0
+0
+1
+0
+0
+�
+���� ,
+Ω{1} =
+�
+����
+0
+0
+0
+0
+0
+0
+−1/c{S}
+2
+1/c{S}
+2
+0
+�
+���� .
+(5.1)
+Interestingly, these coefficients (along with Γ = 0) satisfy all coupling conditions up
+to third order, and the slow base method determined by Ω satisfies all conditions
+up to order two. Thus, we expect MERK2 to have second-order accuracy, even for
+nonlinear f {F }. We confirm this with numerical tests involving nonlinear f {F } in
+Section 6, where we use c{S}
+2
+= 1
+2 since it was unspecified in [13].
+Figure 4a shows the stability regions for MERK2. Because this method is explicit,
+we only plot J {E}
+α,102. Notably, these regions nearly match that of the base explicit
+method for most values of α examined.
+5.2. MERK3. Converting the four-stage, third-order accurate, MERK3 from
+[13] to an IMEX-MRI-SR method, we have
+Ω{0} =
+�
+�������
+0
+0
+0
+0
+c{S}
+2
+0
+0
+0
+2
+3
+0
+0
+0
+1
+0
+0
+0
+�
+�������
+,
+Ω{1} =
+�
+�������
+0
+0
+0
+0
+0
+0
+0
+0
+−
+2
+3c{S}
+2
+2
+3c{S}
+2
+0
+0
+− 3
+2
+0
+3
+2
+0
+�
+�������
+.
+(5.2)
+These satisfy all coupling conditions up to third order, and the corresponding slow
+base method satisfies all conditions up to order three. Thus similarly to MERK2, we
+expect it to show third-order accuracy on problems with nonlinear f {F }. We again
+confirm this result on numerical tests in Section 6 using c{S}
+2
+= 1/2.
+Figure 4b shows the stability regions for MERK3, J {E}
+α,102. Notably, these regions
+show no degradation of stability as α is increased.
+
+4
+2
+Base
+0
+α=10°
+α=45°
+-2
+α=65°
+α=85°
+-4
+-6
+-4
+-2
+0
+Re4
+2
+Base
+0
+α=10°
+α=45°
+-2
+α=65°
+α=85°
+-4
+-6
+-4
+-2
+0
+ReIMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+19
+(a) MERK2
+(b) MERK3
+Fig. 4: J {E}
+α,102 Regions for MERK2 and MERK3
+5.3. MERK4. We may express MERK4 as an IMEX-MRI-SR method with 7
+stages and nΩ = 3. The corresponding coefficients are provided in Appendix D.
+We find that this method satisfies all IMEX-MRI-SR coupling conditions through
+order four, and its slow base method satisfies all order conditions up through fourth
+order, so long as c{S}
+6
+= (3−4c{S}
+5
+)/(4−6c{S}
+5
+). This restriction on c{S}
+6
+is satisfied by
+the choices in [13] of c{S}
+2
+= 1/2, c{S}
+3
+= 1/2, c{S}
+4
+= 1/3, c{S}
+5
+= 5/6, and c{S}
+6
+= 1/3.
+Thus like before, we expect this method to demonstrate fourth-order accuracy for
+nonlinear f {F }, which we confirm numerically in Section 6. Interestingly, the joint
+stability regions J {E}
+α,102 for MERK4 are empty. However, given the reliability of this
+method in [13] and our own results from Section 6, we believe that these empty
+regions more strongly indicate a deficiency in these definitions of joint stability than
+any actual issues with MERK4 itself.
+5.4. MERK5. The IMEX-MRI-SR method that corresponds with MERK5 has
+11 stages and nΩ = 4. These coefficients are given in Appendix E. When analyzing
+this method, we find that when
+c{S}
+9
+= 12 − 15c{S}
+10
+− 15c{S}
+8
++ 20c{S}
+10 c{S}
+8
+15 − 20c{S}
+10
+− 20c{S}
+8
++ 30c{S}
+10 c{S}
+8
+the method satisfies all IMEX-MRI-SR coupling conditions up through fourth-order,
+and its slow base method satisfies all order conditions up through fifth-order. This
+condition is satisfied by the choice in [13] of c{S}
+2
+= 1/2, c{S}
+3
+= 1/2, c{S}
+4
+= 1/3,
+c{S}
+5
+= 1/2, c{S}
+6
+= 1/3, c{S}
+7
+= 1/4, c{S}
+8
+= 7/10, c{S}
+9
+= 1/2, and c{S}
+10
+= 2/3,
+therefore we expect it to have at least fourth-order accuracy without restriction on
+the linearity of f {F }. As we do not have fifth-order IMEX-MRI-SR order conditions,
+we cannot check these for MERK5; however, our numerical tests in Section 6 indeed
+show fifth-order convergence for nonlinear f {F }. As with MERK4, the MERK5 joint
+stability regions J {E}
+α,102 are empty.
+6. Numerical Results. In this section we examine the convergence rates for
+our newly-proposed IMEX-MRI-SR methods from Section 4, as well as for MERK
+
+4
+2
+Base
+0
+α=10°
+α=45°
+-2
+α=65°
+α=85°
+.4
+-6
+-4
+-2
+0
+Re4
+2
+Base
+0
+α=10°
+α=45°
+-2
+α=65°
+α=85°
+-6
+-4
+-2
+0
+Re20
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+methods applied to problems with nonlinear f {F }. We also compare the efficiency of
+our IMEX-MRI-SR methods against both IMEX-MRI-GARK and legacy Lie-Trotter
+and Strang-Marchuk methods from [3].
+6.1. KPR. The Kværnø-Prothero-Robinson problem is a coupled system of
+IVPs which has been widely used for testing multirate algorithms, since it is non-
+linear, non-autonomous, includes stiffness and multirate tuning parameters, and has
+an analytical solution. We use the same formulation and partitioning as in [3],
+�
+�u′
+v′
+�
+� = Λ
+�
+�
+−3+u2−cos(βt)
+2u
+−2+u2−cos(t)
+2v
+�
+� −
+�
+�
+β sin(βt)
+2u
+sin(t)
+2v
+�
+� ,
+�
+�u(0)
+v(0)
+�
+� =
+�
+� 2
+√
+3
+�
+�
+Λ =
+�
+�
+λ{F }
+1−ε
+α (λ{F } − λ{S})
+−αε(λ{F } − λ{S})
+λ{S}
+�
+�
+(6.1)
+for t ∈ [0, 5π/2], with parameters λ{F } = −10, λ{S} = −1, ε = 0.1, α = 1, and
+β = 20. This problem has solution
+u(t) =
+�
+3 + cos(βt),
+v(t) =
+�
+2 + cos(t).
+(6.2)
+We partition the problem as
+f {E} =
+�
+�
+0
+sin(t)
+2v
+�
+� ,
+f {I} =
+�
+�0
+0
+0
+1
+�
+� Λ
+�
+�
+−3+u2−cos(βt)
+2u
+−2+u2−cos(t)
+2v
+�
+�
+f {F } =
+�
+�1
+0
+0
+0
+�
+� Λ
+�
+�
+−3+u2−cos(βt)
+2u
+−2+u2−cos(t)
+2v
+�
+� .
+(6.3)
+In Figure 5, we plot the convergence as H is refined for MERK methods and
+implicit-explicit methods, including all provided IMEX-MRI-SR methods from Sec-
+tion 4, IMEX-MRI-GARK3(a,b), IMEX-MRI-GARK4, Lie-Trotter and Strang-Marchuk.
+We see that all methods converge at their expected rates. Notably, as expected from
+Section 5, the MERK methods show no convergence issues even though f {F } in (6.3)
+is nonlinear.
+In these tests, we combined multirate methods with explicit inner Runge–Kutta
+methods of the same order, with the only exception of pairing a second-order inner
+solver with the first-order Lie-Trotter method. Each of Lie-Trotter, Strang-Marchuk,
+and IMEX-MRI-SR2(1) used the second-order Heun method given by the Butcher
+table
+0
+0
+0
+1
+1
+0
+1/2
+1/2
+. The IMEX-MRI-SR3(2) and IMEX-MRI-GARK3(a,b) methods
+used the third-order method by Bogacki and Shampine [1]. The IMEX-MRI-SR4(3)
+and IMEX-MRI-GARK4 methods used the fourth-order method by Zonneveld [27].
+We measured error at 10 equally-spaced points in the time interval. The estimated
+convergence rates for each method, using a least squares fit of log(Max Error) versus
+log(H), are in the legend parentheses. For the MERK convergence tests, we used
+H = π/2k, k = 2, ..., 9 and for the implicit-explicit method convergence tests, we used
+H = π/2k, k = 4, ..., 11. For all tests we used fast time step size h = H/10. The
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+21
+implicit-explicit methods used a standard Newton-Raphson method with a banded
+linear solver for the implicit solves.
+10
+2
+10
+1
+100
+H
+10
+13
+10
+11
+10
+9
+10
+7
+10
+5
+10
+3
+10
+1
+Max Error
+MERK2 (2.06)
+MERK3 (3.17)
+MERK4 (4.25)
+MERK5 (5.32)
+10
+2
+10
+1
+H
+10
+11
+10
+9
+10
+7
+10
+5
+10
+3
+10
+1
+Max Error
+Lie-Trotter (1.01)
+Strang-Marchuk (1.98)
+IMEX-MRI-SR2(1) (2.01)
+IMEX-MRI-SR3(2) (3.11)
+IMEX-MRI-SR4(3) (3.85)
+IMEX-MRI-GARK3a (3.05)
+IMEX-MRI-GARK3b (3.09)
+IMEX-MRI-GARK4 (4.11)
+Fig. 5: Convergence for the KPR test problem (6.1) for MERK methods (left) and
+implicit-explicit methods (right) using the partitioning (6.3). All methods converge
+at the expected theoretical rates (with measured convergence rates in parentheses),
+including MERK methods using the given nonlinear fast partition.
+6.2. Stiff Brusselator. The stiff brusselator problem is an advection-reaction-
+diffusion system of nonlinear partial differential equations. It is a modification to
+the standard brusselator [7] used in [3], from which we use the same formulation and
+partitioning:
+ut = αuuxx + ρuux + ru(a − (w − 1)u + u2v),
+vt = αvvxx + ρvvx + rv(wu − u2v),
+wt = αwwxx + ρwwx + rw(b − w
+ε
+− wu),
+for t ∈ [0, 3] and x ∈ [0, 1], with stationary boundary conditions
+ut(t, 0) = ut(t, 1) = vt(t, 0) = vt(t, 1) = wt(t, 0) = wt(t, 1),
+and initial conditions
+u(0, x) = a + 0.1 sin(πx),
+v(0, x) = b/a + 0.1 sin(πx),
+w(0, x) = b + 0.1 sin(πx).
+We partition the problem as
+f {I} =
+�
+����
+αuuxx
+αvvxx
+αwwxx
+�
+���� ,
+f {E} =
+�
+����
+ρuux
+ρvvx
+ρwwx
+�
+���� ,
+f {F } =
+�
+����
+ru(a − (w − 1)u + u2v)
+rv(wu − u2v)
+rw( b−w
+ε
+− wu)
+�
+���� ,
+and use second order centered difference approximations for all spatial derivative
+operators. This problem has no analytical solution, so we used MATLAB’s ode15s
+with tight tolerances AbsTol = 10−14 and RelTol = 2.5 × 10−14 to generate reference
+solutions.
+
+22
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+6.2.1. Fixed Time Step. In this section, we compare runtime efficiency of the
+splitting, IMEX-MRI-SR, and IMEX-MRI-GARK methods using fixed time step sizes.
+We use the same fixed parameters αu = αv = αw = 10−2, ρu = ρv = ρw = 10−3,
+ru = rv = rw = 1, a = 0.6, b = 2, and ε = 10−2 with initial conditions
+u(0) = a + 0.1 sin(πx), v(0) = b/a + 0.1 sin(πx), w(0) = b + 0.1 sin(πx),
+for 201 and 801 grid points as in [3]. All methods used fast time steps of h = H/10
+and all methods used the same inner methods and implicit algebraic solvers as in
+Section 6.1.
+100
+101
+102
+103
+Runtime (s)
+10
+10
+10
+8
+10
+6
+10
+4
+10
+2
+Max Error
+201 grid
+101
+102
+103
+104
+Runtime (s)
+10
+10
+10
+8
+10
+6
+10
+4
+10
+2
+Max Error
+801 grid
+Lie-Trotter
+Strang-Marchuk
+IMEX-MRI-SR2(1)
+IMEX-MRI-SR3(2)
+IMEX-MRI-SR4(3)
+IMEX-MRI-GARK3a
+IMEX-MRI-GARK3b
+IMEX-MRI-GARK4
+Fig. 6:
+Efficiency for stiff brusselator problem using 201 grid points (left)
+and 801 grid points (right).
+Estimated least-squares convergence rates be-
+fore settling at the error-floor are (0.76,1.59,2.00,3.09,3.00,3.36,3.25,3.41) and
+(0.72,1.24,2.01,2.90,1.90,2.71,2.69,2.42) for the 201 and 801 grids, respectively, for
+Lie-Trotter, Strang-Marchuk, IMEX-MRI-SR2(1), IMEX-MRI-SR3(2), IMEX-MRI-
+SR4(3), IMEX-MRI-GARK3a, IMEX-MRI-GARK3b, IMEX-MRI-GARK4.
+In Figure 6 we plot the observed maximum solution error over ten equally spaced
+points in the time interval using step sizes of H = 0.1·2−k, k = 0, ..., 10. Both splitting
+methods and both fourth-order methods experience significant order reduction, with
+IMEX-MRI-SR4(3) taking the biggest hit in reducing an entire order of accuracy for
+the 201 grid, and two orders of accuracy for the 801 grid. The second- and third-order
+IMEX-MRI-SR and IMEX-MRI-GARK methods all achieve their expected order for
+the 201 grid. The third-order accurate methods experience only slight order reduction
+for the 801 grid while IMEX-MRI-SR2(1) remains steady at its expected order. All
+methods exhibit an error floor of approximately 10−11, that is likely caused by the
+accuracy of the reference solution.
+The stiffness of this problem highlights the stability limitations of the fourth
+order methods, IMEX-MRI-SR4(3) and IMEX-MRI-GARK4, which was observed in
+[3]. For the 201 grid, IMEX-MRI-SR4(3) and IMEX-MRI-GARK4 were unstable for
+step sizes greater than 1/320 and 1/80 respectively. For the 801 grid, IMEX-MRI-
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+23
+SR4(3) and IMEX-MRI-GARK4 were unstable for step sizes greater than 1/640 and
+1/160, respectively.
+We can see that IMEX-MRI-SR2(1) is far more efficient than the first- and second-
+order splitting methods, providing errors two to three orders of magnitude smaller for
+the same runtimes. It also has a steady rate of error decrease as runtime increases,
+while the splitting methods show periods of stagnation at larger step sizes.
+The third-order and fourth-order methods tend to have similar efficiency on this
+problem, attaining similar error for similar runtimes. The third-order IMEX-MRI-
+GARK methods have a slight edge for the 201 grid and for some error ranges in the
+801 grid, but IMEX-MRI-SR3(2) becomes the most efficient method for the 801 grid
+for errors between approximately 10−7 and 10−11. The third-order methods are all
+more efficient than the fourth-order methods, where IMEX-MRI-SR4(3) maintains a
+slight but consistent edge over IMEX-MRI-GARK4.
+6.2.2. Adaptive Time Step. In this section, we compare work-precision effi-
+ciency for the IMEX-MRI-SR methods in the adaptive time step context. Because
+the splitting and IMEX-MRI-GARK methods do not have embeddings, we omit them
+from these tests. We use the Constant-Constant controller from [5] with the recom-
+mended parameters k1 = 0.42, k2 = 0.44, and the recommended fast error measure-
+ment strategy, LASA-mean. This controller adapts H and M, such that the inner
+time step size h = H/M, in a similar manner to a standard I-controller.
+We use time-varying parameters αu = αv = αw = ρu = ρv = ρw = 6 × 10−5 +
+5 × 10−5 cos(πt), ru = rv = rw = 0.6 + 0.5 cos(4πt) adapted from [5] with the same
+fixed parameters a = 1, b = 3.5, ε = 10−3 and initial conditions
+u(0) = 1.2 + 0.1 sin(πx), v(0) = 3.1 + 0.1 sin(πx), w(0) = 3 + 0.1 sin(πx),
+with 101 grid points. All IMEX-MRI-SR methods used the same inner methods and
+implicit algebraic solvers as in Section 6.1.
+105
+106
+107
+Fast Function Evaluations
+10
+10
+10
+9
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+Max Error
+102
+103
+104
+105
+Implicit Solves
+10
+10
+10
+9
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+Max Error
+IMEX-MRI-SR2(1)
+IMEX-MRI-SR3(2)
+IMEX-MRI-SR4(3)
+Fig. 7: Fast function evaluations (left) and total implicit solves (right) versus the
+observed maximum error for the stiff brusselator problem.
+In Figure 7 we plot the number of fast function evaluations and the number of
+implicit solves, good indicators of overall cost at the fast and slow timescales, versus
+
+24
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+the observed maximum solution error over ten equally spaced points in the time
+interval when running with controller tolerance values of tol = 10−k, k = 1, ..., 9.
+We use tol{S} = tol{F } = 1
+2tol in our tests for simplicity. We note that IMEX-
+MRI-SR3(2) failed the tests with tol = 10−3, 10−5 and IMEX-MRI-SR4(3) failed the
+tests with tol = 10−4, 10−5 due to getting stuck in oscillations between successful
+and failed steps.
+We can see that IMEX-MRI-SR2(1) is much less efficient than the higher order
+methods in terms of fast function evaluations (farthest from the bottom-left corner),
+where IMEX-MRI-SR3(2) is generally the most efficient (closest to the bottom-left
+corner) across the range of errors.
+IMEX-MRI-SR4(3) is generally comparable to
+IMEX-MRI-SR3(2) in terms of fast function evaluations but occasionally gets “stuck,”
+providing approximately the same error value for different total fast function evalu-
+ations, depending on the value of tol. This is likely an indication that IMEX-MRI-
+SR4(3) has a low quality embedding which provides inaccurate error estimates, pos-
+sibly stemming from the embedding coefficients being defined by minimizing ∥ˆτ (4)∥2
+rather than the C-statistic.
+The total number of implicit solves is comparable across all three methods for
+errors larger than approximately 10−5. Below that, IMEX-MRI-SR3(2) and IMEX-
+MRI-SR4(3) are again comparable with IMEX-MRI-SR2(1) falling further behind.
+We see the same phenomenon of IMEX-MRI-SR4(3) getting “stuck” at certain error
+values, achieving the same error for varying total implicit solves.
+7. Conclusions. We introduce a new class of multirate time integration methods
+which builds off of previous work in IMEX-MRI-GARK and MERK methods, serving
+to improve various aspects of each. The proposed class of IMEX-MRI-SR methods
+are flexible, allowing IMEX treatment of the slow time scale while allowing the use
+of any viable IVP solver for the fast time scale. These methods remove the sorted
+abscissae requirement of IMEX-MRI-GARK methods since they start each internal
+stage at the beginning of the time step, thereby dramatically simplifying their order
+conditions and allowing introduction of embeddings. IMEX-MRI-SR methods can
+also be viewed as an extension to MERK methods, in that these allow implicitness at
+the slow time scale and nonlinearity at the fast time scale.
+The convergence theory of IMEX-MRI-SR methods leverages GARK theory [23],
+through which we established order conditions for methods of orders one through four.
+Due to their structural similarity to IMEX-MRI-SR methods, we leveraged these new
+order conditions to analyze the previously-proposed MERK methods without their
+restriction to linear fast partitions. Using this analytical framework, we provided the
+first theoretical justification that these MERK methods (at least up through fourth
+order) should retain their high orders of accuracy even on problems with nonlinear
+fast partitions.
+We analyzed joint stability as in [3], as well as in simplified implicit- or explicit-
+only senses of joint stability. With these reduced definitions of stability, we gain some
+insight into what happens when joint stability regions break down.
+Using this theoretical framework, we provided three new IMEX-MRI-SR methods.
+These included a second-order method with a first-order embedding, and a third-order
+method with a second-order embedding, both derived from scratch by solving all order
+conditions simultaneously. We additionally provided a fourth-order method with a
+third-order embedding, that we derived from the existing LIRK4 ARK method [2].
+We experimentally examined convergence for the three new methods and the ex-
+isting MERK methods on the KPR test problem, finding that they all converge at
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+25
+their expected orders of accuracy. We also provided experimental efficiency results on
+the stiff brusselator problem, a nonlinear advection-reaction-diffusion system of PDEs,
+where we found that our methods are competitive with IMEX-MRI-GARK methods
+and even surpass them in some cases.
+We also found that IMEX-MRI-SR2(1) is
+vastly more efficient than the similarly second-order Strang-Marchuk operator split-
+ting method. We found that in the context of adaptive time stepping, the third- and
+fourth-order IMEX-MRI-SR methods were comparable in work required for achieving
+a given error, while the second-order method lagged behind.
+More work remains to be done on IMEX-MRI-SR methods. Higher order condi-
+tions can be derived which, while tedious due to the number of conditions at higher
+orders, is tractable due to the use of Kronecker product identities. Further analy-
+sis should be done on the factors that most strongly affect joint stability and what
+conditions, if any, can be enforced to ensure a non-empty joint stability region. Ad-
+ditionally, more methods should be derived, in particular an embedded fourth-order
+method with improved joint stability over the method provided here.
+8. Acknowledgments. The authors would like to thank Vu Thai Luan for his
+help in gaining deeper understanding of MERK methods.
+We would also like to
+thank the SMU Center for Research Computing for use of the Maneframe2 computing
+cluster, where we performed all simulations reported in this work
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+ordinary differential equations, Applied Numerical Mathematics, 136 (2019), pp. 183–
+205, https://doi.org/10.1016/j.apnum.2018.10.007, https://doi.org/10.1016/j.apnum.2018.
+10.007.
+[12] A. N. Langville and W. J. Stewart, The kronecker product and stochastic automata net-
+works, Journal of Computational and Applied Mathematics, 167 (2004), pp. 429–447,
+
+26
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+https://doi.org/10.1016/j.cam.2003.10.010, https://doi.org/10.1016/j.cam.2003.10.010.
+[13] V. T. Luan, R. Chinomona, and D. R. Reynolds, A New Class of High-Order Methods
+for Multirate Differential Equations, SIAM Journal on Scientific Computing, 42 (2020),
+pp. A1245–A1268, https://doi.org/10.1137/19M125621X.
+[14] V. T. Luan, R. Chinomona, and D. R. Reynolds, Multirate exponential Rosenbrock methods,
+2021, https://arxiv.org/abs/2106.05385.
+[15] V. T. Luan and A. Ostermann, Explicit exponential runge–kutta methods of high order
+for parabolic problems, Journal of Computational and Applied Mathematics, 256 (2014),
+pp. 168–179, https://doi.org/10.1016/j.cam.2013.07.027, https://doi.org/10.1016/j.cam.
+2013.07.027.
+[16] V. T. Luan and A. Ostermann, Parallel exponential rosenbrock methods, Computers & Math-
+ematics with Applications, 71 (2016), pp. 1137–1150, https://doi.org/10.1016/j.camwa.
+2016.01.020, https://doi.org/10.1016/j.camwa.2016.01.020.
+[17] G. I. Marchuk, Some application of splitting-up methods to the solution of mathematical
+physics problems, Aplikace matematiky, 13 (1968), pp. 103–132, http://eudml.org/doc/
+14518.
+[18] P. Prince and J. Dormand,
+High order embedded runge-kutta formulae,
+Journal of
+Computational
+and
+Applied
+Mathematics,
+7
+(1981),
+pp.
+67–75,
+https://doi.org/
+https://doi.org/10.1016/0771-050X(81)90010-3,
+https://www.sciencedirect.com/science/
+article/pii/0771050X81900103.
+[19] S. Roberts, J. Loffeld, A. Sarshar, C. S. Woodward, and A. Sandu, Implicit multirate
+gark methods, Journal of Scientific Computing, 87 (2021), pp. 1–32.
+[20] S. Roberts, A. Sarshar, and A. Sandu, Coupled multirate infinitesimal gark schemes for
+stiff systems with multiple time scales, SIAM Journal on Scientific Computing, 42 (2020),
+pp. A1609–A1638.
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+definite operators:
+reaction-diffusion systems, Journal of Computational Physics, 203
+(2005), pp. 449–466, https://doi.org/10.1016/j.jcp.2004.09.004, https://doi.org/10.1016/
+j.jcp.2004.09.004.
+[22] A. Sandu, A class of multirate infinitesimal GARK methods, SIAM Journal on Numerical
+Analysis, 57 (2019), pp. 2300–2327, https://doi.org/10.1137/18M1205492.
+[23] A. Sandu and M. G¨unther, A generalized-structure approach to additive Runge–Kutta meth-
+ods, SIAM Journal on Numerical Analysis, 53 (2015), pp. 17–42, https://doi.org/10.1137/
+130943224, https://doi.org/10.1137/130943224.
+[24] A. Sarshar, S. Roberts, and A. Sandu, Design of High-Order Decoupled Multirate GARK
+Schemes, SIAM Journal on Scientific Computing, 41 (2019), pp. A816–A847, https://doi.
+org/10.1137/18M1182875.
+[25] G. Strang, On the construction and comparison of difference schemes, SIAM Journal on
+Numerical Analysis, 5 (1968), pp. 506–517, https://doi.org/10.1137/0705041, https://doi.
+org/10.1137/0705041.
+[26] J. Wensch, O. Knoth, and A. Galant, Multirate infinitesimal step methods for atmospheric
+flow simulation, BIT Numerical Mathematics, 49 (2009), pp. 449–473.
+[27] J. Zonneveld, Automatic integration of ordinary differential equations, Stichting Mathema-
+tisch Centrum. Rekenafdeling, 743 (1963).
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+27
+Appendix A. IMEX-MRI-SR2(1) Coefficients.
+c{S} =
+�
+�������
+0
+3
+5
+4
+15
+1
+�
+�������
+, Ω{0} =
+�
+����������
+0
+0
+0
+0
+3
+5
+0
+0
+0
+14
+165
+2
+11
+0
+0
+− 13
+54
+137
+270
+11
+15
+0
+− 1
+4
+1
+2
+3
+4
+0
+�
+����������
+, Γ =
+�
+����������
+0
+0
+0
+0
+− 11
+23
+11
+23
+0
+0
+− 6692
+52371
+− 18355
+52371
+11
+23
+0
+11621
+90666
+− 215249
+226665
+17287
+50370
+11
+23
+− 31
+12
+− 1
+6
+11
+4
+0
+�
+����������
+.
+(A.1)
+Appendix B. IMEX-MRI-SR3(2) Coefficients.
+c{S} =
+�
+����������
+0
+23
+34
+4
+5
+17
+15
+1
+�
+����������
+, Ω{0} =
+�
+�������������
+0
+0
+0
+0
+0
+23
+34
+0
+0
+0
+0
+71
+70
+− 3
+14
+0
+0
+0
+124
+1155
+4
+7
+5
+11
+0
+0
+162181
+187680
+119
+1380
+11
+32
+− 5
+17
+0
+76355
+74834
+− 46
+31
+67
+34
+− 36
+71
+0
+�
+�������������
+,
+Ω{1} =
+�
+�������������
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+− 14453
+63825
+14453
+63825
+0
+0
+0
+− 2101267877
+1206582300
+− 2476735438
+301645575
+− 13575085
+2098404
+0
+0
+− 762580446799
+588660102960
+11083240219
+4328383110
+− 211274129
+100368304
+89562055
+106641323
+0
+− 3732974
+2278035
+13857574
+2278035
+− 52
+9
+4
+3
+0
+�
+�������������
+Γ =
+�
+�������������
+0
+0
+0
+0
+0
+− 4
+7
+4
+7
+0
+0
+0
+− 2707004
+3127425
+919904
+3127425
+4
+7
+0
+0
+852879271
+703839675
+− 1575000496
+703839675
+5
+11
+4
+7
+0
+43136869
+2019912118
+− 73810600
+1009956059
+− 17653551
+87822266
+− 13993902
+43911133
+4
+7
+− 179
+4140
+799
+14490
+1
+14
+− 1
+12
+0
+�
+�������������
+(B.1)
+Appendix C. IMEX-MRI-SR4(3) Coefficients.
+
+28
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+c{S} =
+�
+����������������
+0
+1
+4
+3
+4
+11
+20
+1
+2
+1
+1
+�
+����������������
+, Ω{0} =
+�
+�������������������
+0
+0
+0
+0
+0
+0
+0
+1
+4
+0
+0
+0
+0
+0
+0
+9
+8
+− 3
+8
+0
+0
+0
+0
+0
+187
+2340
+7
+9
+− 4
+13
+0
+0
+0
+0
+64
+165
+1
+6
+− 3
+5
+6
+11
+0
+0
+0
+1816283
+549120
+− 2
+9
+− 4
+11
+− 1
+6
+− 2561809
+1647360
+0
+0
+0
+7
+11
+− 2203
+264
+10825
+792
+− 85
+12
+841
+396
+0
+1
+400
+49
+12
+43
+6
+− 7
+10
+− 85
+12
+− 2963
+1200
+0
+�
+�������������������
+Ω{1} =
+�
+�������������������
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+− 11
+4
+11
+4
+0
+0
+0
+0
+0
+− 1228
+2925
+− 92
+225
+808
+975
+0
+0
+0
+0
+− 2572
+2805
+167
+255
+199
+136
+− 1797
+1496
+0
+0
+0
+− 1816283
+274560
+253
+36
+− 23
+44
+76
+3
+− 20775791
+823680
+0
+0
+0
+107
+132
+1289
+88
+− 9275
+792
+0
+− 371
+99
+0
+− 1
+200
+− 137
+24
+− 235
+16
+1237
+80
+0
+2963
+600
+0
+�
+�������������������
+Γ =
+�
+�������������������
+0
+0
+0
+0
+0
+0
+0
+− 1
+4
+1
+4
+0
+0
+0
+0
+0
+1
+4
+− 1
+2
+1
+4
+0
+0
+0
+0
+13
+100
+− 7
+30
+− 11
+75
+1
+4
+0
+0
+0
+6
+85
+− 301
+1360
+− 99
+544
+45
+544
+1
+4
+0
+0
+0
+− 9
+4
+− 19
+48
+− 75
+16
+85
+12
+1
+4
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+�
+�������������������
+(C.1)
+Appendix D. MERK4 IMEX-MRI-SR Coefficients.
+We list the non-zero
+coefficients of the MERK5 method’s IMEX-MRI-SR formulation below.
+Ω{0}
+i,1 = c{S}
+i
+, i = 1, ..., s{S}
+(D.1)
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+29
+Ω{1}
+3,1 = −Ω{1}
+3,2 , Ω{1}
+3,2 = c{S}×2
+3
+c{S}
+2
+, Ω{1}
+4,1 = −Ω{1}
+4,2 , Ω{1}
+4,2 = c{S}×2
+4
+c{S}
+2
+,
+Ω{1}
+5,1 = −(Ω{1}
+5,3 + Ω{1}
+5,4 ), Ω{1}
+5,3 = −
+c{S}
+4
+c{S}×2
+5
+c{S}
+3
+(c{S}
+3
+− c{S}
+4
+)
+, Ω{1}
+5,4 =
+c{S}
+3
+c{S}×2
+5
+c{S}
+4
+(c{S}
+3
+− c{S}
+4
+)
+Ω{1}
+6,1 = −(Ω{1}
+6,3 + Ω{1}
+6,4 ), Ω{1}
+6,3 = −
+c{S}
+4
+c{S}×2
+6
+c{S}
+3
+(c{S}
+3
+− c{S}
+4
+)
+, Ω{1}
+6,4 =
+c{S}
+3
+c{S}×2
+6
+c{S}
+4
+(c{S}
+3
+− c{S}
+4
+)
+Ω{1}
+7,1 = −(Ω{1}
+7,5 + Ω{1}
+7,6 ), Ω{1}
+7,5 = −
+c{S}
+6
+c{S}
+5
+(c{S}
+5
+− c{S}
+6
+)
+, Ω{1}
+7,6 =
+c{S}
+5
+c{S}
+6
+(c{S}
+5
+− c{S}
+6
+)
+(D.2)
+Ω{1}
+5,1 = −(Ω{1}
+5,3 + Ω{1}
+5,4 ), Ω{1}
+5,3 =
+1
+c{S}
+3
+(c{S}
+3
+− c{S}
+4
+)
+, Ω{1}
+5,4 = −
+1
+c{S}
+4
+(c{S}
+3
+− c{S}
+4
+)
+Ω{1}
+6,1 = −(Ω{1}
+6,3 + Ω{1}
+6,4 ), Ω{1}
+6,3 =
+1
+c{S}
+3
+(c{S}
+3
+− c{S}
+4
+)
+, Ω{1}
+6,4 = −
+1
+c{S}
+4
+(c{S}
+3
+− c{S}
+4
+)
+Ω{1}
+7,1 = −(Ω{1}
+7,5 + Ω{1}
+7,6 ), Ω{1}
+7,5 =
+1
+c{S}
+5
+(c{S}
+5
+− c{S}
+6
+)
+, Ω{1}
+7,6 = −
+1
+c{S}
+6
+(c{S}
+5
+− c{S}
+6
+)
+(D.3)
+This method attains general fourth-order accuracy when
+c{S}
+6
+= 3 − 4c{S}
+5
+4 − 6c{S}
+5
+.
+(D.4)
+
+30
+A. C. FISH, D. R. REYNOLDS, AND S. B. ROBERTS
+Appendix E. MERK5 IMEX-MRI-SR Coefficients.
+We list the non-zero coefficients of the MERK5 method’s IMEX-MRI-SR formu-
+lation below.
+(E.1)
+Ω{0}
+i,1 = c{S}
+i
+, i = 1, ..., s{S}
+Ω{1}
+3,1 = −Ω{1}
+3,2 , Ω{1}
+3,2 = c{S}×2
+3
+α2,
+Ω{1}
+4,1 = −Ω{1}
+4,2 , Ω{1}
+4,2 = c{S}×2
+4
+α2,
+Ω{1}
+5,1 = −
+�
+Ω{1}
+5,3 + Ω{1}
+5,4
+�
+, Ω{1}
+5,3 = c{S}×2
+5
+α3, Ω{1}
+5,4 = c{S}×2
+5
+α4
+Ω{1}
+6,1 = −
+�
+Ω{1}
+6,3 + Ω{1}
+6,4
+�
+, Ω{1}
+6,3 = c{S}×2
+6
+α3, Ω{1}
+6,4 = c{S}×2
+6
+α4
+Ω{1}
+7,1 = −
+�
+Ω{1}
+7,3 + Ω{1}
+7,4
+�
+, Ω{1}
+7,3 = c{S}×2
+7
+α3, Ω{1}
+7,4 = c{S}×2
+7
+α4
+Ω{1}
+8,1 = −
+�
+Ω{1}
+8,5 + Ω{1}
+8,6 + Ω{1}
+8,7
+�
+, Ω{1}
+8,5 = c{S}×2
+8
+α5, Ω{1}
+8,6 = c{S}×2
+8
+α6, Ω{1}
+8,7 = c{S}×2
+8
+α7
+Ω{1}
+9,1 = −
+�
+Ω{1}
+9,5 + Ω{1}
+9,6 + Ω{1}
+9,7
+�
+, Ω{1}
+9,5 = c{S}×2
+9
+α5, Ω{1}
+9,6 = c{S}×2
+9
+α6, Ω{1}
+9,7 = c{S}×2
+9
+α7
+Ω{1}
+10,1 = −
+�
+Ω{1}
+10,5 + Ω{1}
+10,6 + Ω{1}
+10,7
+�
+, Ω{1}
+10,5 = c{S}×2
+10
+α5, Ω{1}
+10,6 = c{S}×2
+10
+α6, Ω{1}
+10,7 = c{S}×2
+10
+α7
+Ω{1}
+11,1 = −
+�
+Ω{1}
+11,8 + Ω{1}
+11,9 + Ω{1}
+11,10
+�
+, Ω{1}
+11,8 = α8, Ω{1}
+11,9 = α9, Ω{1}
+11,10 = α10
+(E.2)
+where
+α2 =
+1
+c{S}
+2
+, α3 =
+c{S}
+3
+c{S}
+3
+(c{S}
+4
+− c{S}
+3
+)
+, α4 =
+c{S}
+4
+c{S}
+4
+(c{S}
+3
+− c{S}
+4
+)
+α5 =
+c{S}
+6
+c{S}
+7
+c{S}
+5
+(c{S}
+5
+− c{S}
+6
+)(c{S}
+5
+− c{S}
+7
+)
+, α6 =
+c{S}
+5
+c{S}
+7
+c{S}
+6
+(c{S}
+6
+− c{S}
+5
+)(c{S}
+6
+− c{S}
+7
+)
+,
+α7 =
+c{S}
+5
+c{S}
+6
+c{S}
+7
+(c{S}
+7
+− c{S}
+5
+)(c{S}
+7
+− c{S}
+6
+)
+, α8 =
+c{S}
+9
+c{S}
+10
+c{S}
+8
+(c{S}
+8
+− c{S}
+9
+)(c{S}
+8
+− c{S}
+10 )
+α9 =
+c{S}
+8
+c{S}
+10
+c{S}
+9
+(c{S}
+9
+− c{S}
+8
+)(c{S}
+9
+− c{S}
+10 )
+, α10 =
+c{S}
+8
+c{S}
+9
+c{S}
+10 (c{S}
+10
+− c{S}
+8
+)(c{S}
+10
+− c{S}
+9
+)
+(E.3)
+Ω{2}
+5,1 = −
+�
+Ω{2}
+5,3 + Ω{2}
+5,4
+�
+, Ω{2}
+5,3 = c{S}×2
+5
+β3, Ω{2}
+5,4 = −c{S}×2
+5
+β4
+Ω{2}
+6,1 = −
+�
+Ω{2}
+6,3 + Ω{2}
+6,4
+�
+, Ω{2}
+6,3 = c{S}×2
+6
+β3, Ω{2}
+6,4 = −c{S}×2
+6
+β4
+Ω{2}
+7,1 = −
+�
+Ω{2}
+7,3 + Ω{2}
+7,4
+�
+, Ω{2}
+7,3 = c{S}×2
+7
+β3, Ω{2}
+7,4 = −c{S}×2
+7
+β4
+Ω{2}
+8,1 = −
+�
+Ω{2}
+8,5 + Ω{2}
+8,6 + Ω{2}
+8,7
+�
+, Ω{2}
+8,5 = −c{S}×2
+8
+β5, Ω{2}
+8,6 = −c{S}×2
+8
+β6, Ω{2}
+8,7 = −c{S}×2
+8
+β7
+Ω{2}
+9,1 = −
+�
+Ω{2}
+9,5 + Ω{2}
+9,6 + Ω{2}
+9,7
+�
+, Ω{2}
+9,5 = −c{S}×2
+9
+β5, Ω{2}
+9,6 = −c{S}×2
+9
+β6, Ω{2}
+9,7 = −c{S}×2
+9
+β7
+Ω{2}
+10,1 = −
+�
+Ω{2}
+10,5 + Ω{2}
+10,6 + Ω{2}
+10,7
+�
+, Ω{2}
+10,5 = −c{S}×2
+10
+β5, Ω{2}
+10,6 = −c{S}×2
+10
+β6, Ω{2}
+10,7 = −c{S}×2
+10
+β7
+Ω{2}
+11,1 = −
+�
+Ω{2}
+11,8 + Ω{2}
+11,9 + Ω{2}
+11,10
+�
+, Ω{2}
+11,8 = β8, Ω{2}
+11,9 = β9, Ω{2}
+11,10 = β10
+(E.4)
+
+IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS
+31
+where
+β3 =
+1
+c{S}
+3
+(c{S}
+4
+− c{S}
+3
+)
+, β4 =
+1
+c{S}
+4
+(c{S}
+3
+− c{S}
+4
+)
+β5 =
+c{S}
+6
++ c{S}
+7
+c{S}
+5
+(c{S}
+5
+− c{S}
+6
+)(c{S}
+5
+− c{S}
+7
+)
+, β6 =
+c{S}
+5
++ c{S}
+7
+c{S}
+6
+(c{S}
+6
+− c{S}
+5
+)(c{S}
+6
+− c{S}
+7
+)
+,
+β7 =
+c{S}
+5
++ c{S}
+6
+c{S}
+7
+(c{S}
+7
+− c{S}
+5
+)(c{S}
+7
+− c{S}
+6
+)
+, β8 =
+c{S}
+9
++ c{S}
+10
+c{S}
+8
+(c{S}
+8
+− c{S}
+9
+)(c{S}
+8
+− c{S}
+10 )
+β9 =
+c{S}
+8
++ c{S}
+10
+c{S}
+9
+(c{S}
+9
+− c{S}
+8
+)(c{S}
+9
+− c{S}
+10 )
+, β10 =
+c{S}
+8
++ c{S}
+9
+c{S}
+10 (c{S}
+10
+− c{S}
+8
+)(c{S}
+10
+− c{S}
+9
+)
+(E.5)
+Ω{3}
+8,1 = −
+�
+Ω{3}
+8,5 + Ω{3}
+8,6 + Ω{3}
+8,7
+�
+, Ω{3}
+8,5 = c{S}×2
+8
+γ5, Ω{3}
+8,6 = c{S}×2
+8
+γ6, Ω{3}
+8,7 = c{S}×2
+8
+γ7
+Ω{3}
+9,1 = −
+�
+Ω{3}
+9,5 + Ω{3}
+9,6 + Ω{3}
+9,7
+�
+, Ω{3}
+9,5 = c{S}×2
+9
+γ5, Ω{3}
+9,6 = c{S}×2
+9
+γ6, Ω{3}
+9,7 = c{S}×2
+9
+γ7
+Ω{3}
+10,1 = −
+�
+Ω{3}
+10,5 + Ω{3}
+10,6 + Ω{3}
+10,7
+�
+, Ω{3}
+10,5 = c{S}×2
+10
+γ5, Ω{3}
+10,6 = c{S}×2
+10
+γ6, Ω{3}
+10,7 = c{S}×2
+10
+γ7
+Ω{3}
+11,1 = −
+�
+Ω{3}
+11,8 + Ω{3}
+11,9 + Ω{3}
+11,10
+�
+, Ω{3}
+11,8 = γ8, Ω{3}
+11,9 = γ9, Ω{3}
+11,10 = γ10
+(E.6)
+where
+γ5 =
+1
+c{S}
+5
+(c{S}
+5
+− c{S}
+6
+)(c{S}
+5
+− c{S}
+7
+)
+, γ6 =
+1
+c{S}
+6
+(c{S}
+6
+− c{S}
+5
+)(c{S}
+6
+− c{S}
+7
+)
+,
+γ7 =
+1
+c{S}
+7
+(c{S}
+7
+− c{S}
+5
+)(c{S}
+7
+− c{S}
+6
+)
+, γ8 =
+1
+c{S}
+8
+(c{S}
+8
+− c{S}
+9
+)(c{S}
+8
+− c{S}
+10 )
+γ9 =
+1
+c{S}
+9
+(c{S}
+9
+− c{S}
+8
+)(c{S}
+9
+− c{S}
+10 )
+, γ10 =
+1
+c{S}
+10 (c{S}
+10
+− c{S}
+8
+)(c{S}
+10
+− c{S}
+9
+)
+(E.7)
+This method satisfies all available coupling conditions (up through fourth-order)
+and its base method satisfies all order conditions up through fifth-order when
+c{S}
+9
+= 12 − 15c{S}
+10
+− 15c{S}
+8
++ 20c{S}
+10 c{S}
+8
+15 − 20c{S}
+10
+− 20c{S}
+8
++ 30c{S}
+10 c{S}
+8
+.
+(E.8)
+
diff --git a/V9AyT4oBgHgl3EQf8_o-/content/tmp_files/load_file.txt b/V9AyT4oBgHgl3EQf8_o-/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c82e13e2fd4d6ead980b53973884b2999f50fa8d
--- /dev/null
+++ b/V9AyT4oBgHgl3EQf8_o-/content/tmp_files/load_file.txt
@@ -0,0 +1,1646 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf,len=1645
+page_content='IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL  STAGE-RESTART METHODS ∗  ALEX C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH†, DANIEL R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS†, AND STEVEN B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS‡  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Implicit-Explicit (IMEX) methods are flexible numerical time integration methods  which solve an initial-value problem (IVP) that is partitioned into stiff and nonstiff processes with the  goal of lower computational costs than a purely implicit or explicit approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' A complementary form  of flexible IVP solvers are multirate infinitesimal methods for problems partitioned into fast- and  slow-changing dynamics, that solve a multirate IVP by evolving a sequence of “fast” IVPs using any  suitably accurate algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This article introduces a new class of high-order implicit-explicit multi-  rate methods that are designed for multirate IVPs in which the slow-changing dynamics are further  partitioned in an IMEX fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  This new class, which we call implicit-explicit multirate stage-  restart (IMEX-MRI-SR), both improves upon the previous implicit-explicit multirate generalized-  structure additive Runge Kutta (IMEX-MRI-GARK) methods, and extends multirate exponential  Runge Kutta (MERK) methods into the IMEX context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We leverage GARK theory to derive con-  ditions guaranteeing orders of accuracy up to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We provide second-, third-, and fourth-order  accurate example methods and perform numerical simulations demonstrating convergence rates and  computational performance in both fixed-step and adaptive-step settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' multirate time integration, initial-value problems, implicit-explicit methods  AMS subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' 65L20, 65L05, 65L06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Flexible time integration methods for solving systems of initial-  value problems (IVPs) have seen growing interest in recent years, largely due to their  ability to provide highly accurate approximations of the IVP solution with increased  computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' These integrators strive to reduce computational costs by  partitioning the IVP into different components, and then treating each using different  step sizes or numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Some of the primary families of flexible methods in-  clude implicit-explicit (IMEX) partitioning [2, 9, 11, 23], linear-nonlinear partitioning  [8, 13, 14, 15, 16], and multirate partitioning [6, 22, 24, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  IMEX time integration methods solve IVPs in which the right-hand side function  ∗Submitted to the editors DATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Funding: The work of Alex Fish and Daniel Reynolds was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Department  of Energy, Office of Science, Office of Advanced Scientific Computing Research, Scientific Discovery  through Advanced Computing (SciDAC) Program through the FASTMath Institute, under DOE  award DE-SC0021354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The work of Steven Roberts was performed under the auspices of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Department of Energy by  Lawrence Livermore National Laboratory under Contract DEAC52-07NA27344 and was supported  by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Department of Energy, Office of Science, Office of Advanced Scientific Computing Re-  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' LLNL-JRNL-843436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  This document was prepared as an account of work sponsored by an agency of the United States  government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Neither the United States government nor Lawrence Livermore National Security, LLC,  nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or  responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product,  or process disclosed, or represents that its use would not infringe privately owned rights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Reference  herein to any specific commercial product, process, or service by trade name, trademark, manufac-  turer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or  favoring by the United States government or Lawrence Livermore National Security, LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The views  and opinions of authors expressed herein do not necessarily state or reflect those of the United States  government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or  product endorsement purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  †Department of Mathematics, Southern Methodist University, Dallas, TX, USA (afish@smu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='edu,  reynolds@smu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  ‡Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore,  CA, USA (roberts115@llnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='gov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='00865v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='NA]  2 Jan 2023  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  f(t, y(t)) is additively split into stiff {I} and nonstiff {E} processes,  y′(t) = f(t, y) := f {I}(t, y) + f {E}(t, y),  t ≥ t0,  y(t0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1)  IMEX methods then couple two different numerical methods to treat these compo-  nents: f {I} typically uses a stiff but computationally expensive solver, whereas f {E}  may use a cheaper but nonstiff solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' For example, additive Runge–Kutta (ARK)  methods typically combine an A-stable diagonally-implicit Runge–Kutta (DIRK) method  with an explicit Runge–Kutta (RK) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Similarly, multirate methods solve IVPs in which the right-hand side is additively  partitioned into rapidly and slowly evolving dynamics, {F} and {S},  y′(t) = f(t, y) := f {F }(t, y) + f {S}(t, y),  t ≥ t0,  y(t0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2)  Multirate methods then apply numerical methods with different step sizes for each  component to save on computation time while retaining a desired level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  In this work we combine the above approaches to consider a three-way additively  partitioned IVP, wherein the slow partition f {S} from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2) is split in an IMEX fashion,  y′(t) = f {F }(t, y) + f {I}(t, y) + f {E}(t, y),  t ≥ t0,  y(t0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3)  In particular, we add a new class of methods to the ever-growing family of multirate  infinitesimal (MRI) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' These approximate the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3) through  solving a sequence of “fast” IVPs,  v′  i(θ) = f {F }(θ, vi) + gi(θ),  θ ∈ [θ0,i, θf,i],  v(θ0,i) = v0,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4)  The forcing functions gi(θ) incorporate information from the slow dynamics defined  by f {I} and f {E} in a manner defined by the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' MRI methods assume that  these fast IVPs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4) are solved exactly, but in practice these are approximated using  an additional “inner” numerical method with a smaller step size than the multirate  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This inner method can itself further decompose the problem through IMEX,  linear-nonlinear, or multirate approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Our proposed class of methods is called Implicit-Explicit Multirate Infinitesimal  Stage-Restart (IMEX-MRI-SR) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Each stage of an IMEX-MRI-SR method  consists of evolving a fast IVP followed by an implicit solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This allows derivation  of IMEX-MRI-SR methods by extending a base ARK method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We discuss the role of  base methods further in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1, particularly focusing on their role in satisfying  order conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  IMEX-MRI-SR methods are defined by nΩ coefficient matrices Ω{k} ∈ Rs{S}×s{S},  k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', nΩ − 1, a coefficient matrix Γ ∈ Rs{S}×s{S}, and an abscissae vector c{S} ∈  Rs{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Embedded versions of these methods include additional coefficient vectors  ˆω{k} ∈ Rs{S} and ˆγ ∈ Rs{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The algorithm for evolving a solution to an IVP of the  form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 (IMEX-MRI-SR methods for additively partitioned systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  An IMEX-MRI-SR method evolves the solution to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3) from tn to tn +H  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  3  according to the following algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Let: Y {S}  1  := yn  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5a)  For: i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', s{S}  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  Let:  vi(0) := yn,  Solve:  v′  i(θ) = f {F }(tn + θ, vi(θ)) + gi(θ), for θ ∈ [0, c{S}  i  H]  where gi(θ) =  1  c{S}  i  i−1  �  j=1  ωi,j  �  θ  c{S}  i  H  � �  f {E}  j  + f {I}  j  �  Solve:  Y {S}  i  = vi  �  c{S}  i  H  �  + H  i  �  j=1  γi,jf {I}  j  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5b)  Let: yn+1 := Y {S}  s{S},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5c)  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  Let:  ˆv(0) := yn,  Solve:  ˆv(θ) = f {F }(tn + θ, ˆv(θ)) + ˆg(θ), for θ ∈ [0, H]  where ˆg(θ) =  s{S}−1  �  j=1  ˆωj  � θ  H  � �  f {E}  j  + f {I}  j  �  Solve:  ˆyn+1 = ˆv (H) + H  s{S}−1  �  j=1  ˆγjf {I}  j  + Hˆγs{S}f {I}(tn + H, ˆyn+1),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5d)  where f {E}  j  := f {E} �  tn + c{S}  j  H, Y {S}  j  �  and f {I}  j  := f {I} �  tn + c{S}  j  H, Y {S}  j  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Here  yn+1 is the time-evolved approximation to y(tn+H), and ˆyn+1 is an embedded solution  used for temporal error esimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' If temporal error estimation is not needed, then  step (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5d) may be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 (Slow tendency coefficients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The coefficients ωi,j in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5b) are  defined as in [3]  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6)  ωi,j(τ) :=  nΩ−1  �  k=0  ω{k}  i,j τ k,  ωi,j :=  � 1  0  ωi,j(τ)dτ =  nΩ−1  �  k=0  ω{k}  i,j  k + 1,  and we refer to Ω{k}, Ω, and Γ as the s{S} × s{S} matrices containing the coeffi-  cients {ω{k}  i,j }, {ωi,j}, and {γi,j}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' As in [3] we assume the first row of  the coefficient matrices are identically zero, and that Ω{k} and Ω are strictly lower-  triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The embedding functions ˆωj(τ) are defined similarly to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6), with vectors  of coefficients ˆω{k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' In the next subsection, we present  related methods to IMEX-MRI-SR, discussing both their similarities and their limita-  tions that are improved upon by the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' In Section 2 we prove order  conditions for IMEX-MRI-SR methods up to order four, and in Section 3 we examine  their linear stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' In Section 4 we provide embedded IMEX-MRI-SR methods of  orders 2 through 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' In Section 5 we show that a previous class of multirate methods  may be reformulated as IMEX-MRI-SR methods, and we use our convergence theory  to show previously unproven features of those methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We then present numerical  results in Section 6 to validate our convergence theory, and to compare the efficiency  of IMEX-MRI-SR methods against existing methods for problems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  Finally, in Section 7 we conclude this article with a summary of our contributions,  and an outlook toward future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Related methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' To our knowledge, there exist three MRI algorithms  that allow both IMEX partitioning of the slow dynamics and infinitesimal treatment  of the fast dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' These are the first-order accurate “Lie–Trotter” [4, 21] and the  second-order accurate “Strang–Marchuk” [17, 25] operator splitting methods, and  the recent fourth-order IMEX-MRI-GARK [3] method, an IMEX variation of MRI-  GARK [22] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Both Lie–Trotter and Strang–Marchuk operate by sequentially  applying distinct solvers to each component, only communicating with one another  through the initial conditions applied within each sub-solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Variations of both classes  of methods for the IMEX multirate splitting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3) are shown in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' However, due to  their weak “initial condition” coupling, Lie–Trotter and Strang–Marchuk are limited  to at most first and second order accuracy in time, regardless of the order of accuracy  of each component solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  IMEX-MRI-GARK methods are organized similarly to IMEX-MRI-SR, in that  they advance the solution by alternating between evolving fast IVPs and solving im-  plicit algebraic equations involving f {I}, and they use the result from each stage to  provide a contribution to gi(θ) for later stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' However, unlike IMEX-MRI-SR meth-  ods, IMEX-MRI-GARK methods evolve each fast IVP over an interval [ci−1H, ciH],  with an initial condition given as the result of the previous stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  In each stage,  either a fast evolution or implicit solve may occur, with this choice dictated by the  abscissae: when ∆ci := ci − ci−1 > 0 a fast evolution occurs, but when ∆ci = 0 an  algebraic system must be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' In all existing implicit MRI-GARK and IMEX-MRI-  GARK methods, authors have derived schemes by beginning with a base DIRK or  ARK method, and then introduced additional internal stages to ensure an alternating  pattern of ∆ci ̸= 0 followed by ∆ci+1 = 0, thereby ensuring an appropriate structure  [3, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' However, these methods inherently require abscissae vectors c{S} that are  non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This is generally an uncommon feature in RK methods, especially  for orders of accuracy higher than two [2, 9, 10, 11, 19, 20, 23], so there are relatively  few base DIRK and ARK methods available for deriving new MRI-GARK and IMEX-  MRI-GARK methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Additionally, the “padding” process for adding internal stages  to ensure ∆ci = 0 is not obvious, and frequently results in an overly complicated trial-  and-error process to decide where to insert stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' As a result of these challenges, no  authors have successfully created IMEX-MRI-GARK methods with embeddings for  temporal error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  A second class of methods that are closely related to IMEX-MRI-SR are multi-  rate exponential Runge–Kutta (MERK) methods [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' While these do not support  implicitness at the slow time scale (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', f {I} = 0), and they assume that the fast  partition is linear (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', f {F }(t, y) = Ly), their structure matches (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Each internal  stage is computed through evolving a fast IVP over an interval [0, ciH], using a forcing  function that is determined through the values of f {E} at previous slow stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' IMEX-MRI-SR Order Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Similar to IMEX-MRI-GARK meth-  ods, we derive order conditions for IMEX-MRI-SR methods by first representing the  algorithm in GARK form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Due to the 3-component partitioning (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3), we must iden-  tify GARK coefficients A{σ,ν}, b{ν}, and c{ν} for σ ∈ {S, F} and ν ∈ {I, E, F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We  refer to A{F,ν}, ν ∈ {I, E, F} as the fast GARK tables, and to A{S,ν}, ν ∈ {I, E, F}  as the slow GARK tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We assume the fast IVP (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5b) is solved with one step of  a sufficiently accurate Runge–Kutta method having s{F } stages, and defined by the  Butcher table (A{F }, b{F }, c{F }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The jth sub-stage in computing the solution to the  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  5  fast IVP v′  i(θ) is given by  Vi,j = yn + c{S}  i  H  s{F }  �  k=1  a{F }  j,k f {F }  i,k  + H  s{F }  �  k=1  a{F }  j,k  i−1  �  ℓ=1  ωi,ℓ(c{F }  k  )  �  f {E}  ℓ  + f {I}  ℓ  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1)  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', s{S}, and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', s{F }, where f {F }  i,k  := f {F }(tn + c{F }  k  H, Vi,k), f {E}  ℓ  :=  f {E}(tn + c{S}  ℓ  H, Y {S}  ℓ  ) and f {I}  ℓ  := f {I}(tn + c{S}  ℓ  H, Y {S}  ℓ  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The IMEX-MRI-SR  method’s slow stages Y {S}  i  , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' , s{S}, are then computed by  Y {S}  i  = yn + c{S}  i  H  s{F }  �  j=1  b{F }  j  f {F }  i,j  + H  s{F }  �  j=1  b{F }  j  i−1  �  ℓ=1  ωi,ℓ  �  c{F }  j  � �  f {E}  ℓ  + f {I}  ℓ  �  + H  i  �  ℓ=1  γi,ℓf {I}  ℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2)  We leverage GARK theory [23] to construct order conditions as in [3, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Since  the slow partitions share the same stages, Y {S}  i  , these methods have six GARK ma-  trices, A{F,F }, A{F,E}, A{F,I}, A{S,F }, A{S,E}, and A{S,I} and three GARK vectors  b{F }, b{E}, and b{I},  A{F,F }  A{F,E}  A{F,I}  A{S,F }  A{S,E}  A{S,I}  b{F },T  b{E},T  b{I},T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Here, A{F,F } is a square s{SF } × s{SF } matrix (where we define s{SF } = s{S} ·  s{F }), containing the coefficients relating the fast stages {Vi,j} to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' It is  block-diagonal because Vi,j depends only on Vi,k, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', s{F } through the A{F }  coefficients, and never on Vℓj, ℓ ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' These s{F } × s{F } block-diagonal elements are  named A{F,F,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  A{F,E} and A{F,I} are tall s{SF } × s{S} matrices relating the fast stages {Vi,j}  to the explicit and implicit function evaluations of the slow stages {Y {S}  i  }, comprised  of s{S} blocks named A{F,E,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  A{S,F } is a wide s{S} ×s{SF } matrix, containing the coefficients relating the slow  stages {Y {S}  i  } to the fast stages {Vi,j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We name these s{S} total s{S} × s{F } blocks  that comprise A{S,F } as A{S,F,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Each A{S,F,i} contains at most one row of non-zero  entries, located in row i, because Y {S}  i  depends only on Vi,j, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', s{F }, and never  on Vℓ,j, ℓ ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  A{S,E} and A{S,I} are square s{S}×s{S} matrices relating the slow stages {Y {S}  i  }  to the explicit and implicit function evaluations of those slow stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The vectors b{σ} equal the last rows of A{S,σ}, σ ∈ {F, E, I}, because IMEX-  MRI-SR methods have the first-same-as-last (FSAL) property, where the last stage is  used as the solution to the step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  When an embedding (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5d) is included, it will correspond to three additional  GARK vectors, ˆb{F } ˆb{E} and ˆb{I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Due to the structural similarity of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5d) to the  last stage Y {S}  s{S}, the contents of these vectors will only differ from b{F }, b{E} and  b{I} through their dependence on ˆω{k}  j  and ˆγj instead of ω{k}  s{S},j and γs{S},j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Thus  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  the order conditions that follow for the primary GARK matrices and vectors can be  applied to the embedding as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  With the above simplifications, the GARK tableau can be expressed in block-  matrix form as  A{F,F,1}  A{F,E,1}  A{F,I,1}  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  A{F,F,s{S}}  A{F,E,s{S}}  A{F,I,s{S}}  A{S,F,1}  · ·  A{S,F,s{S}}  A{S,E}  A{S,I}  b{F },T  b{E},T  b{I},T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  A GARK method with this tabular structure has stage update formulas  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3a)  Vi,j = yn + H  s{F }  �  k=1  a{F,F,i}  j,k  f {F }  i,j  + H  s{F }  �  k=1  a{F,E,i}  j,k  f {E}  j  + H  s{F }  �  k=1  a{F,I,i}  j,k  f {I}  j  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3b)  Y {S}  i  = yn + H  s{F }  �  j=1  a{S,F,i}  i,j  f {F }  ij  + H  s{F }  �  j=1  A{S,E}  i,j  f {E}  j  + H  s{F }  �  j=1  A{S,I}  i,j  f {I}  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  By matching coefficients in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3a), we identify the fast GARK coefficients,  a{F,F,i}  j,k  = c{S}  i  a{F }  j,k ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4a)  a{F,E,i}  j,ℓ  =  s{F }  �  l=1  a{F }  j,k ωi,ℓ(c{F }  l  ) =  s{F }  �  l=1  nΩ−1  �  k=0  ω{k}  i,ℓ a{F }  j,l c{F }×k  l  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4b)  a{F,I,i}  j,ℓ  = a{F,E,i}  j,ℓ  =  s{F }  �  k=1  b{F }  j,k ωi,ℓ(c{F }  k  ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4c)  where the superscript ×k denotes element-wise exponentiation of a vector by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Con-  verting these to matrix form, we have the fast GARK tables,  A{F,F } = C{S} ⊗ A{F } ∈ Rs{SF }×s{SF },  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5a)  A{F,E} =  nΩ−1  �  k=0  Ω{k} ⊗ A{F }c{F }×k ∈ Rs{SF }×s{S},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5b)  A{F,I} = A{F,E} =  nΩ−1  �  k=0  Ω{k} ⊗ A{F }c{F }×k ∈ Rs{SF }×s{S},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5c)  where ⊗ denotes the Kronecker product and C{σ} = diag(c{σ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We similarly find the slow GARK table coefficients by comparing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3b),  a{S,F,i}  i,j  = c{S}  i  b{F }  j  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6a)  a{S,E}  i,ℓ  =  s{F }  �  k=1  b{F }  k  ωi,ℓ(c{F }  k  ) =  nΩ−1  �  k=0  ω{k}  i,ℓ b{F },T c{F }×k,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6b)  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  7  a{S,I}  i,ℓ  =  s{F }  �  k=1  b{F }  k  ωi,ℓ  �  c{F }  j  �  + γi,ℓ =  nΩ−1  �  k=0  ω{k}  i,ℓ b{F },T c{F }×k + γi,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6c)  Due to the infinitesimal nature of the fast method, we assume that it satisfies all  bushy-tree order conditions,  b{F },T c{F }×k =  1  k + 1,  k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', nΩ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Leveraging this, and examining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6), the slow GARK tables in matrix-form are  A{S,F } = C{S} ⊗ b{F },T ∈ Rs{S}×s{SF },  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7a)  A{S,E} =  nΩ−1  �  k=0  Ω{k}  1  k + 1 = Ω ∈ Rs{S}×s{S},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7b)  A{S,I} =  nΩ−1  �  k=0  Ω{k}  1  k + 1 + Γ = Ω + Γ ∈ Rs{S}×s{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7c)  We additionally define the following variables, knowing that an IMEX-MRI-SR  method has the FSAL property with respect to the slow stages Y {S},  b{F },T = eT  s{S}A{S,F } = (C{S}es{S} ⊗ b{F },T ) = (es{S} ⊗ b{F })T ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='8a)  b{E},T = eT  s{S}A{S,E} = eT  s{S}Ω,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='8b)  b{I},T = eT  s{S}A{S,I} = eT  s{S}(Ω + Γ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='8c)  where es{S} is an s{S}-length vector of all zeroes except a one in the last position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Base Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' If f {F }(t, y) = 0, then an IMEX-MRI-SR method re-  duces to a simple ARK method defined by slow explicit and implicit base methods,  (A{E}, b{E}, c{E}) = (A{S,E}, b{E}, A{S,E}1s{S}),  (A{I}, b{I}, c{I}) = (A{S,I}, b{I}, A{S,I}1s{S}),  where 1s{S} is a vector of ones with length s{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This ARK method has stages  Y {S}  i  = yn + H  i−1  �  j=1  a{E}  i,j f {E}(tn + c{E}  j  H, Y {S}  j  )  + H  i  �  j=1  a{I}  i,j f {I}(tn + c{I}  j  H, Y {S}  j  ),  with the last stage being equal to the solution for the step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We will refer to this ARK  method as the slow base method throughout the derivation of the order conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 (First order conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' As long as the slow base method and the  arbitrary fast method are order one or higher, there are no additional first-order  coupling conditions for a GARK method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Thus, the effective first-order IMEX-MRI-  SR condition is that A{S,E} = Ω and A{S,I} = Ω+Γ form an order one ARK method,  which can be simply achieved if both A{S,E} and A{S,I} have first-order accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 (Deriving IMEX-MRI-SR methods from existing ARK methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Due to the IMEX-MRI-SR structure, a base ARK method should be stiffly-accurate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  otherwise it must first be converted to stiffly-accurate form by appending the b vectors  to the bottom and pad a column of zeros to the right of the ARK’s A matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Kronecker Product Identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' In the ensuing derivations we leverage  the following identities:  (A ⊗ B)T = AT ⊗ BT ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9a)  (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9b)  (A ⊗ v)1{r(A)} = (A1{r(A)}) ⊗ v,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9c)  (v ⊗ A)1{r(A)} = v ⊗ (A1{r(A)}),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9d)  where A, B, C, D are arbitrary matrices with compatible dimemsions, v is an arbitrary  vector, and 1{r(A)} is a vector of ones with length equal to the number of rows of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Identities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9b) can be found in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Identities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9c) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9d) can be  shown through elementary computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' GARK Internal Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 (GARK Internal Consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' An IMEX-MRI-SR method satis-  fies the GARK internal consistency conditions,  c{F,F } = c{F,E} = c{F,I},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='10a)  c{S,F } = c{S,E} = c{S,I},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='10b)  where c{F,ν} = A{F,ν}1s{SF }, c{S,ν} = A{S,ν}1s{S}, ν ∈ {I, E, F}, if the following  conditions hold:  Ω{0}1s{S} = c{S},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='11a)  Ω{k}1s{S} = 0s{S}, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', nΩ − 1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='11b)  Γ1s{S} = 0s{S},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='11c)  where 0s{S} is a vector of zeros with length s{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Computing the fast GARK abscissae from the respective fast GARK ta-  bles,  c{F,F } = A{F,F }1s{SF } = (C{S} ⊗ A{F })1s{SF } = c{S} ⊗ c{F },  c{F,E} = A{F,E}1s{S} =  nΩ−1  �  k=0  Ω{k} ⊗ A{F }c{F }×k1s{S}  = (Ω{0}1s{S}) ⊗ c{F } +  nΩ−1  �  k=1  (Ω{k}1s{S}) ⊗ A{F }c{F }×k,  c{F,I} = A{F,I}1s{S} = A{F,E}1s{S} = c{F,E},  thus c{F,F } = c{F,I} = c{F,E} when Ω01s{S} = c{S} and Ω{k}1s{S} = 0s{S}, k =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', nΩ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Computing the slow GARK abscissae from the slow GARK tables,  c{S,F } = A{S,F }1s{SF } = (C{S} ⊗ b{F })1s{SF } = c{S},  c{S,E} = A{S,E}1s{S} = Ω1s{S} =  nΩ−1  �  k=0  Ω{k}1s{S}  1  k + 1,  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  9  c{S,E} = A{S,I}1s{S} = (Ω + Γ)1s{S} =  nΩ−1  �  k=0  Ω{k}1s{S}  1  k + 1 + Γ1s{S},  and thus c{S,F } = c{S,I} = c{S,E} is satisfied using the same conditions as above,  with the additional constraint that Γ1s{S} = 0s{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 (Second order conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' When the tables that comprise a GARK  method are each at least second-order accurate and internal consistency holds, there  are no additional second-order coupling conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Thus, the internal consistency  conditions act as second-order conditions when the slow base method and arbitrary  fast method are at least second-order accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Higher Order Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 (Third Order Conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' An internally-consistent IMEX-MRI-  SR method with third-order accurate slow base method and with fast method of order  max(3, nΩ + 1) accurate is third-order accurate if the following condition holds:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='12)  eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  c{S} = 1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' From [3], a internally consistent GARK method of this structure with a  third-order accurate slow base method and an order max(3, nΩ + 1) fast method has  four third-order coupling conditions,  b{σ},T A{S,F }c{F } = 1  6,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='13a)  b{F },T A{S,σ}c{S} = 1  6,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='13b)  for σ ∈ {I, E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' An internally-consistent IMEX-MRI-SR method has  c{F } = c{F,F } = c{F,E} = c{F,I} = c{S} ⊗ c{F },  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='14a)  c{S} = c{S,F } = c{S,E} = c{S,I} = c{E} = c{I} = c{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='14b)  Conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='13a) are automatically satisfied for both values of σ:  1  6 = b{σ},T A{S,F }c{F }  = b{σ},T (C{S} ⊗ b{F },T )(c{S} ⊗ c{F }) = 1  3 · 1  2  Conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='13b) reduce to the single condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='12) because A{F,E} = A{F,I} for  an IMEX-MRI-SR method:  1  6 = b{F },T A{F,σ}c{S} = (eT  s{S} ⊗ b{F },T )  �nΩ−1  �  k=0  Ω{k} ⊗ A{F }c{F }×k  �  c{S}  = eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  c{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 (Fourth Order Conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' An IMEX-MRI-SR method satisfying  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 is fourth-order accurate if the slow base method is fourth-order accurate,  the arbitrary fast method is order max(4, nΩ+2) accurate, and the following conditions  hold:  eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 3)  �  c{S} = 1  8,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15a)  eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  C{S}c{S} = 1  12,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15b)  eT  s{S}ΓC{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  c{S} = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15c)  eT  s{S}ΩC{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  c{S} = 1  24,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15d)  eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  Ωc{S} = 1  24,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15e)  eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  Γc{S} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15f)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' From [3], there are 26 fourth-order coupling conditions for a third-order  GARK method with this structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' this further reduces to 21 when A{F,E} = A{F,I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  These GARK coupling conditions are:  b{σ},T C{S}A{S,F }c{F } = 1  8  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16a)  b{σ},T A{S,ν}A{S,F }c{F } = 1  24  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16b)  b{σ},T A{S,F }C{F }c{F } = 1  12  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16c)  b{σ},T A{S,F }A{F,F }c{F } = 1  24  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16d)  b{F },T C{F }A{F,σ}c{S} = 1  8  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16e)  b{F },T A{F,σ}C{S}c{S} = 1  12  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16f)  b{σ},T A{S,F }A{F,ν}c{S} = 1  24  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16g)  b{F },T A{F,F }A{F,σ}c{S} = 1  24  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16h)  b{F },T A{F,σ}A{S,ν}c{S} = 1  24  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16i)  b{F },T A{F,σ}A{S,F }c{F } = 1  24  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16j)  for σ, ν ∈ {I, E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' c{F } and c{S} are defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='14a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='14b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We arrive at the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15) by checking each of the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16) in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  11  The first four conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16a)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16d) are automatically satisfied:  1  8 = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T C{S}A{S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='F }c{F }  = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T C{S}(C{S} ⊗ b{F },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T )(c{S} ⊗ c{F })  = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T C{S}C{S}c{S} 1  2 = 1  4 · 1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  1  24 = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T A{S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='ν}A{S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='F }c{F }  = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T A{ν}(C{S} ⊗ b{F },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T )(c{S} ⊗ c{F })  = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T A{ν}C{S}c{S} 1  2 = 1  12 · 1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  1  12 = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T A{S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='F }C{F }c{F }  = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T (C{S} ⊗ b{F },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T ) diag(c{S} ⊗ c{F })(c{S} ⊗ c{F })  = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T C{S}C{S}c{S} 1  3 = 1  4 · 1  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  1  24 = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T A{S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='F }A{F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='F }c{F }  = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T (C{S} ⊗ b{F },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T )(C{S} ⊗ A{F })(c{S} ⊗ c{F })  = b{σ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='T C{S}C{S}c{S} 1  6 = 1  4 · 1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16e) is not automatically satisfied and simplifies to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15a):  1  8 = b{F },T C{F }A{F,σ}c{S}  = (es{S} ⊗ b{F })T diag(c{S} ⊗ c{F })  �nΩ−1  �  k=0  Ω{k} ⊗ A{F }c{F }×k  �  c{S}  = eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 3)  �  c{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16f) is not automatically satisfied and simplifies to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  1  12 = b{F },T A{F,σ}C{S}c{S}  = (es{S} ⊗ b{F })T  �nΩ−1  �  k=0  Ω{k} ⊗ A{F }c{F }×k  �  C{S}c{S}  = eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  C{S}c{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16g) is not automatically satisfied and reduces to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15c) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15d)  when σ = I, E, respectively, and both conditions are enforced simultaneously,  1  24 = b{σ},T A{S,F }A{F,ν}c{S}  = b{σ},T (C{S} ⊗ b{F })  �nΩ−1  �  k=0  Ω{k} ⊗ A{F }c{F }×k  �  c{S}  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  = b{σ},T C{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  c{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16h) is not automatically satisfied but simplifies to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='12) minus  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15a),  1  24 = b{F },T A{F,F }A{F,σ}c{S}  = (es{S} ⊗ b{F })T (C{S} ⊗ A{F })  �nΩ−1  �  k=0  Ω{k} ⊗ A{F }c{F }×k  �  c{S}  = eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)(k + 3)  �  c{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16i) is not automatically satisfied and reduces to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15e) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15f)  when ν = I, E, respectively, and both conditions are enforced simultaneously:  1  24 = b{F },T A{F,σ}A{S,ν}c{S}  = (es{S} ⊗ b{F })T  �nΩ−1  �  k=0  Ω{k} ⊗ A{F }c{F }×k  �  A{S,ν}c{S}  = eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  A{S,ν}c{S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Finally, condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='16j) simplifies to the same condition as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='15b),  1  24 = b{F },T A{F,σ}A{S,F }c{F }  = (es{S} ⊗ b{F })  �nΩ−1  �  k=0  Ω{k} ⊗ A{F }c{F }×k  �  (C{S} ⊗ b{F })(c{S} ⊗ c{F })  = eT  s{S}  �nΩ−1  �  k=0  Ω{k}  1  (k + 1)(k + 2)  �  C{S}c{S} 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7 (Minimum nΩ for third-order accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' An IMEX-MRI-SR re-  quires at least nΩ = 2 for third-order accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' An IMEX-MRI-SR method with one Ω matrix, Ω{0}, has A{S,E} = Ω = Ω0  and, from the FSAL property, b{E},T = eT  s{S}Ω = eT  s{S}Ω{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' For an IMEX-MRI-SR  method of order p, we assume the slow-explicit base method also satisfies all standard  Runge–Kutta order conditions up to order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Thus, a third-order IMEX-MRI-SR slow-  explicit base method with one Ω satisfies the second order condition b{E},T c{S} = 1  2,  which simplifies to  1  2 =  s{S}  �  j=1  ω{0}  s{S},jc{S}  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  However, the third-order IMEX-MRI-SR coupling condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='12) simplifies to  1  3 =  s{S}  �  j=1  ω{0}  s{S},jc{S}  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  13  Since these cannot hold simultaneously, a third-order IMEX-MRI-SR method with one  Ω matrix is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' There are no such mutually exclusive conditions for nΩ > 1,  and in Section 4 we introduce third and a fourth order methods with nΩ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Linear Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' As in [3, 22], we analyze the stability of IMEX-MRI-SR  methods when applied to the linear, scalar test problem  y′(t) = λ{F }y + λ{E}y + λ{I}y,  t ≥ 0,  y(0) = 1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1)  with λ{F }, λ{E}, λ{I} ∈ C−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' For convenience, we additionally define z{F } = Hλ{F },  z{E} = Hλ{E}, and z{I} = Hλ{I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' When applied to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1), the IVP (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5b) becomes  v′  i(θ) = λ{F }vi(θ) +  1  c{S}  i  i−1  �  j=1  ωi,j  �  θ  c{S}  i  H  �  (λ{E}Y {S}  j  + λ{I}Y {S}  j  )  = λ{F }vi(θ) +  1  c{S}  i  i−1  �  j=1  nΩ−1  �  k=0  ω{k}  i,j  �  θ  c{S}  i  H  �k  (λ{E}Y {S}  j  + λ{I}Y {S}  j  )  for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', s{S}, with θ ∈ [0, c{S}  i  H] and vi(0) = yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The solution to this at  θ = c{S}  i  H is  vi(c{S}  i  H) = ec{S}  i  z{F }yn + (z{E} + z{I})  i−1  �  j=1  nΩ−1  �  k=0  ω{k}  i,j  �� 1  0  ec{S}  i  z{F }(1−t)tkdt  �  Y {S}  j  = ϕ0(c{S}  i  z{F })yn + (z{E} + z{I})  i−1  �  j=1  ηi,j(z{F })Y {S}  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Here we define η as in [3] as a function of the fast eigenvalue z{F },  ηi,j(z{F }) =  nΩ−1  �  k=0  ω{k}  i,j ϕk+1(c{S}  i  z{F }),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2)  where the family of analytical functions {ϕk} are given by [22],  ϕ0(z) = ez,  ϕk(z) =  � 1  0  ez(1−t)tk−1dt,  k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The ith IMEX-MRI-SR stage for the stability problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1) then becomes  Y {S}  i  = ϕ0(c{S}  i  z{F })yn + (z{E} + z{I})  i−1  �  j=1  ηi,j(z{F })Y {S}  j  + z{I}  i  �  j=1  γi,jY {S}  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3)  Concatenating Y = [Y {S}  1  · · Y {S}  s{S}]T and writing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3) in matrix form,  Y = ϕ0(c{S}z{F })yn + (z{E} + z{I})η(z{F })Y + z{I}ΓY  =  �  I − (z{E} + z{I})η(z{F }) − z{I}Γ  �−1  ϕ0(c{S}z{F })yn  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  where  η(z{F }) =  nΩ−1  �  k=0  diag(ϕk+1(c{S}z{F }))Ω{k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Thus the stability function for an IMEX-MRI-SR method applied to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1) is  R(z{F },z{E}, z{I}) =  eT  s{S}  �  I − (z{E} + z{I})η(z{F }) − z{I}Γ  �−1  ϕ0(c{S}z{F }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4)  We consider a few definitions of joint stability for IMEX-MRI-SR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' As  with IMEX-MRI-GARK methods, we consider a region that incorporates all three z,  Jα,ρ,β,ξ = {z{E} ∈ C− : |R(z{F }, z{E}, z{I})| ≤ 1, ∀z{F } ∈ S{F }  α,ρ , ∀z{I} ∈ S{I}  β,ξ },  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5)  where S{σ}  α,ρ = {z{σ} ∈ C− : | arg(z{σ}) − π| ≤ α, |z{σ}| ≤ ρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We note that this  definition of joint stability may be overly-restrictive, as it demands the implicit and  fast parts of the method to be A(β)- and A(α)-stable, respectively, for any joint  stability region to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Thus, we also analyze the implicit and explicit stability  regions of IMEX-MRI-SR methods independently, with stability regions defined as:  J {I}  α,ρ = {z{I} ∈ C− : |R(z{F }, 0, z{I})| ≤ 1, ∀z{F } ∈ S{F }  α,ρ },  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6)  J {E}  α,ρ = {z{E} ∈ C− : |R(z{F }, z{E}, 0)| ≤ 1, ∀z{F } ∈ S{F }  α,ρ }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7)  These are more consistent with standard stability analyses of ARK methods, wherein  explicit and implicit stability are considered separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Example Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We introduce three IMEX-MRI-SR methods of orders  2, 3 and 4, each of which includes an embedding with accuracy one order lower (1, 2  and 3, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=') for temporal error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We note that we designed the embedding  coefficients with efficiency in mind, so that computation of the embedded solutions  do not require an additional implicit solve at the slow time scale (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', ˆγs{S} = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  When presenting the coefficients for each method we use the notation  Ω{k} =  �  �Ω{k}  ˆω{k}  �  � , Γ =  �  �Γ  ˆγ  �  � ,  where Ω{k} and Γ are the matrices of coefficients defining the primary method, and  ˆω{k} and ˆγ are the embedding row of coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' IMEX-MRI-SR2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The first method is second-order with a first-order  embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' It has 4 stages, nΩ = 1, and requires 3 slow nonlinear solves per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The coefficients can be found in Appending A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  To create this method, we used the free coefficients of the primary method to  maximize the size of the stability region defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We used the free coefficients  of the embedded method to minimize the value of the C-statistic from Prince and  Dormand [18],  C(p+1) = ∥ˆτ (p+1) − τ (p+1)∥2  ∥ˆτ (p)∥2  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1)  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  15  where τ (p) is the vector of order condition residuals at order p for the primary method,  ˆτ (p) is the vector of order condition residuals at order p for the embedded method, and  p is the order of the primary method (in this case, p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The C-statistic gives an  estimate of how much the error stemming from unsatisfied (p + 1)-order conditions of  the primary method corrupts the error estimate provided by the embedded method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  This method has large, robust stability regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Figures 1a and 1b show Jα,102,β,104  for α ∈ {10◦, 45◦} and varying β, along with the explicit slow base method’s stability  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We see that the multirate method has a stability region essentially identical  to the explicit slow base method when α = 10◦ for any value of z{I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' When α grows  to 45◦ we can see a decay of the stability region associated with the β = 85◦, while  the regions associated with smaller β values experience negligible decay, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Figure 1c shows J {E}  α,102 for varying α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We see that when z{I} = 0, the stability  region is again large, with only a slight decay in area when α = 85◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Figure 1c shows J {I}  α,102 for varying α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' When z{E} = 0, the multirate method is  A-stable for 0◦ ≤ α ≤ 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' When α grows further to 65◦, the region decays slightly  but is still approximately A(80◦)-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' When α = 85◦, the stability region decays  to an enclosed bubble similar to the regions in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (a) J10◦,102,α,104  (b) J45◦,102,α,104  (c) J {E}  α,102  (d) J {I}  α,102  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' 1: Joint Stability Regions for IMEX-MRI-SR2(1)  4  2  Explicit  β=10°  0  β=45°  2  β=65°  β=85°  4  6  4  2  0  Re4  2  Explicit  β=10°  0  β=45°  2  β=65°  β=85°  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4  6  4  2  0  Re4  2  Base  0  α=10°  α=45°  2  α=65°  α=85°  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4  6  4  2  0  Re6  4  2  Base  m  0  α=10°  2  α=45°  α=65°  4  α=85°  6  10-8  6  4  2  0  2  Re16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' IMEX-MRI-SR3(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Our second method is third-order with a second-  order embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' It has 5 stages, nΩ = 2, and requires 4 nonlinear solves per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The coefficients can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  To create this method, we used the free coefficients for both the method and  embedding as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1, this time using the C-statistic (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1) with p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Figures 2a and 2b show the relatively large joint stability regions Jα,102,β,104  for α = {10◦, 45◦}, respectively, along with the explicit slow base method’s stability  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Again, at α = 10◦ and lower values of β, the method is essentially as stable as  the explicit slow base method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The areas of these stability regions decrease for higher  values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' When α grows to 45◦ in Figure 2b, the stability regions shrink slightly  in comparison with α = 10◦ from Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Figures 2c and 2d show J {E}  α,102 and J {I}  α,102 for varying α, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We see that  when z{I} = 0, the stability region is reasonably large for smaller value of α, but the  region shrinks as α grows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' however, even for α = 85◦, the region retains a good extent  along the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Similar to the second-order method, when z{E} = 0, the  stability region is A-stable for most values of α, only losing A-stability for α = 85◦,  where the region decays to a rather large, yet enclosed, bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (a) J10◦,102,β,104  (b) J45◦,102,β,104  (c) J {E}  α,102  (d) J {I}  α,102  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' 2: Joint Stability Regions for IMEX-MRI-SR3(2)  4  2  Explicit  β=10°  0  β=45°  2  β=65°  β=85°  4  6  4  2  0  Re4  2  Explicit  β=10°  0  β=45°  2  β=65°  β=85°  4  6  4  2  0  Re4  2  Base  α=10°  0  α=45°  2  α=65°  α=85°  4  6  4  2  0  Re30  20  10  Base  m  0  α=10°  10  α=45°  20  α=65°  α=85°  30  50-40-30-20-10  0  10  ReIMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' IMEX-MRI-SR4(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Our final method is fourth-order with a third-order  embedding, has 7 stages, nΩ = 2, and requires 5 nonlinear solves per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The  coefficients can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Due to the number of order conditions involved when simultaneously solving all  IMEX-MRI-SR coupling conditions and base ARK order conditions at fourth-order,  we based this method off of an existing 4(3) ARK method, LIRK4 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Since this was  not stiffly accurate, then as discussed in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 we padded the A{E} and A{I}  matrices with the b vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=',  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2)  A{S,E} =  �  �A{E}  0{6}  bT  0  �  � , A{S,I} =  �  �A{I}  0{6}  bT  0  �  � ,  where 0{6} ∈ R6 is all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Since the last row of A{S,E} and A{S,I} both equal bT , the last row of Γ equals  zero, as follows from the definitions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7b) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' When A{S,I} has a zero in  the bottom right entry (and therefore Γ has a zero in the bottom right entry, from  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7c)), there is no nonlinear solve required to compute this last stage and therefore  the updated time step solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We believe that this negatively affects stability, as  we will show in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  As before, we used the free variables of the IMEX-MRI-SR method to optimize  stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This method had an empty joint stability region defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5) for all of  our attempts to choose or optimize values of the free variables, so we instead optimized  the size of the stability region defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' To optimize the embedded method,  we minimized the 2-norm of the fourth-order condition residuals, ∥ˆτ (4)∥2, to reduce  the overall error in the embedded solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We note that our previous approach of  minimizing the C-statistic (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1) was not possible since we do not yet have the fifth-  order IMEX-MRI-SR coupling conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Figure 3a shows the stability regions J {E}  α,1  for varying α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Unlike the lower-order  methods, this method’s explicit stability region never fully matches that of the explicit  base method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Similarly to the other methods, as α grows the stability region shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Figure 3b shows the stability regions J {I}  α,1 for varying α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Again unlike the lower-  order methods, these regions are not A-stable for α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Because J {I}  α,ρ is never  A-stable for this method, the stability region (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5) is always empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We suspect that  this is primarily due to the lack of an implicit solve in the last stage of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  In future work we plan to investigate this issue in more detail to better understand  the factors that contribute to IMEX-MRI-SR joint stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' MERK Methods as Explicit IMEX-MRI-SR Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' In [13], Luan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' define MERK methods with orders of accuracy spanning two through five by  explicitly defining the abscissae c{S}  i  and the forcing functions gi(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Due to their  similar structure to IMEX-MRI-SR methods, we may analyze MERK methods using  our theory from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Because MERK methods are always explicit, their IMEX-  MRI-SR Γ matrices will be all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We recall that MERK methods are defined under  an assumption that the fast function is linear, f {F }(t, y) = Ly, but that the slow  function can be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' It is thus natural to assume that MERK methods might  only satisfy a subset of the IMEX-MRI-SR order conditions, potentially failing those  that handle nonlinearity in f {F }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  (a) J {E}  α,102  (b) J {I}  α,102  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' 3: Joint Stability Regions for IMEX-MRI-SR4(3)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' MERK2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Converting the three-stage second-order accurate MERK2 method  from [13] into IMEX-MRI-SR form, we have  Ω{0} =  �  ����  0  0  0  c{S}  2  0  0  1  0  0  �  ���� ,  Ω{1} =  �  ����  0  0  0  0  0  0  −1/c{S}  2  1/c{S}  2  0  �  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1)  Interestingly, these coefficients (along with Γ = 0) satisfy all coupling conditions up  to third order, and the slow base method determined by Ω satisfies all conditions  up to order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Thus, we expect MERK2 to have second-order accuracy, even for  nonlinear f {F }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We confirm this with numerical tests involving nonlinear f {F } in  Section 6, where we use c{S}  2  = 1  2 since it was unspecified in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Figure 4a shows the stability regions for MERK2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Because this method is explicit,  we only plot J {E}  α,102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Notably, these regions nearly match that of the base explicit  method for most values of α examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' MERK3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Converting the four-stage, third-order accurate, MERK3 from  [13] to an IMEX-MRI-SR method, we have  Ω{0} =  �  �������  0  0  0  0  c{S}  2  0  0  0  2  3  0  0  0  1  0  0  0  �  �������  ,  Ω{1} =  �  �������  0  0  0  0  0  0  0  0  −  2  3c{S}  2  2  3c{S}  2  0  0  − 3  2  0  3  2  0  �  �������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2)  These satisfy all coupling conditions up to third order, and the corresponding slow  base method satisfies all conditions up to order three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Thus similarly to MERK2, we  expect it to show third-order accuracy on problems with nonlinear f {F }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We again  confirm this result on numerical tests in Section 6 using c{S}  2  = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Figure 4b shows the stability regions for MERK3, J {E}  α,102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Notably, these regions  show no degradation of stability as α is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  4  2  Base  0  α=10°  α=45°  2  α=65°  α=85°  4  6  4  2  0  Re4  2  Base  0  α=10°  α=45°  2  α=65°  α=85°  4  6  4  2  0  ReIMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  19  (a) MERK2  (b) MERK3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' 4: J {E}  α,102 Regions for MERK2 and MERK3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' MERK4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We may express MERK4 as an IMEX-MRI-SR method with 7  stages and nΩ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The corresponding coefficients are provided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We find that this method satisfies all IMEX-MRI-SR coupling conditions through  order four, and its slow base method satisfies all order conditions up through fourth  order, so long as c{S}  6  = (3−4c{S}  5  )/(4−6c{S}  5  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This restriction on c{S}  6  is satisfied by  the choices in [13] of c{S}  2  = 1/2, c{S}  3  = 1/2, c{S}  4  = 1/3, c{S}  5  = 5/6, and c{S}  6  = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Thus like before, we expect this method to demonstrate fourth-order accuracy for  nonlinear f {F }, which we confirm numerically in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Interestingly, the joint  stability regions J {E}  α,102 for MERK4 are empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' However, given the reliability of this  method in [13] and our own results from Section 6, we believe that these empty  regions more strongly indicate a deficiency in these definitions of joint stability than  any actual issues with MERK4 itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' MERK5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The IMEX-MRI-SR method that corresponds with MERK5 has  11 stages and nΩ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' These coefficients are given in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' When analyzing  this method, we find that when  c{S}  9  = 12 − 15c{S}  10  − 15c{S}  8  + 20c{S}  10 c{S}  8  15 − 20c{S}  10  − 20c{S}  8  + 30c{S}  10 c{S}  8  the method satisfies all IMEX-MRI-SR coupling conditions up through fourth-order,  and its slow base method satisfies all order conditions up through fifth-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This  condition is satisfied by the choice in [13] of c{S}  2  = 1/2, c{S}  3  = 1/2, c{S}  4  = 1/3,  c{S}  5  = 1/2, c{S}  6  = 1/3, c{S}  7  = 1/4, c{S}  8  = 7/10, c{S}  9  = 1/2, and c{S}  10  = 2/3,  therefore we expect it to have at least fourth-order accuracy without restriction on  the linearity of f {F }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' As we do not have fifth-order IMEX-MRI-SR order conditions,  we cannot check these for MERK5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' however, our numerical tests in Section 6 indeed  show fifth-order convergence for nonlinear f {F }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' As with MERK4, the MERK5 joint  stability regions J {E}  α,102 are empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Numerical Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' In this section we examine the convergence rates for  our newly-proposed IMEX-MRI-SR methods from Section 4, as well as for MERK  4  2  Base  0  α=10°  α=45°  2  α=65°  α=85°  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4  6  4  2  0  Re4  2  Base  0  α=10°  α=45°  2  α=65°  α=85°  6  4  2  0  Re20  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  methods applied to problems with nonlinear f {F }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We also compare the efficiency of  our IMEX-MRI-SR methods against both IMEX-MRI-GARK and legacy Lie-Trotter  and Strang-Marchuk methods from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' KPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The Kværnø-Prothero-Robinson problem is a coupled system of  IVPs which has been widely used for testing multirate algorithms, since it is non-  linear, non-autonomous, includes stiffness and multirate tuning parameters, and has  an analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We use the same formulation and partitioning as in [3],  �  �u′  v′  �  � = Λ  �  �  −3+u2−cos(βt)  2u  −2+u2−cos(t)  2v  �  � −  �  �  β sin(βt)  2u  sin(t)  2v  �  � ,  �  �u(0)  v(0)  �  � =  �  � 2  √  3  �  �  Λ =  �  �  λ{F }  1−ε  α (λ{F } − λ{S})  −αε(λ{F } − λ{S})  λ{S}  �  �  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1)  for t ∈ [0, 5π/2], with parameters λ{F } = −10, λ{S} = −1, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1, α = 1, and  β = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This problem has solution  u(t) =  �  3 + cos(βt),  v(t) =  �  2 + cos(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2)  We partition the problem as  f {E} =  �  �  0  sin(t)  2v  �  � ,  f {I} =  �  �0  0  0  1  �  � Λ  �  �  −3+u2−cos(βt)  2u  −2+u2−cos(t)  2v  �  �  f {F } =  �  �1  0  0  0  �  � Λ  �  �  −3+u2−cos(βt)  2u  −2+u2−cos(t)  2v  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3)  In Figure 5, we plot the convergence as H is refined for MERK methods and  implicit-explicit methods, including all provided IMEX-MRI-SR methods from Sec-  tion 4, IMEX-MRI-GARK3(a,b), IMEX-MRI-GARK4, Lie-Trotter and Strang-Marchuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We see that all methods converge at their expected rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Notably, as expected from  Section 5, the MERK methods show no convergence issues even though f {F } in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3)  is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  In these tests, we combined multirate methods with explicit inner Runge–Kutta  methods of the same order, with the only exception of pairing a second-order inner  solver with the first-order Lie-Trotter method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Each of Lie-Trotter, Strang-Marchuk,  and IMEX-MRI-SR2(1) used the second-order Heun method given by the Butcher  table  0  0  0  1  1  0  1/2  1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The IMEX-MRI-SR3(2) and IMEX-MRI-GARK3(a,b) methods  used the third-order method by Bogacki and Shampine [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The IMEX-MRI-SR4(3)  and IMEX-MRI-GARK4 methods used the fourth-order method by Zonneveld [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We measured error at 10 equally-spaced points in the time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The estimated  convergence rates for each method, using a least squares fit of log(Max Error) versus  log(H), are in the legend parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' For the MERK convergence tests, we used  H = π/2k, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', 9 and for the implicit-explicit method convergence tests, we used  H = π/2k, k = 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' For all tests we used fast time step size h = H/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  21  implicit-explicit methods used a standard Newton-Raphson method with a banded  linear solver for the implicit solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  10  2  10  1  100  H  10  13  10  11  10  9  10  7  10  5  10  3  10  1  Max Error  MERK2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='06)  MERK3 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='17)  MERK4 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='25)  MERK5 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='32)  10  2  10  1  H  10  11  10  9  10  7  10  5  10  3  10  1  Max Error  Lie-Trotter (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='01)  Strang-Marchuk (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='98)  IMEX-MRI-SR2(1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='01)  IMEX-MRI-SR3(2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='11)  IMEX-MRI-SR4(3) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='85)  IMEX-MRI-GARK3a (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='05)  IMEX-MRI-GARK3b (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='09)  IMEX-MRI-GARK4 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='11)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' 5: Convergence for the KPR test problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1) for MERK methods (left) and  implicit-explicit methods (right) using the partitioning (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' All methods converge  at the expected theoretical rates (with measured convergence rates in parentheses),  including MERK methods using the given nonlinear fast partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Stiff Brusselator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The stiff brusselator problem is an advection-reaction-  diffusion system of nonlinear partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' It is a modification to  the standard brusselator [7] used in [3], from which we use the same formulation and  partitioning:  ut = αuuxx + ρuux + ru(a − (w − 1)u + u2v),  vt = αvvxx + ρvvx + rv(wu − u2v),  wt = αwwxx + ρwwx + rw(b − w  ε  − wu),  for t ∈ [0, 3] and x ∈ [0, 1], with stationary boundary conditions  ut(t, 0) = ut(t, 1) = vt(t, 0) = vt(t, 1) = wt(t, 0) = wt(t, 1),  and initial conditions  u(0, x) = a + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 sin(πx),  v(0, x) = b/a + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 sin(πx),  w(0, x) = b + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 sin(πx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We partition the problem as  f {I} =  �  ����  αuuxx  αvvxx  αwwxx  �  ���� ,  f {E} =  �  ����  ρuux  ρvvx  ρwwx  �  ���� ,  f {F } =  �  ����  ru(a − (w − 1)u + u2v)  rv(wu − u2v)  rw( b−w  ε  − wu)  �  ���� ,  and use second order centered difference approximations for all spatial derivative  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This problem has no analytical solution, so we used MATLAB’s ode15s  with tight tolerances AbsTol = 10−14 and RelTol = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 × 10−14 to generate reference  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  22  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Fixed Time Step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' In this section, we compare runtime efficiency of the  splitting, IMEX-MRI-SR, and IMEX-MRI-GARK methods using fixed time step sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We use the same fixed parameters αu = αv = αw = 10−2, ρu = ρv = ρw = 10−3,  ru = rv = rw = 1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6, b = 2, and ε = 10−2 with initial conditions  u(0) = a + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 sin(πx), v(0) = b/a + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 sin(πx), w(0) = b + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 sin(πx),  for 201 and 801 grid points as in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' All methods used fast time steps of h = H/10  and all methods used the same inner methods and implicit algebraic solvers as in  Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  100  101  102  103  Runtime (s)  10  10  10  8  10  6  10  4  10  2  Max Error  201 grid  101  102  103  104  Runtime (s)  10  10  10  8  10  6  10  4  10  2  Max Error  801 grid  Lie-Trotter  Strang-Marchuk  IMEX-MRI-SR2(1)  IMEX-MRI-SR3(2)  IMEX-MRI-SR4(3)  IMEX-MRI-GARK3a  IMEX-MRI-GARK3b  IMEX-MRI-GARK4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' 6:  Efficiency for stiff brusselator problem using 201 grid points (left)  and 801 grid points (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Estimated least-squares convergence rates be-  fore settling at the error-floor are (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='76,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='59,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='00,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='09,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='00,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='36,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='25,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='41) and  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='72,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='24,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='01,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='90,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='90,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='71,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='69,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='42) for the 201 and 801 grids, respectively, for  Lie-Trotter, Strang-Marchuk, IMEX-MRI-SR2(1), IMEX-MRI-SR3(2), IMEX-MRI-  SR4(3), IMEX-MRI-GARK3a, IMEX-MRI-GARK3b, IMEX-MRI-GARK4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  In Figure 6 we plot the observed maximum solution error over ten equally spaced  points in the time interval using step sizes of H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1·2−k, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Both splitting  methods and both fourth-order methods experience significant order reduction, with  IMEX-MRI-SR4(3) taking the biggest hit in reducing an entire order of accuracy for  the 201 grid, and two orders of accuracy for the 801 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The second- and third-order  IMEX-MRI-SR and IMEX-MRI-GARK methods all achieve their expected order for  the 201 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The third-order accurate methods experience only slight order reduction  for the 801 grid while IMEX-MRI-SR2(1) remains steady at its expected order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' All  methods exhibit an error floor of approximately 10−11, that is likely caused by the  accuracy of the reference solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The stiffness of this problem highlights the stability limitations of the fourth  order methods, IMEX-MRI-SR4(3) and IMEX-MRI-GARK4, which was observed in  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' For the 201 grid, IMEX-MRI-SR4(3) and IMEX-MRI-GARK4 were unstable for  step sizes greater than 1/320 and 1/80 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' For the 801 grid, IMEX-MRI-  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  23  SR4(3) and IMEX-MRI-GARK4 were unstable for step sizes greater than 1/640 and  1/160, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We can see that IMEX-MRI-SR2(1) is far more efficient than the first- and second-  order splitting methods, providing errors two to three orders of magnitude smaller for  the same runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' It also has a steady rate of error decrease as runtime increases,  while the splitting methods show periods of stagnation at larger step sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The third-order and fourth-order methods tend to have similar efficiency on this  problem, attaining similar error for similar runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The third-order IMEX-MRI-  GARK methods have a slight edge for the 201 grid and for some error ranges in the  801 grid, but IMEX-MRI-SR3(2) becomes the most efficient method for the 801 grid  for errors between approximately 10−7 and 10−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The third-order methods are all  more efficient than the fourth-order methods, where IMEX-MRI-SR4(3) maintains a  slight but consistent edge over IMEX-MRI-GARK4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Adaptive Time Step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' In this section, we compare work-precision effi-  ciency for the IMEX-MRI-SR methods in the adaptive time step context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Because  the splitting and IMEX-MRI-GARK methods do not have embeddings, we omit them  from these tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We use the Constant-Constant controller from [5] with the recom-  mended parameters k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='42, k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='44, and the recommended fast error measure-  ment strategy, LASA-mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This controller adapts H and M, such that the inner  time step size h = H/M, in a similar manner to a standard I-controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We use time-varying parameters αu = αv = αw = ρu = ρv = ρw = 6 × 10−5 +  5 × 10−5 cos(πt), ru = rv = rw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 cos(4πt) adapted from [5] with the same  fixed parameters a = 1, b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5, ε = 10−3 and initial conditions  u(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 sin(πx), v(0) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 sin(πx), w(0) = 3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 sin(πx),  with 101 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' All IMEX-MRI-SR methods used the same inner methods and  implicit algebraic solvers as in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  105  106  107  Fast Function Evaluations  10  10  10  9  10  8  10  7  10  6  10  5  10  4  10  3  Max Error  102  103  104  105  Implicit Solves  10  10  10  9  10  8  10  7  10  6  10  5  10  4  10  3  Max Error  IMEX-MRI-SR2(1)  IMEX-MRI-SR3(2)  IMEX-MRI-SR4(3)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' 7: Fast function evaluations (left) and total implicit solves (right) versus the  observed maximum error for the stiff brusselator problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  In Figure 7 we plot the number of fast function evaluations and the number of  implicit solves, good indicators of overall cost at the fast and slow timescales, versus  24  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  the observed maximum solution error over ten equally spaced points in the time  interval when running with controller tolerance values of tol = 10−k, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We use tol{S} = tol{F } = 1  2tol in our tests for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We note that IMEX-  MRI-SR3(2) failed the tests with tol = 10−3, 10−5 and IMEX-MRI-SR4(3) failed the  tests with tol = 10−4, 10−5 due to getting stuck in oscillations between successful  and failed steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We can see that IMEX-MRI-SR2(1) is much less efficient than the higher order  methods in terms of fast function evaluations (farthest from the bottom-left corner),  where IMEX-MRI-SR3(2) is generally the most efficient (closest to the bottom-left  corner) across the range of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  IMEX-MRI-SR4(3) is generally comparable to  IMEX-MRI-SR3(2) in terms of fast function evaluations but occasionally gets “stuck,”  providing approximately the same error value for different total fast function evalu-  ations, depending on the value of tol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' This is likely an indication that IMEX-MRI-  SR4(3) has a low quality embedding which provides inaccurate error estimates, pos-  sibly stemming from the embedding coefficients being defined by minimizing ∥ˆτ (4)∥2  rather than the C-statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The total number of implicit solves is comparable across all three methods for  errors larger than approximately 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Below that, IMEX-MRI-SR3(2) and IMEX-  MRI-SR4(3) are again comparable with IMEX-MRI-SR2(1) falling further behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We see the same phenomenon of IMEX-MRI-SR4(3) getting “stuck” at certain error  values, achieving the same error for varying total implicit solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We introduce a new class of multirate time integration methods  which builds off of previous work in IMEX-MRI-GARK and MERK methods, serving  to improve various aspects of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The proposed class of IMEX-MRI-SR methods  are flexible, allowing IMEX treatment of the slow time scale while allowing the use  of any viable IVP solver for the fast time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' These methods remove the sorted  abscissae requirement of IMEX-MRI-GARK methods since they start each internal  stage at the beginning of the time step, thereby dramatically simplifying their order  conditions and allowing introduction of embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' IMEX-MRI-SR methods can  also be viewed as an extension to MERK methods, in that these allow implicitness at  the slow time scale and nonlinearity at the fast time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  The convergence theory of IMEX-MRI-SR methods leverages GARK theory [23],  through which we established order conditions for methods of orders one through four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Due to their structural similarity to IMEX-MRI-SR methods, we leveraged these new  order conditions to analyze the previously-proposed MERK methods without their  restriction to linear fast partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Using this analytical framework, we provided the  first theoretical justification that these MERK methods (at least up through fourth  order) should retain their high orders of accuracy even on problems with nonlinear  fast partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We analyzed joint stability as in [3], as well as in simplified implicit- or explicit-  only senses of joint stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' With these reduced definitions of stability, we gain some  insight into what happens when joint stability regions break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Using this theoretical framework, we provided three new IMEX-MRI-SR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  These included a second-order method with a first-order embedding, and a third-order  method with a second-order embedding, both derived from scratch by solving all order  conditions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We additionally provided a fourth-order method with a  third-order embedding, that we derived from the existing LIRK4 ARK method [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We experimentally examined convergence for the three new methods and the ex-  isting MERK methods on the KPR test problem, finding that they all converge at  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  25  their expected orders of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We also provided experimental efficiency results on  the stiff brusselator problem, a nonlinear advection-reaction-diffusion system of PDEs,  where we found that our methods are competitive with IMEX-MRI-GARK methods  and even surpass them in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We also found that IMEX-MRI-SR2(1) is  vastly more efficient than the similarly second-order Strang-Marchuk operator split-  ting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' We found that in the context of adaptive time stepping, the third- and  fourth-order IMEX-MRI-SR methods were comparable in work required for achieving  a given error, while the second-order method lagged behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  More work remains to be done on IMEX-MRI-SR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Higher order condi-  tions can be derived which, while tedious due to the number of conditions at higher  orders, is tractable due to the use of Kronecker product identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Further analy-  sis should be done on the factors that most strongly affect joint stability and what  conditions, if any, can be enforced to ensure a non-empty joint stability region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ad-  ditionally, more methods should be derived, in particular an embedded fourth-order  method with improved joint stability over the method provided here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' The authors would like to thank Vu Thai Luan for his  help in gaining deeper understanding of MERK methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We would also like to  thank the SMU Center for Research Computing for use of the Maneframe2 computing  cluster, where we performed all simulations reported in this work  REFERENCES  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
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+page_content=' Roberts, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Sandu, Design of High-Order Decoupled Multirate GARK  Schemes, SIAM Journal on Scientific Computing, 41 (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' A816–A847, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
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+page_content='  [25] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Strang, On the construction and comparison of difference schemes, SIAM Journal on  Numerical Analysis, 5 (1968), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' 506–517, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
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+page_content='  org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
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+page_content='  [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Wensch, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Knoth, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Galant, Multirate infinitesimal step methods for atmospheric  flow simulation, BIT Numerical Mathematics, 49 (2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' 449–473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Zonneveld, Automatic integration of ordinary differential equations, Stichting Mathema-  tisch Centrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Rekenafdeling, 743 (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  27  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' IMEX-MRI-SR2(1) Coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  c{S} =  �  �������  0  3  5  4  15  1  �  �������  , Ω{0} =  �  ����������  0  0  0  0  3  5  0  0  0  14  165  2  11  0  0  − 13  54  137  270  11  15  0  − 1  4  1  2  3  4  0  �  ����������  , Γ =  �  ����������  0  0  0  0  − 11  23  11  23  0  0  − 6692  52371  − 18355  52371  11  23  0  11621  90666  − 215249  226665  17287  50370  11  23  − 31  12  − 1  6  11  4  0  �  ����������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
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+page_content='�������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1)  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' MERK4 IMEX-MRI-SR Coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We list the non-zero  coefficients of the MERK5 method’s IMEX-MRI-SR formulation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Ω{0}  i,1 = c{S}  i  , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=', s{S}  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1)  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  29  Ω{1}  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −Ω{1}  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 = c{S}×2  3  c{S}  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −Ω{1}  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 = c{S}×2  4  c{S}  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Ω{1}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −(Ω{1}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 + Ω{1}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 = −  c{S}  4  c{S}×2  5  c{S}  3  (c{S}  3  − c{S}  4  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 =  c{S}  3  c{S}×2  5  c{S}  4  (c{S}  3  − c{S}  4  )  Ω{1}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −(Ω{1}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 + Ω{1}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 = −  c{S}  4  c{S}×2  6  c{S}  3  (c{S}  3  − c{S}  4  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 =  c{S}  3  c{S}×2  6  c{S}  4  (c{S}  3  − c{S}  4  )  Ω{1}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −(Ω{1}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 + Ω{1}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 = −  c{S}  6  c{S}  5  (c{S}  5  − c{S}  6  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 =  c{S}  5  c{S}  6  (c{S}  5  − c{S}  6  )  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2)  Ω{1}  5,1 = −(Ω{1}  5,3 + Ω{1}  5,4 ), Ω{1}  5,3 =  1  c{S}  3  (c{S}  3  − c{S}  4  )  , Ω{1}  5,4 = −  1  c{S}  4  (c{S}  3  − c{S}  4  )  Ω{1}  6,1 = −(Ω{1}  6,3 + Ω{1}  6,4 ), Ω{1}  6,3 =  1  c{S}  3  (c{S}  3  − c{S}  4  )  , Ω{1}  6,4 = −  1  c{S}  4  (c{S}  3  − c{S}  4  )  Ω{1}  7,1 = −(Ω{1}  7,5 + Ω{1}  7,6 ), Ω{1}  7,5 =  1  c{S}  5  (c{S}  5  − c{S}  6  )  , Ω{1}  7,6 = −  1  c{S}  6  (c{S}  5  − c{S}  6  )  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3)  This method attains general fourth-order accuracy when  c{S}  6  = 3 − 4c{S}  5  4 − 6c{S}  5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4)  30  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' FISH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' REYNOLDS, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' ROBERTS  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' MERK5 IMEX-MRI-SR Coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  We list the non-zero coefficients of the MERK5 method’s IMEX-MRI-SR formu-  lation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1)  Ω{0}  i,1 = c{S}  i  , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' s{S}  Ω{1}  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −Ω{1}  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 = c{S}×2  3  α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Ω{1}  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −Ω{1}  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2 = c{S}×2  4  α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  Ω{1}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{1}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 + Ω{1}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 = c{S}×2  5  α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 = c{S}×2  5  α4  Ω{1}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{1}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 + Ω{1}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 = c{S}×2  6  α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 = c{S}×2  6  α4  Ω{1}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{1}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 + Ω{1}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 = c{S}×2  7  α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 = c{S}×2  7  α4  Ω{1}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{1}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 + Ω{1}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 + Ω{1}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 = c{S}×2  8  α5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 = c{S}×2  8  α6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7 = c{S}×2  8  α7  Ω{1}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{1}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 + Ω{1}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 + Ω{1}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 = c{S}×2  9  α5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 = c{S}×2  9  α6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7 = c{S}×2  9  α7  Ω{1}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{1}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 + Ω{1}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 + Ω{1}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 = c{S}×2  10  α5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 = c{S}×2  10  α6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7 = c{S}×2  10  α7  Ω{1}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{1}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='8 + Ω{1}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9 + Ω{1}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='10  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='8 = α8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9 = α9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{1}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='10 = α10  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='2)  where  α2 =  1  c{S}  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' α3 =  c{S}  3  c{S}  3  (c{S}  4  − c{S}  3  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' α4 =  c{S}  4  c{S}  4  (c{S}  3  − c{S}  4  )  α5 =  c{S}  6  c{S}  7  c{S}  5  (c{S}  5  − c{S}  6  )(c{S}  5  − c{S}  7  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' α6 =  c{S}  5  c{S}  7  c{S}  6  (c{S}  6  − c{S}  5  )(c{S}  6  − c{S}  7  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  α7 =  c{S}  5  c{S}  6  c{S}  7  (c{S}  7  − c{S}  5  )(c{S}  7  − c{S}  6  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' α8 =  c{S}  9  c{S}  10  c{S}  8  (c{S}  8  − c{S}  9  )(c{S}  8  − c{S}  10 )  α9 =  c{S}  8  c{S}  10  c{S}  9  (c{S}  9  − c{S}  8  )(c{S}  9  − c{S}  10 )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' α10 =  c{S}  8  c{S}  9  c{S}  10 (c{S}  10  − c{S}  8  )(c{S}  10  − c{S}  9  )  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3)  Ω{2}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{2}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 + Ω{2}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 = c{S}×2  5  β3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 = −c{S}×2  5  β4  Ω{2}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{2}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 + Ω{2}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 = c{S}×2  6  β3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 = −c{S}×2  6  β4  Ω{2}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{2}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 + Ω{2}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='3 = c{S}×2  7  β3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4 = −c{S}×2  7  β4  Ω{2}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{2}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 + Ω{2}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 + Ω{2}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 = −c{S}×2  8  β5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 = −c{S}×2  8  β6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7 = −c{S}×2  8  β7  Ω{2}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{2}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 + Ω{2}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 + Ω{2}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 = −c{S}×2  9  β5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 = −c{S}×2  9  β6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7 = −c{S}×2  9  β7  Ω{2}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{2}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 + Ω{2}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 + Ω{2}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 = −c{S}×2  10  β5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 = −c{S}×2  10  β6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7 = −c{S}×2  10  β7  Ω{2}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{2}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='8 + Ω{2}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9 + Ω{2}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='10  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='8 = β8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9 = β9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{2}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='10 = β10  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='4)  IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS  31  where  β3 =  1  c{S}  3  (c{S}  4  − c{S}  3  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' β4 =  1  c{S}  4  (c{S}  3  − c{S}  4  )  β5 =  c{S}  6  + c{S}  7  c{S}  5  (c{S}  5  − c{S}  6  )(c{S}  5  − c{S}  7  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' β6 =  c{S}  5  + c{S}  7  c{S}  6  (c{S}  6  − c{S}  5  )(c{S}  6  − c{S}  7  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  β7 =  c{S}  5  + c{S}  6  c{S}  7  (c{S}  7  − c{S}  5  )(c{S}  7  − c{S}  6  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' β8 =  c{S}  9  + c{S}  10  c{S}  8  (c{S}  8  − c{S}  9  )(c{S}  8  − c{S}  10 )  β9 =  c{S}  8  + c{S}  10  c{S}  9  (c{S}  9  − c{S}  8  )(c{S}  9  − c{S}  10 )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' β10 =  c{S}  8  + c{S}  9  c{S}  10 (c{S}  10  − c{S}  8  )(c{S}  10  − c{S}  9  )  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5)  Ω{3}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{3}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 + Ω{3}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 + Ω{3}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 = c{S}×2  8  γ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 = c{S}×2  8  γ6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7 = c{S}×2  8  γ7  Ω{3}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{3}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 + Ω{3}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 + Ω{3}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 = c{S}×2  9  γ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 = c{S}×2  9  γ6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7 = c{S}×2  9  γ7  Ω{3}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{3}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 + Ω{3}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 + Ω{3}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='5 = c{S}×2  10  γ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6 = c{S}×2  10  γ6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7 = c{S}×2  10  γ7  Ω{3}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='1 = −  �  Ω{3}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='8 + Ω{3}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9 + Ω{3}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='10  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='8 = γ8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='9 = γ9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content=' Ω{3}  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='10 = γ10  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='6)  where  γ5 =  1  c{S}  5  (c{S}  5  − c{S}  6  )(c{S}  5  − c{S}  7  )  , γ6 =  1  c{S}  6  (c{S}  6  − c{S}  5  )(c{S}  6  − c{S}  7  )  ,  γ7 =  1  c{S}  7  (c{S}  7  − c{S}  5  )(c{S}  7  − c{S}  6  )  , γ8 =  1  c{S}  8  (c{S}  8  − c{S}  9  )(c{S}  8  − c{S}  10 )  γ9 =  1  c{S}  9  (c{S}  9  − c{S}  8  )(c{S}  9  − c{S}  10 )  , γ10 =  1  c{S}  10 (c{S}  10  − c{S}  8  )(c{S}  10  − c{S}  9  )  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='7)  This method satisfies all available coupling conditions (up through fourth-order)  and its base method satisfies all order conditions up through fifth-order when  c{S}  9  = 12 − 15c{S}  10  − 15c{S}  8  + 20c{S}  10 c{S}  8  15 − 20c{S}  10  − 20c{S}  8  + 30c{S}  10 c{S}  8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
+page_content='8)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQf8_o-/content/2301.00865v1.pdf'}
diff --git a/VdE2T4oBgHgl3EQfXwdZ/content/tmp_files/2301.03847v1.pdf.txt b/VdE2T4oBgHgl3EQfXwdZ/content/tmp_files/2301.03847v1.pdf.txt
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@@ -0,0 +1,4069 @@
+Evaluating the Performance of
+Low-Cost PM2.5 Sensors in Mobile
+Settings
+Priyanka deSouza1,2*, An Wang3, Yuki Machida3, Tiffany Duhl4, Simone Mora3, Prashant
+Kumar5,6, Ralph Kahn7, Carlo Ratti3, John L Durant4, Neelakshi Hudda4
+1: Department of Urban and Regional Planning, University of Colorado Denver, CO 80217,
+USA
+2: CU Population Center, University of Colorado Boulder, Boulder CO 80309, USA
+3: MIT Senseable City Lab, Cambridge MA 02139, USA
+4: Department of Civil and Environmental Engineering, Tufts University, Medford, MA 02155,
+USA
+5: Global Centre for Clean Air Research (GCARE), School of Sustainability, Civil and
+Environmental Engineering, Faculty of Engineering and Physical Sciences, University of
+Surrey, Guildford GU2 7XH, Surrey, UK
+6: Institute for Sustainability, University of Surrey, Guildford GU2 7XH, Surrey, UK
+7: NASA Goddard Space Flight Center, Greenbelt MD 20771, USA
+*Corresponding author: priyanka.desouza@ucdenver.edu
+Abstract
+Low-cost sensors (LCS) for measuring air pollution are increasingly being deployed in
+mobile applications but questions concerning the quality of the measurements remain
+unanswered. For example, what is the best way to correct LCS data in a mobile setting?
+Which factors most significantly contribute to differences between mobile LCS data and
+higher-quality instruments? Can data from LCS be used to identify hotspots and generate
+generalizable pollutant concentration maps? To help address these questions we deployed
+low-cost PM2.5 sensors (Alphasense OPC-N3) and a research-grade instrument (TSI
+DustTrak) in a mobile laboratory in Boston, MA, USA. We first collocated these instruments
+with stationary PM2.5 reference monitors (Teledyne T640) at nearby regulatory sites. Next,
+using the reference measurements, we developed different models to correct the OPC-N3
+and DustTrak measurements, and then transferred the corrections to the mobile setting. We
+observed that more complex correction models appeared to perform better than simpler
+models in the stationary setting; however, when transferred to the mobile setting, corrected
+OPC-N3 measurements agreed less well with corrected DustTrak data. In general,
+corrections developed using minute-level collocation measurements transferred better to the
+mobile setting than corrections developed using hourly-averaged data. Mobile laboratory
+speed, OPC-N3 orientation relative to the direction of travel, date, hour-of-the-day, and road
+class together explain a small but significant amount of variation between corrected OPC-N3
+and DustTrak measurements during the mobile deployment. Persistent hotspots identified by
+the OPC-N3s agreed with those identified by the DustTrak. Similarly, maps of PM2.5
+distribution produced from the mobile corrected OPC-N3 and DustTrak measurements
+1
+
+agreed well. These results suggest that identifying hotspots and developing generalizable
+maps of PM2.5 are appropriate use-cases for mobile LCS data.
+1 Introduction
+Exposure to fine particulate matter (PM2.5; mass concentration of particles having an
+aerodynamic diameter of ≤2.5 μm) is a major environmental health risk in cities around the
+world1–6. Air quality monitoring is critical for developing effective pollution management plans;
+however, reference air quality monitors have high capital and operating costs. Even in the
+United States, which has one of the densest monitoring networks in the world, less than a
+third of US counties have a regulatory, also termed reference, monitor7. Such a sparse
+network cannot adequately characterize the spatial variation of air pollution in complex urban
+environments where pollutant concentrations can vary significantly at distances as small as
+a few 10s of meters8,9.
+Dense networks of low-cost sensors (LCS) (<USD $2500 as defined by the US EPA Air
+Sensor Toolbox10) are increasingly being used to fill PM2.5 concentration data gaps in cities
+around the world11–16. In particular, there is growing interest in using LCS in mobile
+monitoring applications to capture the granular spatial variations in pollution in cities at a
+relatively low-cost. However, the measurements from LCS are error-prone and need to be
+corrected prior to data interpretation17–20.
+Due to the growing use of LCS, regulatory agencies in the United States and Europe have
+developed protocols to evaluate LCS in the lab and in the field. Although these protocols
+offer an important point of reference to evaluate LCS, considerable challenges remain in
+using them. For example, the agreement of LCS measurements with those from reference
+monitors varies with  relative humidity (RH), temperature (T), as well as aerosol
+concentration, size distribution, and composition. Mobile deployments of LCS may introduce
+additional uncertainties due to vibrations, wind speed and vehicular disturbances of PM2.5
+mixing patterns. Therefore, LCS need to be systematically evaluated against reference
+monitors to appropriately correct high-frequency mobile measurements as well as to assess
+appropriate use-cases for mobile LCS data.
+To date, little research has evaluated LCS in mobile settings. We found only one study that
+systematically evaluated the performance of LCS for PM2.5 (PurpleAir PA-II monitors) against
+measurements from collocated reference instruments in a mobile platform21. This study
+found that the fidelity of the PurpleAir measurements decreased with increasing air velocity
+around the unit.
+We were interested in evaluating the transferability of correction models developed based on
+a stationary collocation experiment to a mobile deployment of a popular LCS, the
+Alphasense Optical Particle Counter (OPC-N3). The impact of correction model choice on
+the performance of an LCS in a mobile setting has not previously been reported in the
+literature. In addition, we evaluate the performance of corrected Alphasense OPC-N3s under
+different driving conditions and in different urban microenvironments. We report on which
+factors (i.e., sensor orientation, vehicle speed, road class, and meteorology) contribute to
+differences between the corrected OPC-N3 and PM2.5 measurements made using a research
+2
+
+grade monitor (TSI DustTrak). Finally, we evaluate the feasibility of using mobile LCS for two
+common use-cases: detecting PM2.5 hotspots and producing generalizable maps of PM2.5
+concentrations over a sampling area.
+2 Materials and methods
+2.1 City Scanner: A low-cost sensor suite
+As part of the City Scanner project at the Senseable City Lab, Massachusetts Institute of
+Technology, we developed a customizable low-cost sensor suite for mobile applications22,23.
+For this study, we used two City Scanner sensor suites each equipped with an Alphasense
+OPC-N324 optical particle counter to measure particulate matter, a SHT2025 to measure
+temperature and humidity, a GPS sensor, and flash memory. Data was transmitted to a
+server via the Global System for Mobile Communication (GSM) and accessed remotely.
+2.1.1 Low-cost PM2.5 sensors: Alphasense OPC-N3
+The OPC measures the intensity of light scattered by each particle as sampled air is
+illuminated by a laser beam. The amount of light scattered is a function of particle size. The
+manufacturer uses monodisperse particles of known size to calibrate the OPC, deriving
+counts in 24 particle-size bins ranging from 0.35 to 40 μm. An Alphasense data reduction
+algorithm converts the particle counts to PM2.5 mass concentration. A brief overview of past
+evaluations of the OPC-N3 is provided in section S1 of the Supplementary Information (SI).
+We used a several-years-old OPC-N3 (referred to as OPC 1) and a new OPC-N3 (referred
+to as OPC 2) in this experiment.
+2.2 High-cost portable PM2.5 monitor: TSI DustTrak™
+We used a TSI DustTrak II 8533 as a reference instrument for the mobile monitoring
+experiments. The DustTrak uses a laser photometer to determine real-time PM mass
+concentration based on 900 light scattering. The DustTrak II 853326 detects particles in the
+size range from 0.1 to 15 μm in diameter, and has a mass concentration resolution of ±0.1%
+or 1 μg/m3 (whichever is greater). DustTraks report PM1, PM2.5, PM4, and PM10. We selected
+the DustTrak because it has a high reported precision27.
+2.3 Experimental Design
+We collocated the two CityScanners at a stationary monitoring site containing a reference
+monitor and we developed correction models (also referred to as calibration models) for the
+OPC-N3s using the reference measurements as the gold standard. We separately
+collocated a DustTrak monitor with a reference monitor in another stationary reference
+monitoring site and developed a correction model for the DustTrak. We then mounted the
+two CityScanners and the DustTrak in our mobile laboratory and collected measurements
+while driving prescribed routes in Boston. We evaluated the transferability of different
+OPC-N3 correction models to the mobile setting by comparing corrected OPC-N3
+measurements with corrected DustTrak data. We also evaluated how much of the variation
+3
+
+between the corrected mobile OPC-N3 and DustTrak measurements were due to factors
+such as orientation of the OPC-N3, vehicle speed, sky view factor (fraction of visible sky that
+provides an indication of street geometry and building density28), meteorology, time-of-day
+and day. We then evaluated the performance of the corrected mobile OPC-N3
+measurements vis a vis the DustTrak for two use-cases.
+2.3.1 Performance evaluation in a stationary setting under
+well-characterized ambient conditions
+We collocated two Alphasense OPC-N3s and a reference monitor between April 11, 2022
+11:40 AM and April 27, 2022 4:36 PM at the EPA Speciation Trends Network reference site
+(Site ID:25-025-0042; maintained by the Massachusetts Department of Environmental
+Protection (MADEP)) in Nubian Square, Boston, which is representative of urban
+background. This site is within the study area covered by our mobile monitoring campaign.
+We collocated the DustTrak at a MADEP reference site located adjacent to Interstate-93 in
+Boston (Site ID: 25-025-0044; also within the study area) between June 29, 2022, 8:50 AM
+and July 5, 2022, 8:30 AM. The reference monitor at both MADEP sites was a Teledyne
+T640 monitor that had a sampling frequency of 1 minute.
+The sampling frequencies of the OPC-N3s and DustTrak during stationary collocation were 5
+seconds and 1 minute, respectively. We averaged the OPC-N3 data every 1 minute and
+merged the minute-averaged data with coincident 1-minute measurements from the
+reference monitor. Similarly, we merged the DustTrak measurements with coincident
+reference measurements. We also averaged the OPC-N3 and DustTrak measurements
+every hour and merged the hourly-averaged data with coincident hourly-averaged
+measurements from the reference monitors. We had a total of 22,882 minute-averaged, and
+386 hourly-averaged coincident OPC and reference measurements. We had a total of 8,618
+minute-averaged, and 143 hourly-averaged coincident DustTrak and reference
+measurements.
+We evaluated the two OPC-N3s performance vis-a-vis each other using two metrics: (1)
+Pearson Correlation Coefficient (R), and (2) the root-mean-square-error (RMSE). We chose
+to also evaluate the RMSE because R might be a misleading indicator of sensor
+performance when PM2.5 concentrations are close to the limit of detection of the instruments.
+In addition, we compared the sensor measurements in the stationary setting after using
+empirical corrections (described in the next section) to correct the OPC-N3 and DustTrak
+PM2.5 measurements using the reference measurements.
+2.3.1.1 Defining the Models Used to Correct the OPCs
+The correction models were developed to predict the true PM2.5 concentrations as accurately
+as possible. The reference measurements, considered as true PM2.5 concentrations, were
+the dependent variable in the models.
+We evaluated 17 correction models that included different combinations of temperature (T),
+relative humidity (RH) and dewpoint (D) (Table S1). We used T and RH reported by each
+CityScanner mobile sensor suite as it most closely represents the conditions experienced by
+4
+
+the OPCs. R obtained when comparing minute-averaged T and RH experienced by the
+collocated OPCs were both ~0.95. The R observed when comparing minute-averaged T and
+RH from the OPCs with that reference site were 0.92 and 0.90, respectively (Figure S1). We
+derived (D) from T and RH reported by the CityScanner sensor suite using the
+weathermetrics package in the programming language R29, as D has been shown to be a
+good proxy of particle hygroscopic growth30,31.
+Sixteen of the 17 correction models were multivariate regression models that were also
+applied in recent papers18,30 to correct networks of low-cost sensors. As T, RH, and D are not
+independent (Figure S2), the 16 linear regression models include adding the meteorological
+conditions as interaction terms, instead of additive terms. The seventeenth correction model
+relies on the machine learning technique: Random Forest (RF). Machine learning models
+can capture more complex nonlinear effects, such as unknown relationships between
+additional spatial and temporal variables. We opted to use the RF as it has been widely used
+in correcting LCS32,33. A description of the RF technique is described in detailed in SI
+Section S2. We used a 10-fold cross-validation (CV) method to avoid overfitting in the RF
+model. For more details on the cross-validation methods refer to SI Section S3.
+We derived separate correction models for each OPC for minute-averaged and
+hourly-averaged stationary-site and used the models to correct the mobile OPC data. We
+report the R and RMSE when comparing predicted measurements derived using each
+correction model with the corresponding reference measurements.
+2.3.1.2 Correcting the DustTrak
+We applied three widely-used correction methods34 to correct the DustTrak data, including a
+simple bias correction, a RH-adjusted empirical equation, and an RH-adjusted linear
+regression model described in SI Section S4. We evaluated the R and RMSE to compare
+corrected DustTrak measurements using each of these three methods with corresponding
+reference measurements.
+We applied the correction developed using minute-averaged data to the DustTrak data in the
+mobile setting, as this frequency best matches the frequency of measurements of the mobile
+(5 s). In this experiment, we use corrected DustTrak data as our gold standard for the mobile
+deployment. In supplementary analyses, we used the correction developed using
+hourly-averaged collocation data to correct DustTrak measurements in the mobile setting.
+2.3.2 Performance evaluation of the OPC measurements in a mobile
+setting
+We evaluated the transferability of each correction technique for the OPCs to a mobile
+setting by examining if there was an improvement in R and RMSE in comparing the
+corrected versus uncorrected OPC measurements with the coincident corrected DustTrak
+data. We also evaluated if the distribution of T and RH during the collocation period were
+representative of the mobile deployment (SI Figures S3, S4).
+We collocated the two CityScanners alongside the DustTrak in a mobile laboratory35. The
+mobile laboratory was an electric vehicle with a battery-powered electricity supply for the
+5
+
+DustTrak; thus obviating the need for measurement correction due to self-sampling of
+emissions from the vehicle; it is described in detail elsewhere35. The DustTrak was placed
+inside the mobile laboratory and plumbed into the main plenum via a 45-cm-long conductive
+tube. The two City Scanners were placed atop the mobile lab.
+City Scanner 1 containing OPC 1 was deployed for ~ 58 hours over 12 unique dates
+between Feb 11, 2022 and April 5, 2022. City Scanner 2 containing OPC 2 was deployed for
+~ 14 hours over 3 unique dates between March 23, 2022 and April 5, 2022. The orientation
+of the inlet of OPC 1 relative to the direction of the vehicle (parallel or perpendicular) was
+varied during the course of the experiment (Table S2). The sampling routes were selected to
+encounter a wide range of microenvironments such as highways, tunnels, and commercial
+and residential streets.
+The sampling frequencies of the OPC-N3s and the DustTrak were 5 s. We had a total of
+34,127 coincident OPC 1 and DustTrak mobile measurements over 12 separate mobile
+monitoring runs, and 7,972 coincident OPC 1, OPC 2 and DustTrak measurements over 3
+separate mobile monitoring runs. We present descriptive statistics of each run (Table S2).
+Maps of the sampling route for each run are displayed in SI Figure S5.
+We compared corrected mobile OPC measurements using the different techniques proposed
+in Table S1, and corrected DustTrak measurements (Table 1). We report which correction
+model transferred best to the mobile setting. In SI Table S3 in Supplementary Information,
+we compare corrected OPC and DustTrak measurements using mobile DustTrak data
+corrected using hourly-averaged collocated measurements in the stationary setting.
+We carried out an Analysis of Variance (ANOVA) test to determine which factors most
+explained the absolute difference, and difference between the corrected and uncorrected
+OPC measurements, and the corrected OPC and DustTrak measurements. We considered
+RH, T, time-of-day, day-of-run, vehicle speed, inlet orientation, spatial covariates such as
+class of road the measurement was made on, and skyview factor (SVF) due to buildings in
+the 100-m radius around each mobile measurement location. For 1,529 (4.5%) and 3875
+(11.4%) mobile OPC 1 measurements, we did not have SVF and road class data,
+respectively. For 308 (3.9%) and 1,533 (19.2%) mobile OPC 2 measurements, we did not
+have SVF and road class data, respectively. Given the large number of measurements
+missing road class information, we did not include road class in our main analysis, but
+instead included it in supplementary analyses (SI Table S4).
+2.4 Evaluating use-cases of mobile LCS
+We evaluated two use-cases. First, we studied the identification of hotspots and overlap in
+persistent hotspots as measured by low-cost versus research-grade instruments. Hotspots
+were defined as the top one percentile of raw PM2.5 concentrations measured by each
+instrument. Persistent hotspots were defined as PM2.5 measurements in the top one
+percentile of all measurements in the same location over multiple days. Hierarchical
+clustering was used to identify hotspots from the highest PM2.5 concentrations reported by
+each instrument in the same spatial area using a distance cut-off of 100 m. Thereupon we
+determined the number of measurements (n) in each cluster, as well as the number of
+unique runs (nr) over which measurements in each cluster were made, and lastly defined a
+6
+
+persistent hotspot as a cluster for which n > 5 and nr > 1. Uncorrected PM2.5 measurement
+from OPC 1 were compared to DustTrak measurements to evaluate this use-case, as
+previous work has indicated that raw PM2.5 measurements from low-cost sensors can be
+used to detect hotspots8. In this analysis, we used only measurements for OPC 1 because it
+was in operation for 12 days as opposed to 3 days for OPC 2.
+The second use-case was the development of ‘generalizable’ maps of PM2.5 concentrations
+for the study domain. The maps are produced according to the following steps:
+(1) Calculating a background correction to compare PM2.5 measurements made at the
+same location but on different days and at different times.
+(2) Applying the background correction to the PM2.5 concentrations from each device.
+(3) Dividing the sampling area into 50 m x 50 m grid cells and then calculating the
+median background-corrected PM2.5 concentration for each grid cell in the
+study-area. We opted to use a 50 m resolution as it is an order of magnitude greater
+than the GPS error (6 m), but small enough to derive granular maps of PM2.5
+concentrations for the study area.
+We evaluated the median generalizable PM2.5 concentrations, and sampling error, for each
+grid cell in the sampling area (3,824 grid cells had OPC 1 measurements, 1,763 grid cells
+had OPC 2 measurements) for the corrected and uncorrected data from the OPCs and the
+corrected measurements from the DustTrak. Section S5 in the SI contains more information
+about how such ‘generalizable’ PM2.5 concentrations and sampling errors were calculated.
+We examined correlations between the maps of generalizable PM2.5 concentrations
+produced across (1) all 30 m road segments, (2) only segments where measurements were
+made over more than a single run and the normalized standard error in the generalizable
+PM2.5 concentrations calculated from the sampling variability of each instrument was < 20%.
+All analyses were conducted using the software R. In all analyses, p-values < 0.05 were
+taken to represent statistical significance.
+3 Results and Discussion
+During the collocation, RH spanned the range of conditions observed in the mobile
+deployment, although some temperatures during the mobile deployment were higher (>
+250C) than during the stationary collocation (≤ 250C) (Figures S3-4).
+3.1 Evaluating Corrections Developed During the Collocation
+The R and RMSE obtained when comparing the uncorrected, minute-averaged DustTrak
+and reference monitoring data were 0.81 and 7.8 μg/m3, respectively. When repeating this
+calculation using hourly-averaged uncorrected data we obtained an R and RMSE of 0.77
+and 8.8 μg/m3, respectively.  Overall, we found that a simple bias correction of data from the
+DustTrak (Equation 1 in SI Section S4) resulted in the best performance, yielding an R and
+RMSE of 0.77 and 3.3 μg/m3, and 0.81 and 2.0 μg/m3 using hourly- and minute-averaged
+data, respectively.
+7
+
+As expected, the uncorrected measurements from the newer OPC, OPC 2, were more
+strongly correlated with reference measurements than that of the older OPC, OPC 1. When
+comparing OPC measurements corrected using the different correction functions outlined in
+Table S1 (and visualized in Figure S6) with that of the corresponding reference PM2.5 data in
+the stationary setting, we observed that the RF model yielded the best performance (R > 0.9,
+RMSE < 1 μg/m3) using both hourly- and minute-averaged data for both OPCs. More
+complex linear regression models (Models 13, 14, 15 and 16) outperformed simpler models
+(Models 1 - 12) (Table S1).
+3.1 Evaluating transferability of OPC corrections developed in a
+stationary setting to the mobile deployment
+For corrections developed using minute-averaged collocation data, although the RF model
+yielded the best performance in the stationary setting, it yielded a worse performance (R=
+0.17 for OPC 1 and R = 0.16 for OPC 2) than even the uncorrected OPC measurements (R
+= 0.26 for OPC 1 and R =0.20 for OPC 2) when comparing OPC and DustTrak
+measurements in the mobile setting (Table 1; visualized in Figure S7). For OPC1, more
+complex linear regression models (Models 10 -16) also yielded worse R and RMSE when
+applied to the mobile setting than uncorrected OPC data. For OPC 2, Models 10 to 15
+yielded a small improvement in performance relative to the raw data.
+For both OPCs, Model 6 (which corrected for RH, R and the interaction term between RH
+and D) yielded the largest improvement in R and RMSE (R and RMSE comparing OPC 1
+and the DustTrak were 0.29 and 5.6 μg/m3, R and RMSE comparing OPC 2 with the
+DustTrak were 0.59 and 3.9 μg/m3), and also led to the best fit between the corrected
+measurements of the two OPCs themselves (R and RMSE were 0.65 and 1.4 μg/m3). The
+higher correlation observed between OPC 2 and the DustTrak is likely since OPC 2 was a
+newer OPC and was not degraded.
+We obtained similar findings when evaluating the transferability of hourly-averaged
+correction models (Table 1; visualized in Figures S8 and S9). Model 6 resulted in the
+biggest improvement of R for both OPCs in the mobile setting. However, for correction
+models developed using hourly-averaged stationary OPC data, it appears that most
+corrections resulted in a dramatic increase in RMSE for OPC 1, compared to using
+uncorrected measurements (Table 1). This is likely because the hourly-averaged data used
+to develop the bias correction of OPC 1 did not include the sharp increases in PM2.5 we
+observed in minute-averaged measurements during the stationary collocation, which were
+also observed during the mobile experiment. (Figures S10 and S11 display collocated OPC
+1 and reference data at the minute- and hourly-averaged resolutions, respectively; Figures
+S12 and S13 displays collocated OPC 2 and reference data at the minute- and
+hourly-averaged levels, respectively). Thus, the corrections developed using
+hourly-averaged measurements under-correct OPC 1. The spikes observed in the OPC 2
+PM2.5 concentrations in the stationary setting were smaller, likely because it is a new OPC
+and is less error (spike)-prone. However, for both OPCs, using minute-averaged data at the
+reference site to correct the raw OPC data across all models yielded the best comparison
+between the corrected OPC and DustTrak measurements.
+8
+
+We repeated the exercise of comparing corrected-OPC and DustTrak measurements with
+the DustTrak data corrected using hourly-averaged PM2.5 concentrations instead of
+minute-averaged measurements from the collocation in supplementary analyses (Table S3).
+Model 6 developed using minute-averaged OPC collocation data resulted in the most
+striking improvement in R and RMSE when comparing corrected OPC 2 and DustTrak data.
+Model 6 also resulted in a smaller improved performance for OPC 1 (Model 4 yielded a
+marginally smaller RMSE). Overall, this supplementary analysis confirmed that Model 6
+appeared to be a robust choice for correcting the OPCs in a mobile setting.
+This analysis represents one of the first attempts to provide guidance on correcting LCS for
+use in mobile deployments. We found that although more complex machine learning
+algorithms appeared to perform best at the stationary collocation site, simpler models
+appeared to transfer better to the mobile setting. This is likely because more complex
+models are overfit to the stationary site, even after appropriate cross-validation, due to
+differences in conditions such as meteorology or aerosol type at the stationary site
+compared to during the mobile deployment. Importantly, we also observed that corrections
+developed using high-resolution measurements at the stationary co-location site, that were
+similar to the frequency of data collection during the mobile deployment, also transferred
+best to the mobile setting. We note however, that the best correction model only yielded a
+low to moderate correlation (R = 0.29 when comparind the DustTrak with OPC 1 and 0.59
+when comparing the DustTrak data with OPC 2) between the corrected OPC and DustTrak
+data.
+Table 1: Comparing corrected OPC measurements using different correction models derived
+from minute- and hourly-averaged collocated data with that of the corresponding (from the
+correction model developed using 1 minute-averaged collocation data) DustTrak data in our
+mobile deployment. In Column Model: s1-s15 and b are empirically derived from regression
+analyses. Comparisons using OPC data corrected with Model 6 and the DustTrake data in
+the mobile setting are highlighted.
+Mobile deployment
+ID
+Name
+Model
+Correction derived from minute-averaged OPC
+data
+Correction derived from hourly-averaged OPC
+data
+DustTrak and
+OPC 1
+DustTrak and
+OPC 2
+OPC 1 and
+OPC2
+DustTrak and
+OPC 1
+DustTrak and
+OPC 2
+OPC 1 and
+OPC 2
+R
+RMSE
+R
+RMSE
+R
+RMSE
+R
+RMSE
+R
+RMSE
+R
+RMSE
+Raw OPC  measurements
+0
+Raw
+0.26
+6.3
+0.20
+6.6
+0.41
+1.7
+0.26
+6.3
+0.20
+6.6
+0.41
+1.7
+Multivariate Regression
+1
+Linear
+PM2.5, corrected = PM2.5
+s1 +
+×
+b
+0.26
+6.9
+0.20
+3.4
+0.41
+1.9
+0.26
+16.9
+0.20
+3.5
+0.41
+4.4
+2
++RH
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + b
+×
+0.26
+5.8
+0.20
+3.4
+0.41
+1.5
+0.26
+16.7
+0.20
+3.5
+0.41
+4.3
+3
++T
+PM2.5, corrected = PM2.5
+s1 +
+×
+0.26
+7.2
+0.31
+3.3
+0.46
+1.9
+0.26
+17.7
+0.31
+3.4
+0.46
+4.6
+9
+
+T
+s2 + b
+×
+4
++D
+PM2.5, corrected = PM2.5
+s1 +
+×
+D
+s2 + b
+×
+0.28
+5.0
+0.42
+3.7
+0.51
+1.2
+0.27
+12.6
+0.39
+3.8
+0.48
+3.3
+5
++RH x
+T
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + T
+s3 + RH
+T
+×
+×
+×
+s4 + b
+×
+0.27
+5.3
+0.33
+3.5
+0.46
+1.3
+0.26
+14.9
+0.31
+3.6
+0.45
+3.9
+6
++RH x
+D
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + D
+s3 + RH
+D
+×
+×
+×
+s4 + b
+×
+0.29
+5.6
+0.59
+3.9
+0.65
+1.4
+0.27
+15.6
+0.56
+4.0
+0.55
+4.1
+7
++D x T
+PM2.5, corrected = PM2.5
+s1 +
+×
+D
+s2 + T
+s3 + D
+T
+×
+×
+×
+×
+s4 + b
+0.28
+5.2
+0.48
+3.7
+0.55
+1.2
+0.27
+13.9
+0.45
+3.7
+0.51
+3.6
+8
++RH x
+T x D
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + T
+s3 + D
+s4
+×
+×
+×
++ RH
+T
+s5 + RH
+D x
+×
+×
+×
+s6 + T
+D
+s7 + RH
+T
+×
+×
+×
+D
+s8 + b
+×
+×
+0.25
+5.7
+0.11
+3.8
+0.40
+1.41
+0.26
+16.4
+0.08
+3.8
+0.38
+4.2
+9
+PM x
+RH
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + RH
+PM2.5
+s3
+×
+×
+×
++ b
+0.28
+5.4
+0.31
+3.6
+0.41
+1.7
+0.26
+14.8
+0.26
+3.9
+0.39
+4.0
+10
+PM x D
+PM2.5, corrected = PM2.5
+s1 +
+×
+D
+s2 + D
+PM2.5
+s3 + b
+×
+×
+×
+0.17
+18.4
+0.28
+4.6
+0.53
+6.7
+0.19
+36.5
+0.27
+5.1
+0.53
+13.0
+11
+PM x T
+PM2.5, corrected = PM2.5
+s1 +
+×
+T
+s2 + T
+PM2.5
+s3 + b
+×
+×
+×
+0.25
+8.1
+0.26
+3.6
+0.39
+2.6
+0.26
+18.0
+0.24
+3.9
+0.38
+4.9
+12
+PM x
+nonline
+ar RH
+PM2.5, corrected = PM2.5
+s1 +
+×
+s2 +
+𝑅𝐻
+2
+(1−𝑅𝐻) ×
+𝑅𝐻
+2
+(1−𝑅𝐻) ×
+PM2.5
+s3 + b
+×
+0.22
+19.2
+0.20
+4.4
+0.42
+6.0
+0.23
+40.4
+0.20
+5.0
+0.42
+11.9
+13
+PM x
+RH x T
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + T
+s3 +  PM2.5
+×
+×
+RH
+s4 + PM2.5
+T
+s5
+×
+×
+×
+×
++ RH
+T
+s6 + PM2.5
+×
+×
+×
+RH
+T
+s7 +  b
+×
+×
+0.18
+24.0
+0.27
+5.3
+0.52
+7.8
+0.20
+44.2
+0.25
+6.9
+0.49
+14.8
+14
+PM x
+RH x D
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + D
+s3 +  PM2.5
+×
+×
+RH
+s4 + PM2.5
+D
+×
+×
+×
+×
+s5 + RH
+D
+s6 + PM2.5
+×
+×
+×
+RH
+D
+s7 +  b
+×
+×
+0.21
+17.3
+0.28
+4.5
+0.41
+5.9
+0.21
+42.3
+0.23
+6.9
+0.45
+14.8
+15
+PM x T
+x D
+PM2.5, corrected = PM2.5
+s1 +
+×
+T
+s2 + D
+s3 +  PM2.5
+×
+×
+×
+T
+s4 + PM2.5
+D
+s5 + T
+×
+×
+×
+D
+s6 + PM2.5
+T
+D
+×
+×
+×
+×
+s7 +  b
+×
+0.17
+43.6
+0.29
+5.7
+0.55
+9.8
+0.19
+56.0
+0.27
+7.9
+0.54
+16.8
+16
+PM x
+RH x T
+x D
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + T
+s3 +  D
+s4
+×
+×
+×
++ PM2.5
+RH
+s5 + PM2.5
+×
+×
+T x s6 + T
+RH
+s7 +
+×
+×
+×
+PM2.5
+D
+s8 + D
+RH
+×
+×
+×
+×
+s9 + D
+T
+s10 + PM2.5
+×
+×
+×
+0.20
+33.1
+0.17
+5.4
+0.47
+5.3
+0.11
+101.2
+0.19
+8.8
+0.60
+39.3
+10
+
+RH
+T
+s11 + PM2.5
+RH
+×
+×
+×
+D
+s12 + PM2.5
+D
+T
+×
+×
+×
+×
+s13 + D
+RH
+T
+s14 +
+×
+×
+×
+×
+PM2.5
+RH
+T
+D
+s15
+×
+×
+×
+×
++ b
+Machine Learning (10-fold CV)
+17
+Rando
+m
+Forest
+PM2.5, corrected = f(PM2.5, T,
+RH)
+0.17
+3.6
+0.16
+4.1
+0.09
+1.4
+0.25
+2.7
+0.17
+4.0
+0.14
+2.3
+3.2 Evaluating factors that best explained differences between
+corrected OPC and DustTrak and OPC 1 and OPC 2
+measurements
+The results from our ANOVA analyses are displayed in Table 2. The bulk of the variation in
+the absolute difference/difference between the uncorrected OPC measurements remains
+unexplained. However, when evaluating factors that contributed to the variation in
+differences between corrected OPC measurements, we see T explains a moderate amount
+of variation (11.01% from 0.09% when looking at the absolute difference between corrected
+versus uncorrected OPC measurements; and 14.25 % from 1.11%, when looking at the
+difference between corrected versus uncorrected OPC measurements), followed by
+time-of-day (~3% from 0.2 - 0.3%). Model 6 corrects OPC measurements using RH and D
+(but not explicitly T, although D and T are related). Thus, this result could arise from our
+choice of correction model (Table 2).
+When evaluating factors that contribute to the differences/ absolute differences between
+corrected OPC and DustTrak measurements, we found that for OPC 1 >90% of the variation
+was unexplained by the factors considered. For OPC 2, T explained >20% of the variation
+(Table 2). This could be because Model 6 did not adequately correct OPC 2 measurements
+for T as T during the collocation was not representative of T during the mobile deployment;
+whereas for OPC 1, as the mobile deployment occurred over more days, the T experienced
+was more representative (Figure S3). It is thus important to ensure that the environmental
+conditions during collocation are representative of conditions experienced during the mobile
+deployment. Another potential explanation is that degradation in the performance of OPC 1
+due to aging was responsible for the largest fraction of the difference in OPC 1 and DustTrak
+measurements, rendering the impact of T small in comparison.
+Date contributed next to the variation in the absolute difference/difference between corrected
+OPC 1 and DustTrak measurements, corresponding to 6.1% and 11.5%, respectively. For
+OPC 2, hour of day contributed marginally more to variation than day, accounting for ~4-5%
+of the variation in the difference and absolute difference of the corrected OPC 2 and
+DustTrak measurements (Table 2). Aerosol composition varies over the course of a day, and
+between days. Our results suggest that this variation does not greatly impact the difference
+between corrected-OPC and DustTrak results.
+SVF, speed and orientation, even when they significantly contributed to variation of the
+differences between OPC and DustTrak measurements, or coincident OPC measurements,
+11
+
+only accounted for < 1% of total variation. In supplementary analyses, we also considered
+road class and found that it also significantly contributed to a small proportion (< 5%) of total
+variation in these outcomes (Table S4).
+Our analysis thus suggests, although vehicle speed, orientation of the OPC-N3, and sky
+view factor of the road the measurement was made on (a proxy for the street canyon effect)
+impacted the variation of the mobile corrected OPC-N3 measurements, their impact was
+small and do not greatly impact our mobile monitoring results. The large unexplained
+variation (> 80%) in the difference between the corrected measurements of the two OPCs is
+due to the difference in age of the two instruments. The performance of the older OPC which
+was in operation for several years prior to the experiment has likely degraded over time.
+More work is needed to understand how aging impacts the performance of LCS. The
+majority of variation (> 80%) in the difference between the corrected measurements from
+OPC 1 and the DustTrak is also unexplained, likely due to the same reason.
+Table 2: Sum of Squares (Explained variation (%) in the difference and absolute difference
+between uncorrected and corrected OPC measurements, and corrected OPC and DustTrak
+PM2.5 measurements) from an ANOVA analysis. The T and RH used in ANOVA analyses
+comparing the OPC measurements came from City Scanner corresponding to OPC 1.
+When comparing OPC 1 and OPC 2 measurements with DustTrak data, we used T and RH
+data corresponding to the City Scanner corresponding to OPC 1 and 2, respectively.
+OPC
+No
+Orientation
+RH
+T
+Speed
+Sky view
+factor
+Day
+Hour
+Residuals
+Total Sum
+of Squares
+Absolute
+difference
+between the
+uncorrected
+OPC PM2.5
+measurements
+1
+-
+33*
+(0.17%)
+18*
+(0.09%)
+1
+(0.01%)
+6
+(0.03%)
+174*
+(0.90%)
+61*
+(0.31%)
+19,087
+(98.49%)
+19,380
+Absolute
+difference
+between the
+corrected OPC
+PM2.5
+measurements
+using Model 6
+(derived from
+minute-average
+d collocated
+measurements)
+2
+-
+55*
+(0.51%)
+1,194*
+(11.01%)
+0
+(0.00%)
+8*
+(0.07%)
+80*
+(0.74%)
+414*
+(3.82%)
+9,097
+(83.86%)
+10,848
+Absolute
+difference
+between the
+corrected
+DustTrak
+measurements
+and the OPC
+PM2.5
+measurements
+1
+3
+318*
+(0.05%)
+1,252*
+(0.18%)
+206*
+(0.03%)
+429*
+(0.06%)
+1,171*
+(0.17%)
+41,850*
+(6.07%)
+5,672*
+(0.82%)
+638,062
+(92.6%)
+688,960
+2
+4
+-
+129*
+(0.34%)
+15,247*
+(39.77%)
+45*
+(0.12%)
+4
+(0.01%)
+1,476*
+(3.85%)
+1,739*
+(4.54%)
+19,697
+(51.38%)
+38,337
+Difference
+between the
+uncorrected
+OPC PM2.5
+measurements
+5
+-
+16*
+(0.08%)
+228*
+(1.11%)
+1
+(0.00%)
+14*
+(0.07%)
+261*
+(1.27%)
+40*
+(0.20%)
+19,924
+(97.27%)
+20,484
+Difference
+between the
+corrected OPC
+PM2.5
+measurements
+using Model 6
+(derived from
+6
+-
+29*
+(0.23%)
+1,828*
+(14.25%)
+3
+(0.02%)
+26*
+(0.20%)
+234*
+(1.82%)
+337*
+(2.63%)
+10,370
+(80.85%)
+12,827
+12
+
+minute-average
+d collocated
+measurements)
+Difference
+between the
+corrected
+DustTrak
+measurements
+and the OPC
+PM2.5
+measurements
+1
+7
+3,362*
+(0.33%)
+104,460*
+(10.20%)
+51,605*
+(5.04%)
+35
+(0.00%)
+533*
+(0.05%)
+117,502*
+(11.47%)
+8,046*
+(0.79%)
+738,963
+(72.13%)
+1,024,506
+2
+8
+-
+122*
+(0.31%)
+15,192*
+(38.37%)
+50*
+(0.13%)
+1
+(0.00%)
+1,352*
+(3.41%)
+1,640*
+(4.14%)
+21,239
+(53.64%)
+39,596
+Note: The road class data we used had the following categories: 1) Motorways, 2) Major
+Roads, 3) Other Major Roads, 4) Secondary Roads, 5) Local Connecting Roads, 6) Local
+Roads of High Importance, 7) Local Roads, 8) Local Roads of Minor Importance, 9) Other
+Roads. SVF is a dimensionless parameter that indicates how much sky is obstructed by
+buildings. When the SVF is 0, the sky is totally obstructed. When the SVF is 1, there are no
+obstructions. The SVF for the sampling route was calculated elsewhere using Google Street
+View Imagery28
+3.3 Evaluating the use of mobile OPCs to detect hotspots
+Figure 1 displays hotspots identified using uncorrected OPC 1 and DustTrak measurements.
+We note that although there appear to be differences in the hotspots identified by each
+instrument, the most persistent hotspots corresponding to clusters that contain more than
+five measurements in the top percentile made in the same 100 m radius overlap. This result
+is remarkable despite concerns of degradation of measurements from OPC 1. This suggests
+that mobile OPCs can be used to identify important hotspots.
+13
+
+Figure 1: Clusters of high PM2.5 concentrations observed by OPC 1 and the DustTrak that
+include a minimum of 5 PM2.5 observations in the top percentile recorded by each
+instrument. The legend displays the number of measurements in the top percentile
+corresponding to each cluster. The size of each cluster is proportional to the number of runs
+over which a high measurement was observed in the same 100 m radius.
+3.4 Evaluating the use of mobile OPCs to produce
+generalizable maps of PM2.5 along the study route
+Figure S15 displays a 50 m
+50 m grid drawn across the study area. Grid cells in which
+×
+mobile measurements were made are highlighted. Figure 2 displays maps of generalizable
+PM2.5 concentrations from mobile OPC 1, OPC 2 and DustTrak measurements. R and RMSE
+comparing generalizable OPC 1 concentrations and the corresponding DustTrak
+measurements were 0.39 and 4.9 μg/m3, respectively. These metrics signify an improvement
+from comparing unaggregated corrected mobile OPC 1 and DustTrak measurements where
+the R and RMSE were 0.29 and 5.6, respectively (Table 1). R and RMSE comparing
+generalizable OPC 2 concentrations and the corresponding DustTrak measurements were
+0.60 and 3.8 μg/m3, respectively. These metrics also signified an improvement from
+comparing mobile OPC 1 and DustTrak measurements where the R and RMSE were 0.59
+and 3.9 μg/m3, respectively (Table 1).
+The R and RMSE of generalizable OPC 1 and DustTrak measurements for only the 813 (out
+of 3,824) grid cells where the normalized standard error in the generalizable OPC 1 PM2.5
+concentrations calculated due to sampling variability from each instrument were < 20%, and
+measurements on the segment were made over more than single run improved to 0.38 and
+14
+
+Overlap
+HotspotDustTrak(#measurements)
+5
+6
+13
+84
+HotspotOPC1(#measurements)
+5
+11
+25
+StamenToner2.8 μg/m3, respectively. The R and RMSE of generalizable OPC 2 and DustTrak
+measurements for only the 1,268 (out of 1,622) grid cells where the normalized standard
+error in the generalizable PM2.5 concentrations calculated due to sampling variability from
+each instrument were < 20%, and measurements on the segment were made over more
+than single run were improved: 0.59 and 3.6 μg/m3, respectively (Figure S16). The reduced
+RMSE and similar R suggests that aggregating mobile LCS measurements to produce
+generalizable maps can reduce the error in these instruments. The error is smaller on
+segments with stable generalizable PM2.5 concentrations.
+Figure 2: Generalizable PM2.5 concentrations (median background-corrected PM2.5
+concentrations displayed at the center of the corresponding 50 m
+50 m grid cell across the
+×
+study area) produced using A) OPC 1 measurements, B) DustTrak measurements
+coincident with OPC 1, C) OPC 2 measurements, D) DustTrak measurements coincident
+with OPC 2.
+4 Limitations and Future Directions
+15
+
+median background-crrected OPC1PM2.5
+0-1
+A)
+ctedDustTrak (OPC1)
+2-4
+4-6
+4-6
+B)
+6-8
+6-8
+8-10
+8-10
+10-12
+10-12
+12-220
+：12-220
+Stamen Toner
+Stamen Toner
+OPC2PM2.5
+medianbackgroun
+correctedDustTrak(OPC2)
+1-2
+C)
+D)
+2-4
+4-6
+4-6
+8-9
+6-8
+8-10
+8-10
+10-12
+10-12
+12-40
+Stamen Toner
+Stamen TonerThis experiment had several limitations:
+1) We performed the LCS testing in a single season, during daytime hours. The
+labor-intensive nature of this experiment precluded conducting long-term tests in
+more seasons and hours of the day. We hope to conduct longer-term experiments in
+the future.
+2) We evaluated a single LCS: the Alphasense OPC-N3. In future experiments we aim
+to test a wider array of LCS. We also aim to evaluate LCS that measure other
+pollutants in addition to PM2.5.
+3) Although a diverse set of LCS installation scenarios was tested in this experiment,
+there are many configurations (orientation of the LCS relative to the direction of
+travel, vehicle speeds etc.) that a user might choose to implement that differ
+significantly from those tested in this evaluation protocol; thus the results of this
+experiment may have limited application to other configurations. In future
+experiments, we aim to evaluate other configurations over a broader range of
+experimental conditions.
+This experiment represents one of the first attempts to provide guidance on calibrating LCS
+for use in mobile settings. In general, we recommend that researchers use simpler
+calibration models to correct mobile LCS data, as these models transfer better from
+stationary to mobile deployments. This paper also represents one of the first attempts to
+evaluate the feasibility of using mobile LCS data for two key-use cases: the identification of
+hotspots and the production of generalizable maps of PM2.5 concentrations. We find that LCS
+can be used to identify persistent hotspots. We also find that LCS can be used to develop
+reasonably accurate generalizable maps of PM2.5 in cities. Finally, by using OPCs of different
+ages, our paper represents one of the first efforts to show that the age of the LCS plays an
+important role in mobile experiments, and more work is needed to understand the impact of
+age on LCS performance.
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+18
+
+Supplementary Information
+S1: Performance of the OPC-N3
+The Air Quality Sensor Performance Evaluation Center (AQ-SPEC) of the South Coast Air
+Quality Management District (SCAQMD) reported that in a laboratory evaluation23 of three
+OPC-N3 sensors, 5-min average  PM1, PM2.5 and PM10 measurements were highly
+correlated (R2 > 0.99) with measurements from a reference monitor, and the accuracy of the
+sensors was consistent over the range of PM1 (accuracy was 11-14%), PM2.5 (accuracy was
+17-24%), and PM10 (accuracy was 4-5%). In a field-setting, the correlations between the
+sensors and the reference monitor (5-min averages) were lower: the R2 was ~ 0.80, 0.60,
+0.50 for PM1, PM2.5, and PM10, respectively.
+S2: Description of Random Forest
+Random forest (RF): RF is a decision-tree-based machine learning algorithm that
+has been shown to perform well in air quality predictions. Briefly, to generate a
+random forest model, the user specifies the maximum number of trees that make up
+the forest. Each tree is constructed using a bootstrapped random sample from the
+training data set. The origin node of the decision tree is split into sub-nodes by
+considering a random subset of the possible explanatory variables. Trees are split
+based on which of the explanatory variables in each subset is the strongest predictor
+of the outcome. This process of node splitting is repeated until a terminal node is
+reached1. For our random forest models, the terminal node was specified using a
+minimum node size of five data points per node.
+S3 10-fold Cross-Validation
+Our 10-fold cross-validation (CV) exercise occurred in the following manner: The model
+fitting dataset was randomly split into 10 groups, with each group containing about 10% of
+the grid cells. In each cross-validation iteration, we select nine groups of the grid cells. All of
+the data from these grid cells are used to fit the model and make predictions of the
+remaining group. We repeated this process 10 times, until every group was predicted.
+The CV process was used to tune two major RF hyperparameters: the number of decision
+trees (ntree) and the number of predictors randomly tried at each split (mtry). Briefly, the
+optimal mtry was determined by optimizing the “pseudo R2”: the fraction of variance of the
+PM2.5c measurements explained by the “out of bag” (OOB) predictions. An OOB prediction is
+based only on trees in the random-forest ensemble that were not trained using the PM2.5c
+observations being predicted. Each mtry value was used to train 10 random-forests. The
+average pseudo R2 was used to select the best mtry value.
+19
+
+S4: Calibrating the DustTrak
+We compared the results from using three widely-used calibration methods, including the
+simple linear regression model in Equation (1), a widely-used the RH-adjusted empirical
+equation proposed by Laulainen (1993)2 in Equation (2), and the RH-adjusted linear
+regressio model in Equation (3).
+……..(1)
+𝑃𝑀2.5, 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 =  𝑠1 +  𝑠2 × 𝑃𝑀2.5, 𝐷𝑢𝑠𝑡𝑇𝑟𝑎𝑘
+s1, and s2 are empirically determined parameters determine from a regression analysis.
+PM2.5, DustTrak and PM2.5,reference are the coincident 1-minute averaged PM2.5 concentrations
+reported by the DustTrak and reference site respectively. The R and RMSE obtained were
+0.81 and 2.0 μg/m3, respectively. In supplementary analyses, we also re-ran these
+regressions using hourly-averaged data. The R and RMSE obtained were 0.77 and 3.3
+μg/m3, respectively.
+…… (2)
+𝑂𝑅 =  𝑠1 +  𝑠2 ×
+(𝑅𝐻/100)
+2
+(1 − 𝑅𝐻/100)
+In Equation (2) OR is the overestimation ratio of the DustTrak PM2.5 relative to the reference
+monitor and is
+We used RH measurements from the reference station for this
+𝑃𝑀2.5, 𝐷𝑢𝑠𝑡𝑇𝑟𝑎𝑘
+𝑃𝑀2.5, 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒
+correction. s1 and s2 are empirically determined parameters from a regression analysis.
+Many studies have reported  s1 =1 and  s2 = 0.25. Using minute-averaged data, we obtained
+an s1 = 1, and an s2 = 0.13, and an R and RMSE corresponding to 0.81 and 6.8 μg/m3,
+respectively. In supplementary analyses, we also re-ran these regressions using
+hourly-averaged data, where obtained an R and RMSE of 0.78 and 7.8 μg/m3, respectively.
+…… (3)
+𝑂𝑅 =  𝑠1 +  𝑠2 × 𝑅𝐻
+The R and RMSE obtained from Equation 3 were 0.79 and 7.7 μg/m3, while that obtained
+from using hourly-averaged data were 0.80 and 6.6 μg/m3, respectively.
+S5 Developing maps of generalizable PM2.5 concentrations for
+the time-period of the mobile deployment in the sampling area
+S5.1 Background Correction
+We estimated background contribution for the uncorrected and corrected data for each OPC
+and the DustTrak using two different methods:
+1) We adopted the lowest 10th percentile of the PM2.5 concentrations measured by the
+instrument under consideration for a given hour during the run as the fixed
+background value for that run. 
+2) We used a time-series, spline-of-minimums approach, presented by Brantley et al.
+(2013), to estimate background PM2.5 for each sampling run. We did this by: (a)
+20
+
+Applying a rolling 30-second mean to smooth the measurements, (b) Dividing the
+time
+series
+into
+discrete
+10-minute
+segments
+and
+locating
+the
+minimum
+concentration in each segment, and (c) Fitting a smooth, thin-plate regression spline
+through the minimum concentrations.
+We made the assumption that the background is temporally varying but spatially uniform.
+After estimating the background PM2.5 for each instrument, we performed a background
+time-of-day correction or standardization using Equations 4 and 5 to account for the period
+during which the vehicle operated:
+PM2.5c,i= PM2.5,OPC i - PM2.5, bkg,i + PM2.5, bkg,median      …………....…………………... (4)
+where PM2.5,OPC,i is the calibrated measurement for event i, PM2.5,bkg,i is the contemporaneous
+background value of pollution over the entire region, and PM2.5, bkg,median is the median of the
+PM2.5,bkg values for the sampling run. By subtracting the time-of-day-run-resolved regional
+background
+from
+the
+pollution
+measurement,
+we
+can
+now
+compare
+background-corrected/local PM2.5 concentrations measured during different runs over space. 
+When the background PM2.5 measurement value was estimated to be more than that of the
+concentration measured, we applied a multiplicative background-correction factor as
+displayed in Equation 5.
+PM2.5c,i= PM2.5,OPC, i x PM2.5, bkg,median / PM2.5,bkg,i........................................ (5)
+Unlike Equation 4, this assures non-negative PM2.5c,i values, based on considering the
+background value assessed during the measurement hour as a fraction of the day’s median
+background, thus reducing the contribution of possible measurement error due to treating
+the hourly value as an absolute quantity.
+We found that both methods to calculate the background-corrected PM2.5 for each instrument
+produced similar background-corrected values. The median differences between the two
+background-corrected concentrations for the uncorrected and corrected OPC 1, OPC 2 and
+the DustTrak measurements were < 5 %. Figure S14 displays the background-corrected
+OPC 1, OPC 2, and DustTrak data during the mobile deployment and confirms that both
+methods to derive background PM2.5 concentrations yield similar results. The median
+differences between the raw and each background-corrected concentration for the OPC 1,
+OPC 2 and the DustTrak were between 0 - 5%. Given the minimal differences in median
+PM2.5c from the different methods, we chose the splines-of-minimum approach to obtain
+background concentrations for both pollutants. This is supported by previous research that
+found this approach to be an effective way to account for background concentrations for a
+range of pollutants over a variety of meteorological conditions and sampling routes (Brantley
+et al., 2013).
+S5.2 Calculating the generalizable PM2.5 concentrations across
+the sampling route
+21
+
+We divided the sampling area into grid cells of 50 m in length, using the ‘Create Grid’ QGIS
+Tool. We aggregate the background-corrected PM2.5 concentrations made over different runs
+and derive the sampling error using the method proposed in deSouza et al., (2020).
+First, we snapped each background-corrected PM2.5 measurement to the grid cell that it fell
+in. This allows measurements made in the same grid cell to be analyzed as a group.
+Second, we selected the median (which is outlier resistant) of all background-corrected PM2.5
+measurements made on a given road segment as the ‘generalizable PM2.5 concentration’ for
+that segment. Third, we used a set of bootstrap resampling procedures to quantify the effect
+of sample-to-sample variability and of sampling error on the median concentration for each
+road segment. As a metric of precision, we used the ratio of the standard error of median
+concentration to the median concentration itself.
+22
+
+Figures
+Figure S1: Pair-wise correlations between T and RH recorded by OPC 1, OPC 2, and by the
+reference monitoring station
+23
+
+T Reference
+RHReference
+RH OPC 1
+RHOPC2
+2
+T OPC
+OPC
+TReference
+0.92
+0.92
+0.58
+-0.54
+-0.52
+TOPC1
+0.92
+0.94
+0.53
+-0.6
+-0.58
+TOPC2
+0.92
+0.94
+1
+0.58
+0.61
+-0.63
+RHReference
+-0.58
+-0.53
+-0.58
+0.92
+0.89
+RHOPC1
+-0.54
+0.6
+-0.61
+0.92
+0.96
+RH OPC 2
+0.52
+-0.58
+-0.63
+0.89
+0.96
+-
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1Figure S2: Pairwise correlations between T, RH and D for A) All mobile measurements
+made by the CityScanner platform corresponding to OPC 1, B) All coincident mobile
+24
+
+RH OPC 1
+TOPC1
+DOPC1
+A)
+RH OPC 1
+1
+0.37
+0.77
+T OPC 1
+0.37
+1
+0.87
+D OPC 1
+0.77
+0.87
+1
+-0.8
+-0.6
+-0.4
+-0.2 
+-1
+0.2 
+0.4 
+0.6 
+8°0
+RHOPC1
+RH OPC 2
+DOPC1
+DOPC2
+TOPC 1
+TOPC2
+B)
+RH OPC 1
+1
+0.99
+0.4
+0.55
+0.41
+-0.45
+RH OPC 2
+0.99
+1
+0.44
+0.58
+0.37
+0.43
+D OPC 1
+0.4
+0.44
+1
+0.98
+0.66
+0.61
+D OPC 2
+0.55
+0.58
+0.98
+1
+0.52
+0.48
+T OPC 1
+0.41
+-0.37
+0.66
+0.52
+1
+0.98
+TOPC2
+-0.45
+-0.43
+0.61
+0.48
+0.98
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0.2 
+0.4 
+0.6
+8:0measurements made by both CityScanner platforms for the time period where both OPC1
+and OPC2 were operational
+Figure S3: Comparing the distribution of  Minute-level A) RH and B) T during the collocation
+of the OPCs in comparison to during the mobile monitoring deployment using violin and
+boxplots
+25
+
+A)
+100
+75
+50
+25
+-0
+Co-Location
+OPC1Mobile
+OPC2Mobile
+B)
+30
+20
+Temperature 
+10
+0
+Co-Location
+OPC1Mobile
+OPC2 MobileFigure S4: Comparing the distribution of  Minute-level A) RH and B) T during the collocation
+of the DustTrak in comparison to during the mobile monitoring deployment using violin and
+boxplots
+Figure S5: Spatial extent of each of the 12 mobile monitoring runs
+26
+
+A)
+100-
+75
+RH
+50
+25
+0 :
+Co-Location
+DustTrak Mobile
+B)
+30
+Temperature (C)
+20
+10
+0
+Co-Location
+DustTrak MobileMobile Monitoring Runs
+2022-02-11
+2022-02-16
+2022-02-17
+2022-02-23
+2022-02-24
+2022-03-02
+2022-03-03
+2022-03-08
+2022-03-10
+2022-03-23
+2022-03-30
+2022-04-05Figure S6: Pairwise correlations between OPC 1, OPC 2, corrected OPC 1 and OPC 2, and
+reference measurements during the stationary collocation experiment
+27
+
+Model
+2
+1 Model
+3
+OPC
+L
+Model
+4
+OPC
+Model
+5
+OPC
+O
+1
+Model
+OPC1
+6
+OPC
+1 Model
+OPC1Model1
+OPC 1Model 2
+0.9
+OP
+Model
+OPC1Model3
+9
+OP
+OPC
+1Model
+10
+OPC1Model4
+0.670.670.8
+Model
+Model 11
+OPC1Model5
+1
+OPC
+1Model12
+OPC1Model6
+OPC1Model7
+OPC
+OPC
+14
+OPC1Model8
+OPC
+1 Model
+15
+OPC 1Model 9
+Model
+1Model 16
+OPC1Model10
+do
+Model 17
+OPC 1 Model 11
+OPC 1Model 12
+OPC
+L
+OPC1Model13
+OPC
+2Model 1
+OPC1Model14
+OPC
+2
+2Model
+OPC1Model15
+OPC1Model16
+4
+n
+OPC
+2 Model
+OPC1Model17
+5
+OPC
+2 Model
+OPC2
+6
+OPC
+2 Model
+OPC 2Model 1
+OPC2Model2
+OPC
+Model
+8
+C
+Model
+OPC2Model3
+9
+d
+2
+Model
+OPC2Model4
+2Model11
+OPC2Model5
+2
+12
+OPC2Model6
+2
+2 Model
+2Model 13
+OPC2Model7
+OP
+OPC2Model14
+OPC2Model8
+PC
+15
+OPC2Model9
+2Model
+OPC2Model16
+OPC2Model10
+OPC2Model17
+OPC2Model11
+OPC
+OPC2Model12
+OPC2Model13
+OPC
+Reference
+OPC2Model14
+OPC2Model15
+OPC 2Model 16
+OPC2Model17
+Reference
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8Figure S7: Pairwise correlations between OPC 1, OPC 2, corrected OPC 1 and OPC 2, and
+corrected DustTrak measurements during the mobile deployment
+28
+
+Model
+1 Model 2
+3
+OPC
+Model
+4
+OPC
+Model
+C
+5
+Model
+OPC1
+1
+1
+6
+OPC
+1Model
+OPC1Model1
+1
+OPC1Model7
+OPC1Model2
+OPC 1Model8
+1
+OPO
+OPC1Model3
+0.980.900.98
+9
+OP
+OPC
+OPC1Model
+10
+OPC1Model4
+0.940
+1 Model
+Model11
+OPC1Model5
+OPC1Model12
+OPC1Model6
+13
+OPC1Model7
+9210
+OPC1Model
+de
+14
+OPC1Model8
+OPC
+OPC 1 Model 
+OPC1Model15
+OPC1Model9
+097
+16
+OPC1Model10
+1Model
+OPC1Model11
+OPC1Model12
+OPC1Model13
+OPC2Model1
+OPC1Model14
+OPC
+2
+C
+OPC2Model
+OPC 1 Model 15
+2
+13
+OP
+OPC
+Model
+OPC1Model16
+OPC1Model17
+5
+2
+2 Model
+OPC2
+6
+OPC 2 Model 1
+OPC
+Model
+2Model
+OPC2Model2
+8O
+C
+2 Model
+OPC2Model3
+2
+do
+OPC
+2Model
+2Model10
+OPC2Model4
+OPC
+2Model11
+OPC2Model5
+C
+12
+OPC2Model6
+OP
+Ode
+2 Model
+OPC2Model7
+OPC2 Model 14
+OPC2Model8
+OPC
+OPC2Model15
+OPC2Model9
+do
+OPC2Model16
+OPC2Model10
+de
+2Model17
+OPC2Model11
+OPC2Model12
+OPC2Model13
+OPC2Model 14
+OPC
+rak
+OPC2Model15
+sna
+OPC2Model16
+OPC2Model17
+DustTrak
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1Figure S8: Pairwise correlations between OPC 1, OPC 2, corrected (using hourly-averaged
+collocated measurements) OPC 1 and OPC 2, and corrected DustTrak measurements
+during the mobile deployment
+29
+
+Model
+OPC1Model2
+OPC1Model3
+OPC
+OPC 1Model 4
+OPC1Model5
+OPC1
+9.
+1Model
+OPC1Model1
+OPC1Model
+OPC 1Model 2
+1 Model
+OPC 1Model3
+9
+OPC
+Model
+OPC1Model10
+OPC 1Model 4
+OPC1Model11
+OPC1Model5
+OPC
+112
+OPC 1Model 6
+OPC
+1Model
+OPC1Model13
+OPC1Model7
+14
+OPC1Model8
+1Model
+OPC 1 Model 9
+OPC
+OPC 1 Model 16
+OPC1Model10
+1Model 17
+OPC1Model11
+OPC
+OPC1Model12
+OPC1Model13
+OPC1Model14
+2
+OPC
+2 Model
+OPC1Model15
+2
+13
+OPC
+Model
+OPC1Model16
+4
+2 Model
+OPC1Model17
+2Model5
+do
+OPC2
+2
+6
+OP
+2Model
+OPC2Model 1
+OPC
+2Model
+OPC2Model2
+Model 8
+OPC2Model3
+OPC
+2 Model
+C
+10
+OPC2Model4
+OP
+2Model
+OPC 2 Model 11
+OPC2Model5
+C
+do
+2 Model 12
+OPC2Model6
+odo
+Model13
+OPC2Model7
+OPC2Model 14
+OPC2Model8
+OPC
+OPC2Model9
+PC
+21
+16
+OPC2Model10
+OPC2Model
+OPC2Model17
+OPC 2Model 11
+OPC2Model12
+OPC2Model13
+OPC2Model14
+ustTrak
+OPC2Model15
+OPC2Model16
+OPC2Model17
+DustTrak
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1Figure S9: Pairwise correlations between OPC 1, corrected (using minute-averaged and
+hourly-averaged collocated measurements) OPC 1, and corrected DustTrak measurements
+during the mobile deployment
+30
+
+1Model2
+OPC
+4
+1Model
+5
+OPC
+1Model
+1Model6
+OPC1
+1
+OPC1Model1
+OPC
+1Model7
+OPC
+OPC1Model8
+OPC1Model2
+OPC1Model3
+OPC
+1 Model
+OPC1Model10
+OPC1Model4
+OPC
+Model 11
+OPC1Model5
+OPC
+OPC1Model6
+Model 13
+OPC1Model7
+OPC
+14
+OPC1Model8
+086'086'060
+Model
+Model 15
+OPC1Model9
+OPC
+16
+OPC1Model10
+1
+1Model
+1 Model 17
+OPC
+1HourModel
+OPC1Model11
+2
+OPC1Model12
+OPO
+OPC1
+3
+1Hour Model
+OPC1Model13
+HourModel4
+OPC
+5
+OPC1Model14
+OPC
+Hour Model
+OPC1Model15
+OPC1HourModel6
+6
+720.720.7
+OPC1Model16
+OPC
+Hour Model
+OPC1Model17
+OPC 1HourModel 1
+9
+Hour Model 
+10
+OPC 1HourModel 2
+1HourModel
+11
+OPC1HourModel3
+OPC
+Hour Model
+12
+OPC 1HourModel4
+HourModel
+OPC1HourModel13
+OPC1HourModel5
+OPC
+OPC1HourModel14
+OPC1HourModel6
+do
+L
+OPC 1 Hour Model 15 
+OPC
+OPC1HourModel7
+OPC1HourModel8
+ode
+OPC1HourModel17
+OPC 1 Hour Model 9
+OPC
+OPC1HourModel10
+OPC
+OPC1HourModel11
+OPC1HourModel12
+OPC1HourModel13
+OPC1HourModel14
+DustTrak
+OPC1HourModel15
+OPC1HourModel16
+OPC1HourModel17
+DustTrak
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1Figure S10: Reference, OPC 1, corrected OPC 1 measurements using the different
+calibration models (1 -17) described in Table 1 at the minute-averaged time resolution.
+31
+
+30 
+20
+10
+30 
+20
+10 
+Ref
+30
+15,16,17,
+20
+10
+40
+20
+10
+0
+20
+10
+30
+20
+20
+30
+20
+10
+25
+10
+Apr13
+Apr15
+Apr 17
+Apr19
+Apr21
+Apr 23
+Apr 25
+Apr 27Figure S11: Reference, OPC 1, corrected OPC 1 measurements using the different
+calibration models (1 -17) described in Table 1 at the hourly-averaged time resolution.
+32
+
+0.4
+.2
+Apr13
+Apr15
+Apr17
+Apr19
+Apr 21
+Apr23
+Apr25
+Apr 27Figure S12: Reference, OPC 2, corrected OPC 2 measurements using the different
+calibration models (1 -17) described in Table 1 at the minute-averaged time resolution.
+33
+
+10
+15
+10
+15
+10
+15
+10
+15
+10
+15
+10
+Ret
+11
+822505
+4320
+5250
+Apr13
+Apr 15
+Apr 17
+Apr19
+Apr21
+Apr 23
+Apr 25
+Apr27Figure S13: Reference, OPC 2, corrected OPC 2 measurements using the different
+calibration models (1 -17) described in Table 1 at the hourly-averaged time resolution.
+34
+
+Re
+Apr 13
+Apr 15
+Apr17
+Apr19
+Apr 21
+Apr 23
+Apr 25
+Apr 27Figure S14: Comparing background PM2.5 concentrations calculated for each instrument
+during the course of the mobile deployment
+Figure S15: 50 m x 50 m grid cells drawn over the sampling area. The grid cells in which
+mobile PM2.5 measurements were made are highlighted.
+35
+
+A)OPC 1
+B)OPC 2
+Methods
+Methods
+3.0
+Spline of minimums
+Spline of minimums
+LowesttenthPercentile
+Lowest tenth Percentile
+2.5
+2.0 -
+0
+10000
+20000
+30000
+0
+2000
+4000
+6000
+8000
+10-
+C)DustTrakco-locatedwithOPC1
+D)DustTrakco-locatedwithOPC2
+8:
+Methods
+Methods
+6
+Spline of minimums
+6
+Splineofminimums
+LowesttenthPercentile
+LowesttenthPercentile
+4
+2
+10000
+20000
+30000
+0
+2000
+4000
+6000
+8000-71.24-71.22
+-71.20
+-71.18
+-71.16
+-71.14
+-71.12
+-71.10
+-71.08-71.06
+-71.04
+-71.02
+-71.00
+-70.98
+-70.96
+-70.94-70.92
+：-70.90-70.88
+42.44
+MA1A
+42.42
+Nahant
+1132
+Ariingtor
+42.40
+42.38
+Waitham
+Watertoy
+42.36
+Newton
+42.34
+Newtol
+ChestnutHil
+42.32
+wtonHighands
+lewto
+Upp
+42.30
+HamHeight
+42.28e
+WestRoxbu
+grid on mobile route
+42.26
+grid
+OpenStreetMap
+Dedham
+42.24
+0
+2.5
+5 km36
+
+median background-corrected OPC 1 (stable)
+1-2
+2-4
+A)
+4-6
+6-8
+8-10
+10-12
+12-20
+StamenToner
+median background-corrected OPC 2 (stable)
+1-2
+2-4
+B)
+4-6
+6-8
+8- 10
+10 -12
+Stamen TonerFigure S16: Generalizable PM2.5 concentrations (median background-corrected calibrated
+PM2.5 concentrations in 50 m x 50 m grid cells across the study area) produced using A)
+OPC 1 measurements, B) OPC 2 measurements for grid cells where the normalized error in
+the median PM2.5 concentration was < 20% and measurements in that grid cell were made
+over > 1 day
+Tables
+Table S1: Performance of the correction models using hourly-averaged and
+minute-averaged data as captured using root mean square error (RMSE), and Pearson
+correlation (R). A 10-fold CV was used to prevent overfitting in the machine learning models.
+Separate correction models were developed for the two OPCs used in this study. In Column
+Model: s1-s15 and b are empirically derived from regression analyses.
+ID
+Name
+Model
+Minute-averaged
+Hourly-averaged
+OPC 1
+OPC 2
+OPC 1
+OPC 2
+R
+RMSE
+(μg/m3)
+R
+RMSE
+(μg/m3)
+R
+RMSE
+(μg/m3)
+R
+RMSE
+(μg/m3)
+Raw OPC measurements
+0
+Raw
+0.31
+5.0
+0.46
+4.0
+0.50
+5.0
+0.51
+3.9
+Multivariate Regression
+1
+Linear
+PM2.5, corrected = PM2.5
+s1 + b
+×
+0.31
+1.8
+0.46
+1.7
+0.50
+1.6
+0.51
+1.6
+2
++RH
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + b
+×
+0.35
+1.8
+0.46
+1.7
+0.50
+1.6
+0.51
+1.6
+3
++T
+PM2.5, corrected = PM2.5
+s1 + T
+×
+s2 + b
+×
+0.33
+1.8
+0.48
+1.7
+0.52
+1.6
+0.53
+1.6
+4
++D
+PM2.5, corrected = PM2.5
+s1 + D
+×
+s2 + b
+×
+0.44
+1.7
+0.49
+1.7
+0.53
+1.6
+0.53
+1.6
+5
++RH x T
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + T
+s3 + RH
+T
+×
+×
+×
+s4 + b
+×
+0.48
+1.7
+0.53
+1.6
+0.58
+1.5
+0.58
+1.5
+6
++RH x D
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + D
+s3 + RH
+D
+×
+×
+×
+s4 + b
+×
+0.49
+1.7
+0.56
+1.6
+0.59
+1.5
+0.60
+1.5
+7
++D x T
+PM2.5, corrected = PM2.5
+s1 + D
+×
+s2 + T
+s3 + D
+T
+s4 +
+×
+×
+×
+×
+b
+0.47
+1.7
+0.53
+1.6
+0.56
+1.5
+0.57
+1.5
+8
++RH x T x
+D
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + T
+s3 + D
+s4 +
+×
+×
+×
+RH
+T
+s5 + RH
+D x s6 +
+×
+×
+×
+T
+D
+s7 + RH
+T
+D
+×
+×
+×
+×
+×
+s8 + b
+0.55
+1.6
+0.61
+1.5
+0.66
+1.4
+0.66
+1.4
+9
+PM x RH
+PM2.5, corrected = PM2.5
+s1 +
+×
+0.43
+1.7
+0.52
+1.6
+0.54
+1.6
+0.58
+1.5
+37
+
+RH
+s2 + RH
+PM2.5
+s3 +
+×
+×
+×
+b
+10
+PM x D
+PM2.5, corrected = PM2.5
+s1 + D
+×
+s2 + D
+PM2.5
+s3 + b
+×
+×
+×
+0.50
+1.7
+0.56
+1.58
+0.63
+1.4
+0.63
+1.4
+11
+PM x T
+PM2.5, corrected = PM2.5
+s1 + T
+×
+s2 + T
+PM2.5
+s3 + b
+×
+×
+×
+0.34
+1.8
+0.51
+1.6
+0.52
+1.6
+0.58
+1.5
+12
+PM x
+nonlinear
+RH
+PM2.5, corrected = PM2.5
+s1 +
+×
+s2 +
+PM2.5
+𝑅𝐻
+2
+(1−𝑅𝐻) ×
+𝑅𝐻
+2
+(1−𝑅𝐻) ×
+s3 + b
+×
+0.43
+1.7
+0.59
+1.5
+0.63
+1.4
+0.68
+1.3
+13
+PM x RH x
+T
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + T
+s3 +  PM2.5
+×
+×
+×
+RH
+s4 + PM2.5
+T
+s5 +
+×
+×
+×
+RH
+T
+s6 + PM2.5
+RH
+×
+×
+×
+×
+T
+s7 +  b
+×
+0.56
+1.6
+0.67
+1.4
+0.70
+1.3
+0.76
+1.2
+14
+PM x RH x
+D
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + D
+s3 +  PM2.5
+×
+×
+×
+RH
+s4 + PM2.5
+D
+s5 +
+×
+×
+×
+RH
+D
+s6 + PM2.5
+RH
+×
+×
+×
+×
+D
+s7 +  b
+×
+0.54
+1.6
+0.66
+1.4
+0.68
+1.4
+0.76
+1.2
+15
+PM x T x D
+PM2.5, corrected = PM2.5
+s1 + T
+×
+s2 + D
+s3 +  PM2.5
+T
+×
+×
+×
+×
+s4 + PM2.5
+D
+s5 + T
+D
+×
+×
+×
+s6 + PM2.5
+T
+D
+s7 +
+×
+×
+×
+×
+b
+0.55
+1.6
+0.66
+1.4
+0.68
+1.4
+0.76
+1.2
+16
+PM x RH x
+T x D
+PM2.5, corrected = PM2.5
+s1 +
+×
+RH
+s2 + T
+s3 +  D
+s4 +
+×
+×
+×
+PM2.5
+RH
+s5 + PM2.5
+T
+×
+×
+×
+x s6 + T
+RH
+s7 + PM2.5
+×
+×
+×
+D
+s8 + D
+RH
+s9 + D
+×
+×
+×
+×
+T
+s10 + PM2.5
+RH
+T
+×
+×
+×
+×
+s11 + PM2.5
+RH
+D
+s12 +
+×
+×
+×
+PM2.5
+D
+T
+s13 + D
+×
+×
+×
+×
+RH
+T
+s14 + PM2.5
+RH
+×
+×
+×
+T
+D
+s15 + b
+×
+×
+×
+0.61
+1.5
+0.69
+1.4
+0.73
+1.26
+0.78
+1.2
+Machine Learning (10-fold CV)
+17
+Random
+Forest
+PM2.5, corrected = f(PM2.5, T, RH)
+0.97
+0.45
+0.99
+0.29
+0.92
+0.77
+0.94
+0.69
+Table S2: Mobile monitoring summary data.
+Date
+Sampl
+ing
+time
+(Numb
+er of
+measu
+remen
+ts)
+Sampling
+area
+Vehicle
+Mean
+Speed
+(m/s)
+PM2.5
+Mean (Median)
+μg/m3
+(
+10)
+×
+RH
+Mean (Median)
+(%)
+Temperature
+Mean (Median)
+(0C)
+OPC position vis a vis vehicle
+direction
+R (RMSE
+(μg/m3))
+comparing
+corrected
+5-second
+DustTrak and
+uncorrected
+OPC
+measurements
+OPC1
+OPC2
+OPC1
+OPC2
+OPC1
+OPC2
+OPC 1
+OPC 2
+02/11/202
+09:18-
+“South
+5.2
+9
+-
+27.8
+-
+17.6
+-
+OPC 1: flow rate parallel to
+0.31
+-
+38
+
+2
+13:59
+(3,351)
+route”: I-93
+tunnels,
+Downtown
+Boston,
+South
+Boston,
+Dorchester,
+Roxbury,
+North
+Quincy
+(3)
+(27.6)
+(17.5)
+direction of vehicle
+(4.5)
+2/16/2022
+13:10-
+16:35
+(2,458)
+“North
+route”:
+Everett,
+Revere,
+Chelsea,
+Winthrop,
+East Boston
+5.3
+12
+(5)
+-
+40.0
+(39.8)
+-
+12.1
+(12.0)
+-
+OPC 1: flow rate parallel to
+direction of vehicle
+0.37
+(5.8)
+-
+2/17/2022
+09:15-
+13:42
+(3,183)
+South route
+5.5
+21
+(14)
+-
+42.0
+(42.0)
+-
+22.8
+(22.8)
+-
+OPC 1: flow rate parallel to
+direction of vehicle
+0.54
+(5.4)
+-
+2/23/2022
+08:50-
+12:58
+(2,976)
+North route
+4.7
+14
+(12)
+-
+55.5
+(57.9)
+-
+23.4
+(22.5)
+-
+OPC 1: flow rate parallel to
+direction of vehicle
+0.21
+(6.3)
+-
+2/24/2022
+09:21-
+13:20
+(2,869)
+South route
+4.8
+4
+(2)
+-
+26.8
+(26.8)
+-
+4.0
+(3.6)
+-
+OPC 1: flow rate parallel to
+direction of vehicle
+0.29
+(7.3)
+-
+3/2/2022
+13:33-
+18:26
+(3,440)
+South route
++ extra run
+through
+1-93
+tunnels
+6.5
+5
+(2)
+-
+34.2
+(33.7)
+-
+11.9
+(11.9)
+-
+OPC 1: flow rate parallel to
+direction of vehicle
+0.25
+(6.3)
+-
+3/3/2022
+13:39-
+16:23
+(1,952)
+North route
+6.6
+5
+(2)
+-
+24.7
+(24.1)
+-
+10.6
+(10.2)
+-
+OPC 1: flow rate parallel to
+direction of vehicle
+0.22
+(3.4)
+-
+3/8/2022
+16:16-
+20:05
+(2,707)
+North route
+3.3
+2
+(1)
+-
+24.5
+(24.3)
+-
+11.8
+(11.9)
+-
+OPC 1: flow rate parallel to
+direction of vehicle
+0.27
+(5.6)
+-
+3/10/2022
+12:22-
+16:52
+(3,220)
+South route
+5.5
+9
+(3)
+-
+31.1
+(30.7)
+-
+16.5
+(16.6)
+-
+OPC 1: flow rate perpendicular to
+direction of vehicle
+0.24
+(8.3)
+-
+39
+
+3/23/2022
+11:29 -
+14:40
+(2,298)
+South route
+5.6
+2
+(1)
+5
+(3)
+29.5
+(29.2)
+23.3
+(23.8)
+15.5
+(15.7)
+16.7
+(16.6)
+OPC 1: flow rate perpendicular to
+direction of vehicle
+OPC 2: flow rate parallel to the
+direction of the vehicle
+0.15
+(6.0)
+0.45
+(5.6)
+3/30/2022
+12:06 -
+16:02
+(2,828)
+South route
+6.1
+2
+(1)
+9
+(5)
+19.2
+(18.9)
+13.3
+(12.7)
+16.6
+(16.8)
+18.2
+(18.6)
+OPC 1: flow rate perpendicular to
+direction of vehicle
+OPC 2: flow rate parallel to the
+direction of the vehicle
+0.33
+(4.7)
+0.44
+(4.1)
+4/05/2022
+10:27 -
+14:24
+(2,845)
+South route
+5.4
+4
+(3)
+7
+(4)
+22.6
+(22.9)
+16.4
+(16.4)
+22.8
+(22.1)
+24.3
+(23.8)
+OPC 1: flow rate perpendicular to
+direction of vehicle
+OPC 2: flow rate parallel to the
+direction of the vehicle
+0.33
+(8.9)
+0.24
+(8.8)
+*Note that due to the low uncorrected OPC PM2.5 measurements, we report PM2.5
+concentrations
+10.
+×
+Table S3: Comparing corrected OPC measurements using different calibration models
+derived from minute- and hourly-averaged collocated data with that of the corresponding
+corrected (from the calibration model developed using hourly-averaged collocation data)
+DustTrak data in our mobile deployment. The R and RMSE when comparing the corrected
+DustTrak measurements using minute-averaged and hourly-averaged data during the
+collocation were 1 and 0.69 μg/m3, respectively.
+ID
+Name
+Model
+Calibration derived from minute-averaged
+data
+Calibration derived from hourly-averaged
+data
+DustTrak and OPC 1
+DustTrak and OPC 2
+DustTrak and OPC 1
+DustTrak and OPC 2
+R
+RMSE
+(μg/m3)
+R
+RMSE
+(μg/m3)
+R
+RMSE
+(μg/m3)
+R
+RMSE
+(μg/m3)
+Raw OPC measurements
+0
+Raw
+0.26
+6.5
+0.20
+6.9
+0.26
+6.5
+0.20
+6.9
+Multivariate Regression
+1
+Linear
+PM2.5, corrected = PM2.5 ×
+s1 + b
+0.26
+7.0
+0.20
+4.0
+0.26
+16.8
+0.20
+4.0
+2
++RH
+PM2.5, corrected = PM2.5 ×
+s1 + RH
+s2 + b
+×
+0.26
+5.9
+0.20
+4.0
+0.26
+16.6
+0.20
+4.0
+3
++T
+PM2.5, corrected = PM2.5 ×
+s1 + T
+s2 + b
+×
+0.26
+7.3
+0.31
+3.9
+0.26
+17.6
+0.31
+3.9
+40
+
+4
++D
+PM2.5, corrected = PM2.5 ×
+s1 + D
+s2 + b
+×
+0.29
+5.2
+0.42
+4.2
+0.27
+12.6
+0.39
+4.3
+5
++RH x T
+PM2.5, corrected = PM2.5 ×
+s1 + RH
+s2 + T
+s3
+×
+×
++ RH
+T
+s4 + b
+×
+×
+0.27
+5.5
+0.33
+4.1
+0.26
+14.9
+0.31
+4.1
+6
++RH x D
+PM2.5, corrected = PM2.5 ×
+s1 + RH
+s2 + D
+s3
+×
+×
++ RH
+D
+s4 + b
+×
+×
+0.29
+5.7
+0.59
+4.4
+0.27
+15.6
+0.56
+4.4
+7
++D x T
+PM2.5, corrected = PM2.5 ×
+s1 + D
+s2 + T
+s3 +
+×
+×
+D
+T
+s4 + b
+×
+×
+0.28
+5.4
+0.48
+4.2
+0.27
+13.8
+0.45
+4.2
+8
++RH x T
+x D
+PM2.5, corrected = PM2.5 ×
+s1 + RH
+s2 + T
+s3
+×
+×
++ D
+s4 + RH
+T
+×
+×
+×
+s5 + RH
+D x s6 + T
+×
+D
+s7 + RH
+T
+×
+×
+×
+×
+D
+s8 + b
+×
+0.25
+5.9
+0.11
+4.3
+0.26
+16.3
+0.08
+4.4
+9
+PM x
+RH
+PM2.5, corrected = PM2.5 ×
+s1 + RH
+s2 + RH
+×
+×
+PM2.5
+s3 + b
+×
+0.28
+5.6
+0.31
+4.1
+0.26
+14.8
+0.26
+4.3
+10
+PM x D
+PM2.5, corrected = PM2.5 ×
+s1 + D
+s2 + D
+×
+×
+PM2.5
+s3 + b
+×
+0.17
+18.4
+0.28
+4.9
+0.19
+36.4
+0.27
+5.4
+11
+PM x T
+PM2.5, corrected = PM2.5 ×
+s1 + T
+s2 + T
+×
+×
+PM2.5
+s3 + b
+×
+0.25
+8.2
+0.26
+4.1
+0.26
+17.9
+0.24
+4.3
+12
+PM x
+nonlinea
+r RH
+PM2.5, corrected = PM2.5 ×
+s1 +
+s2 +
+𝑅𝐻
+2
+(1−𝑅𝐻) ×
+PM2.5
+s3 +
+𝑅𝐻
+2
+(1−𝑅𝐻) ×
+×
+b
+0.22
+19.2
+0.20
+4.8
+0.23
+40.3
+0.20
+5.3
+13
+PM x
+RH x T
+PM2.5, corrected = PM2.5 ×
+s1 + RH
+s2 + T
+s3
+×
+×
++  PM2.5
+RH
+s4 +
+×
+×
+PM2.5
+T
+s5 + RH
+×
+×
+T
+s6 + PM2.5
+×
+×
+×
+RH
+T
+s7 +  b
+×
+×
+0.18
+23.9
+0.27
+5.6
+0.20
+44.1
+0.25
+7.1
+14
+PM x
+RH x D
+PM2.5, corrected = PM2.5 ×
+s1 + RH
+s2 + D
+s3
+×
+×
++  PM2.5
+RH
+s4 +
+×
+×
+PM2.5
+D
+s5 + RH
+×
+×
+D
+s6 + PM2.5
+×
+×
+×
+RH
+D
+s7 +  b
+×
+×
+0.21
+17.3
+0.28
+4.8
+0.21
+42.2
+0.23
+7.1
+15
+PM x T
+x D
+PM2.5, corrected = PM2.5 ×
+s1 + T
+s2 + D
+s3 +
+×
+×
+PM2.5
+T
+s4 +
+×
+×
+PM2.5
+D
+s5 + T
+×
+×
+×
+D
+s6 + PM2.5
+T
+×
+×
+×
+D
+s7 +  b
+×
+0.17
+43.5
+0.29
+5.9
+0.19
+55.9
+0.27
+8.0
+41
+
+16
+PM x
+RH x T x
+D
+PM2.5, corrected = PM2.5 ×
+s1 + RH
+s2 + T
+s3
+×
+×
++  D
+s4 + PM2.5
+×
+×
+RH
+s5 + PM2.5
+T x
+×
+×
+s6 + T
+RH
+s7 +
+×
+×
+PM2.5
+D
+s8 + D
+×
+×
+×
+RH
+s9 + D
+T
+s10
+×
+×
+×
++ PM2.5
+RH
+T
+×
+×
+×
+s11 + PM2.5
+RH
+D
+×
+×
+s12 + PM2.5
+D
+T
+×
+×
+×
+s13 + D
+RH
+T
+×
+×
+×
+×
+s14 + PM2.5
+RH
+T
+×
+×
+D
+s15 + b
+×
+×
+0.20
+33.1
+0.17
+5.7
+0.11
+101.2
+0.19
+8.9
+Machine Learning (10-fold CV)
+17
+Random
+Forest
+PM2.5, corrected = f(PM2.5,
+T, RH)
+0.17
+4.1
+0.16
+4.6
+0.25
+3.3
+0.17
+4.5
+Table S4: Sum of Squares (Explained variation (%) in the difference and absolute difference
+between uncorrected and corrected OPC measurements, and corrected OPC and DustTrak
+PM2.5 measurements) from an ANOVA analysis. The T and RH used in ANOVA analyses
+comparing the OPC measurements came from City Scanner corresponding to OPC 1.
+When comparing OPC 1 and OPC 2 measurements with DustTrak data, we used T and RH
+data corresponding to the City Scanner corresponding to OPC 1 and 2, respectively.
+OPC
+N
+o
+Orientation
+RH
+T
+Speed
+Sky view
+factor
+Day
+Hour
+Road
+Class
+Residuals
+Total Sum
+of Squares
+Absolute
+difference
+between the
+uncorrected
+OPC PM2.5
+measurements
+1
+-
+0
+(0.00%)
+46*
+(0.25%)
+0
+(0.00%)
+0
+(0.00%)
+151*
+(0.81%)
+139*
+(0.75%)
+35*
+(0.19%)
+18,195
+(98.00%)
+18,566
+Absolute
+difference
+between the
+corrected OPC
+PM2.5
+measurements
+using Model 6
+(derived from
+minute-average
+d collocated
+measurements)
+3
+-
+6*
+(0.07%)
+1,111*
+(12.20%)
+0
+(0.00%)
+6*
+(0.07%)
+57*
+(0.63%)
+398*
+(4.37%)
+278*
+(3.05%)
+7,252
+979.62%)
+9,108
+Absolute
+difference
+between the
+corrected
+DustTrak
+measurements
+and the OPC
+PM2.5
+measurements
+1
+5
+115*
+(0.02%)
+1,091*
+(0.18%)
+74
+(0.01%)
+461*
+(0.07%)
+2,254*
+(0.36%)
+40,990*
+(6.63%)
+8,161*
+(1.32%)
+1,755*
+(0.28%)
+563,533
+(91.12%)
+618,434
+2
+6
+-
+602*
+(1.72%)
+13,232*
+(37.74%)
+39*
+(0.11%)
+37*
+(0.11%)
+1,009*
+(2.88%)
+2,468*
+(7.04%)
+195*
+(0.56%)
+17,478
+(49.85%)
+35,060
+Difference
+between the
+uncorrected
+OPC PM2.5
+measurements
+9
+-
+7
+(0.036%)
+330*
+(1.69%)
+0
+(0.00%)
+0
+(0.00%)
+198*
+(1.01%)
+59*
+(0.30%)
+20
+(0.10%)
+18,953
+(96.86%)
+19,567
+Difference
+between the
+corrected OPC
+PM2.5
+11
+-
+9*
+(0.08%)
+1,832*
+(16.64%)
+1
+(0.01%)
+8*
+(0.07%)
+229*
+(2.08%)
+210*
+(1.91%)
+152*
+(1.38%)
+8,568
+(77.83%)
+11,009
+42
+
+measurements
+using Model 6
+(derived from
+minute-average
+d collocated
+measurements)
+Difference
+between the
+corrected
+DustTrak
+measurements
+and the OPC
+PM2.5
+measurements
+1
+13
+1,091*
+(0.12%)
+123,356*
+(13.45%)
+39,338*
+(4.29%)
+1
+(0.00%)
+128*
+(0.01%)
+96,875*
+(10.56%)
+6,209*
+(0.68%)
+1,482*
+(0.16%)
+648,465
+(70.72%)
+916,945
+2
+14
+-
+587*
+(1.61%)
+13,231*
+(36.34%)
+43*
+(0.12%)
+44*
+(0.12%)
+892*
+(2.45%)
+2,280*
+(6.26%)
+182*
+(0.50%)
+19,147
+(52.59%)
+36,406
+43
+
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new file mode 100644
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+page_content='Evaluating the Performance of  Low-Cost PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 Sensors in Mobile  Settings  Priyanka deSouza1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' An Wang3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Yuki Machida3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Tiffany Duhl4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Simone Mora3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Prashant  Kumar5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' Ralph Kahn7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Carlo Ratti3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' John L Durant4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Neelakshi Hudda4  1: Department of Urban and Regional Planning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' University of Colorado Denver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' CO 80217,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  USA  2: CU Population Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Boulder CO 80309,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' USA  3: MIT Senseable City Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Cambridge MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' USA  4: Department of Civil and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Tufts University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Medford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' MA 02155,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  USA  5: Global Centre for Clean Air Research (GCARE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' School of Sustainability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Civil and  Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Faculty of Engineering and Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' University of  Surrey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Guildford GU2 7XH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Surrey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' UK  6: Institute for Sustainability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' University of Surrey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Guildford GU2 7XH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Surrey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' UK  7: NASA Goddard Space Flight Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Greenbelt MD 20771,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' USA  Corresponding author: priyanka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='desouza@ucdenver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='edu  Abstract  Low-cost sensors (LCS) for measuring air pollution are increasingly being deployed in  mobile applications but questions concerning the quality of the measurements remain  unanswered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' For example, what is the best way to correct LCS data in a mobile setting?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Which factors most significantly contribute to differences between mobile LCS data and  higher-quality instruments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Can data from LCS be used to identify hotspots and generate  generalizable pollutant concentration maps?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' To help address these questions we deployed  low-cost PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 sensors (Alphasense OPC-N3) and a research-grade instrument (TSI  DustTrak) in a mobile laboratory in Boston, MA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We first collocated these instruments  with stationary PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 reference monitors (Teledyne T640) at nearby regulatory sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Next,  using the reference measurements, we developed different models to correct the OPC-N3  and DustTrak measurements, and then transferred the corrections to the mobile setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We  observed that more complex correction models appeared to perform better than simpler  models in the stationary setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' however, when transferred to the mobile setting, corrected  OPC-N3 measurements agreed less well with corrected DustTrak data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In general,  corrections developed using minute-level collocation measurements transferred better to the  mobile setting than corrections developed using hourly-averaged data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Mobile laboratory  speed, OPC-N3 orientation relative to the direction of travel, date, hour-of-the-day, and road  class together explain a small but significant amount of variation between corrected OPC-N3  and DustTrak measurements during the mobile deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Persistent hotspots identified by  the OPC-N3s agreed with those identified by the DustTrak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Similarly, maps of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  distribution produced from the mobile corrected OPC-N3 and DustTrak measurements  1  agreed well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' These results suggest that identifying hotspots and developing generalizable  maps of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 are appropriate use-cases for mobile LCS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  1 Introduction  Exposure to fine particulate matter (PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' mass concentration of particles having an  aerodynamic diameter of ≤2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 μm) is a major environmental health risk in cities around the  world1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Air quality monitoring is critical for developing effective pollution management plans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  however, reference air quality monitors have high capital and operating costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Even in the  United States, which has one of the densest monitoring networks in the world, less than a  third of US counties have a regulatory, also termed reference, monitor7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Such a sparse  network cannot adequately characterize the spatial variation of air pollution in complex urban  environments where pollutant concentrations can vary significantly at distances as small as  a few 10s of meters8,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Dense networks of low-cost sensors (LCS) (<USD $2500 as defined by the US EPA Air  Sensor Toolbox10) are increasingly being used to fill PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentration data gaps in cities  around the world11–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In particular, there is growing interest in using LCS in mobile  monitoring applications to capture the granular spatial variations in pollution in cities at a  relatively low-cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' However, the measurements from LCS are error-prone and need to be  corrected prior to data interpretation17–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Due to the growing use of LCS, regulatory agencies in the United States and Europe have  developed protocols to evaluate LCS in the lab and in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Although these protocols  offer an important point of reference to evaluate LCS, considerable challenges remain in  using them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' For example, the agreement of LCS measurements with those from reference  monitors varies with  relative humidity (RH), temperature (T), as well as aerosol  concentration, size distribution, and composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Mobile deployments of LCS may introduce  additional uncertainties due to vibrations, wind speed and vehicular disturbances of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  mixing patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Therefore, LCS need to be systematically evaluated against reference  monitors to appropriately correct high-frequency mobile measurements as well as to assess  appropriate use-cases for mobile LCS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  To date, little research has evaluated LCS in mobile settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We found only one study that  systematically evaluated the performance of LCS for PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 (PurpleAir PA-II monitors) against  measurements from collocated reference instruments in a mobile platform21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This study  found that the fidelity of the PurpleAir measurements decreased with increasing air velocity  around the unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We were interested in evaluating the transferability of correction models developed based on  a stationary collocation experiment to a mobile deployment of a popular LCS, the  Alphasense Optical Particle Counter (OPC-N3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The impact of correction model choice on  the performance of an LCS in a mobile setting has not previously been reported in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In addition, we evaluate the performance of corrected Alphasense OPC-N3s under  different driving conditions and in different urban microenvironments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We report on which  factors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=', sensor orientation, vehicle speed, road class, and meteorology) contribute to  differences between the corrected OPC-N3 and PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurements made using a research  2  grade monitor (TSI DustTrak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Finally, we evaluate the feasibility of using mobile LCS for two  common use-cases: detecting PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 hotspots and producing generalizable maps of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  concentrations over a sampling area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2 Materials and methods  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1 City Scanner: A low-cost sensor suite  As part of the City Scanner project at the Senseable City Lab, Massachusetts Institute of  Technology, we developed a customizable low-cost sensor suite for mobile applications22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  For this study, we used two City Scanner sensor suites each equipped with an Alphasense  OPC-N324 optical particle counter to measure particulate matter, a SHT2025 to measure  temperature and humidity, a GPS sensor, and flash memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Data was transmitted to a  server via the Global System for Mobile Communication (GSM) and accessed remotely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1 Low-cost PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 sensors: Alphasense OPC-N3  The OPC measures the intensity of light scattered by each particle as sampled air is  illuminated by a laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The amount of light scattered is a function of particle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The  manufacturer uses monodisperse particles of known size to calibrate the OPC, deriving  counts in 24 particle-size bins ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='35 to 40 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' An Alphasense data reduction  algorithm converts the particle counts to PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 mass concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' A brief overview of past  evaluations of the OPC-N3 is provided in section S1 of the Supplementary Information (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We used a several-years-old OPC-N3 (referred to as OPC 1) and a new OPC-N3 (referred  to as OPC 2) in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2 High-cost portable PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 monitor: TSI DustTrak™  We used a TSI DustTrak II 8533 as a reference instrument for the mobile monitoring  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The DustTrak uses a laser photometer to determine real-time PM mass  concentration based on 900 light scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The DustTrak II 853326 detects particles in the  size range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1 to 15 μm in diameter, and has a mass concentration resolution of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1%  or 1 μg/m3 (whichever is greater).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' DustTraks report PM1, PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, PM4, and PM10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We selected  the DustTrak because it has a high reported precision27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3 Experimental Design  We collocated the two CityScanners at a stationary monitoring site containing a reference  monitor and we developed correction models (also referred to as calibration models) for the  OPC-N3s using the reference measurements as the gold standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We separately  collocated a DustTrak monitor with a reference monitor in another stationary reference  monitoring site and developed a correction model for the DustTrak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We then mounted the  two CityScanners and the DustTrak in our mobile laboratory and collected measurements  while driving prescribed routes in Boston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We evaluated the transferability of different  OPC-N3 correction models to the mobile setting by comparing corrected OPC-N3  measurements with corrected DustTrak data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We also evaluated how much of the variation  3  between the corrected mobile OPC-N3 and DustTrak measurements were due to factors  such as orientation of the OPC-N3, vehicle speed, sky view factor (fraction of visible sky that  provides an indication of street geometry and building density28), meteorology, time-of-day  and day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We then evaluated the performance of the corrected mobile OPC-N3  measurements vis a vis the DustTrak for two use-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1 Performance evaluation in a stationary setting under  well-characterized ambient conditions  We collocated two Alphasense OPC-N3s and a reference monitor between April 11, 2022  11:40 AM and April 27, 2022 4:36 PM at the EPA Speciation Trends Network reference site  (Site ID:25-025-0042;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' maintained by the Massachusetts Department of Environmental  Protection (MADEP)) in Nubian Square, Boston, which is representative of urban  background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This site is within the study area covered by our mobile monitoring campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We collocated the DustTrak at a MADEP reference site located adjacent to Interstate-93 in  Boston (Site ID: 25-025-0044;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' also within the study area) between June 29, 2022, 8:50 AM  and July 5, 2022, 8:30 AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The reference monitor at both MADEP sites was a Teledyne  T640 monitor that had a sampling frequency of 1 minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  The sampling frequencies of the OPC-N3s and DustTrak during stationary collocation were 5  seconds and 1 minute, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We averaged the OPC-N3 data every 1 minute and  merged the minute-averaged data with coincident 1-minute measurements from the  reference monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Similarly, we merged the DustTrak measurements with coincident  reference measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We also averaged the OPC-N3 and DustTrak measurements  every hour and merged the hourly-averaged data with coincident hourly-averaged  measurements from the reference monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We had a total of 22,882 minute-averaged, and  386 hourly-averaged coincident OPC and reference measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We had a total of 8,618  minute-averaged, and 143 hourly-averaged coincident DustTrak and reference  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We evaluated the two OPC-N3s performance vis-a-vis each other using two metrics: (1)  Pearson Correlation Coefficient (R), and (2) the root-mean-square-error (RMSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We chose  to also evaluate the RMSE because R might be a misleading indicator of sensor  performance when PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations are close to the limit of detection of the instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  In addition, we compared the sensor measurements in the stationary setting after using  empirical corrections (described in the next section) to correct the OPC-N3 and DustTrak  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurements using the reference measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1 Defining the Models Used to Correct the OPCs  The correction models were developed to predict the true PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations as accurately  as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The reference measurements, considered as true PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations, were  the dependent variable in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We evaluated 17 correction models that included different combinations of temperature (T),  relative humidity (RH) and dewpoint (D) (Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We used T and RH reported by each  CityScanner mobile sensor suite as it most closely represents the conditions experienced by  4  the OPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' R obtained when comparing minute-averaged T and RH experienced by the  collocated OPCs were both ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The R observed when comparing minute-averaged T and  RH from the OPCs with that reference site were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='92 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='90, respectively (Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We  derived (D) from T and RH reported by the CityScanner sensor suite using the  weathermetrics package in the programming language R29, as D has been shown to be a  good proxy of particle hygroscopic growth30,31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Sixteen of the 17 correction models were multivariate regression models that were also  applied in recent papers18,30 to correct networks of low-cost sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' As T, RH, and D are not  independent (Figure S2), the 16 linear regression models include adding the meteorological  conditions as interaction terms, instead of additive terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The seventeenth correction model  relies on the machine learning technique: Random Forest (RF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Machine learning models  can capture more complex nonlinear effects, such as unknown relationships between  additional spatial and temporal variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We opted to use the RF as it has been widely used  in correcting LCS32,33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' A description of the RF technique is described in detailed in SI  Section S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We used a 10-fold cross-validation (CV) method to avoid overfitting in the RF  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' For more details on the cross-validation methods refer to SI Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We derived separate correction models for each OPC for minute-averaged and  hourly-averaged stationary-site and used the models to correct the mobile OPC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We  report the R and RMSE when comparing predicted measurements derived using each  correction model with the corresponding reference measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2 Correcting the DustTrak  We applied three widely-used correction methods34 to correct the DustTrak data, including a  simple bias correction, a RH-adjusted empirical equation, and an RH-adjusted linear  regression model described in SI Section S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We evaluated the R and RMSE to compare  corrected DustTrak measurements using each of these three methods with corresponding  reference measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We applied the correction developed using minute-averaged data to the DustTrak data in the  mobile setting, as this frequency best matches the frequency of measurements of the mobile  (5 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In this experiment, we use corrected DustTrak data as our gold standard for the mobile  deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In supplementary analyses, we used the correction developed using  hourly-averaged collocation data to correct DustTrak measurements in the mobile setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2 Performance evaluation of the OPC measurements in a mobile  setting  We evaluated the transferability of each correction technique for the OPCs to a mobile  setting by examining if there was an improvement in R and RMSE in comparing the  corrected versus uncorrected OPC measurements with the coincident corrected DustTrak  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We also evaluated if the distribution of T and RH during the collocation period were  representative of the mobile deployment (SI Figures S3, S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We collocated the two CityScanners alongside the DustTrak in a mobile laboratory35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The  mobile laboratory was an electric vehicle with a battery-powered electricity supply for the  5  DustTrak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' thus obviating the need for measurement correction due to self-sampling of  emissions from the vehicle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' it is described in detail elsewhere35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The DustTrak was placed  inside the mobile laboratory and plumbed into the main plenum via a 45-cm-long conductive  tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The two City Scanners were placed atop the mobile lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  City Scanner 1 containing OPC 1 was deployed for ~ 58 hours over 12 unique dates  between Feb 11, 2022 and April 5, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' City Scanner 2 containing OPC 2 was deployed for  ~ 14 hours over 3 unique dates between March 23, 2022 and April 5, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The orientation  of the inlet of OPC 1 relative to the direction of the vehicle (parallel or perpendicular) was  varied during the course of the experiment (Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The sampling routes were selected to  encounter a wide range of microenvironments such as highways, tunnels, and commercial  and residential streets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  The sampling frequencies of the OPC-N3s and the DustTrak were 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We had a total of  34,127 coincident OPC 1 and DustTrak mobile measurements over 12 separate mobile  monitoring runs, and 7,972 coincident OPC 1, OPC 2 and DustTrak measurements over 3  separate mobile monitoring runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We present descriptive statistics of each run (Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Maps of the sampling route for each run are displayed in SI Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We compared corrected mobile OPC measurements using the different techniques proposed  in Table S1, and corrected DustTrak measurements (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We report which correction  model transferred best to the mobile setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In SI Table S3 in Supplementary Information,  we compare corrected OPC and DustTrak measurements using mobile DustTrak data  corrected using hourly-averaged collocated measurements in the stationary setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We carried out an Analysis of Variance (ANOVA) test to determine which factors most  explained the absolute difference, and difference between the corrected and uncorrected  OPC measurements, and the corrected OPC and DustTrak measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We considered  RH, T, time-of-day, day-of-run, vehicle speed, inlet orientation, spatial covariates such as  class of road the measurement was made on, and skyview factor (SVF) due to buildings in  the 100-m radius around each mobile measurement location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' For 1,529 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5%) and 3875  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4%) mobile OPC 1 measurements, we did not have SVF and road class data,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' For 308 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9%) and 1,533 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2%) mobile OPC 2 measurements, we did not  have SVF and road class data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Given the large number of measurements  missing road class information, we did not include road class in our main analysis, but  instead included it in supplementary analyses (SI Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4 Evaluating use-cases of mobile LCS  We evaluated two use-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' First, we studied the identification of hotspots and overlap in  persistent hotspots as measured by low-cost versus research-grade instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Hotspots  were defined as the top one percentile of raw PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations measured by each  instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Persistent hotspots were defined as PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurements in the top one  percentile of all measurements in the same location over multiple days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Hierarchical  clustering was used to identify hotspots from the highest PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations reported by  each instrument in the same spatial area using a distance cut-off of 100 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Thereupon we  determined the number of measurements (n) in each cluster, as well as the number of  unique runs (nr) over which measurements in each cluster were made, and lastly defined a  6  persistent hotspot as a cluster for which n > 5 and nr > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Uncorrected PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurement  from OPC 1 were compared to DustTrak measurements to evaluate this use-case, as  previous work has indicated that raw PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurements from low-cost sensors can be  used to detect hotspots8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In this analysis, we used only measurements for OPC 1 because it  was in operation for 12 days as opposed to 3 days for OPC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  The second use-case was the development of ‘generalizable’ maps of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations  for the study domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The maps are produced according to the following steps:  (1) Calculating a background correction to compare PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurements made at the  same location but on different days and at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  (2) Applying the background correction to the PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations from each device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  (3) Dividing the sampling area into 50 m x 50 m grid cells and then calculating the  median background-corrected PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentration for each grid cell in the  study-area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We opted to use a 50 m resolution as it is an order of magnitude greater  than the GPS error (6 m), but small enough to derive granular maps of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  concentrations for the study area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We evaluated the median generalizable PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations, and sampling error, for each  grid cell in the sampling area (3,824 grid cells had OPC 1 measurements, 1,763 grid cells  had OPC 2 measurements) for the corrected and uncorrected data from the OPCs and the  corrected measurements from the DustTrak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Section S5 in the SI contains more information  about how such ‘generalizable’ PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations and sampling errors were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We examined correlations between the maps of generalizable PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations  produced across (1) all 30 m road segments, (2) only segments where measurements were  made over more than a single run and the normalized standard error in the generalizable  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations calculated from the sampling variability of each instrument was < 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  All analyses were conducted using the software R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In all analyses, p-values < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='05 were  taken to represent statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  3 Results and Discussion  During the collocation, RH spanned the range of conditions observed in the mobile  deployment, although some temperatures during the mobile deployment were higher (>  250C) than during the stationary collocation (≤ 250C) (Figures S3-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1 Evaluating Corrections Developed During the Collocation  The R and RMSE obtained when comparing the uncorrected, minute-averaged DustTrak  and reference monitoring data were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='81 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8 μg/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' When repeating this  calculation using hourly-averaged uncorrected data we obtained an R and RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='77  and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8 μg/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Overall, we found that a simple bias correction of data from the  DustTrak (Equation 1 in SI Section S4) resulted in the best performance, yielding an R and  RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='77 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3 μg/m3, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='81 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0 μg/m3 using hourly- and minute-averaged  data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  7  As expected, the uncorrected measurements from the newer OPC, OPC 2, were more  strongly correlated with reference measurements than that of the older OPC, OPC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' When  comparing OPC measurements corrected using the different correction functions outlined in  Table S1 (and visualized in Figure S6) with that of the corresponding reference PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 data in  the stationary setting, we observed that the RF model yielded the best performance (R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9,  RMSE < 1 μg/m3) using both hourly- and minute-averaged data for both OPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' More  complex linear regression models (Models 13, 14, 15 and 16) outperformed simpler models  (Models 1 - 12) (Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1 Evaluating transferability of OPC corrections developed in a  stationary setting to the mobile deployment  For corrections developed using minute-averaged collocation data, although the RF model  yielded the best performance in the stationary setting, it yielded a worse performance (R=  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='17 for OPC 1 and R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='16 for OPC 2) than even the uncorrected OPC measurements (R  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26 for OPC 1 and R =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20 for OPC 2) when comparing OPC and DustTrak  measurements in the mobile setting (Table 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' visualized in Figure S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' For OPC1, more  complex linear regression models (Models 10 -16) also yielded worse R and RMSE when  applied to the mobile setting than uncorrected OPC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' For OPC 2, Models 10 to 15  yielded a small improvement in performance relative to the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  For both OPCs, Model 6 (which corrected for RH, R and the interaction term between RH  and D) yielded the largest improvement in R and RMSE (R and RMSE comparing OPC 1  and the DustTrak were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='29 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6 μg/m3, R and RMSE comparing OPC 2 with the  DustTrak were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='59 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9 μg/m3), and also led to the best fit between the corrected  measurements of the two OPCs themselves (R and RMSE were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='65 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4 μg/m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The  higher correlation observed between OPC 2 and the DustTrak is likely since OPC 2 was a  newer OPC and was not degraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We obtained similar findings when evaluating the transferability of hourly-averaged  correction models (Table 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' visualized in Figures S8 and S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Model 6 resulted in the  biggest improvement of R for both OPCs in the mobile setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' However, for correction  models developed using hourly-averaged stationary OPC data, it appears that most  corrections resulted in a dramatic increase in RMSE for OPC 1, compared to using  uncorrected measurements (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This is likely because the hourly-averaged data used  to develop the bias correction of OPC 1 did not include the sharp increases in PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 we  observed in minute-averaged measurements during the stationary collocation, which were  also observed during the mobile experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' (Figures S10 and S11 display collocated OPC  1 and reference data at the minute- and hourly-averaged resolutions, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Figures  S12 and S13 displays collocated OPC 2 and reference data at the minute- and  hourly-averaged levels, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Thus, the corrections developed using  hourly-averaged measurements under-correct OPC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The spikes observed in the OPC 2  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations in the stationary setting were smaller, likely because it is a new OPC  and is less error (spike)-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' However, for both OPCs, using minute-averaged data at the  reference site to correct the raw OPC data across all models yielded the best comparison  between the corrected OPC and DustTrak measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  8  We repeated the exercise of comparing corrected-OPC and DustTrak measurements with  the DustTrak data corrected using hourly-averaged PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations instead of  minute-averaged measurements from the collocation in supplementary analyses (Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Model 6 developed using minute-averaged OPC collocation data resulted in the most  striking improvement in R and RMSE when comparing corrected OPC 2 and DustTrak data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Model 6 also resulted in a smaller improved performance for OPC 1 (Model 4 yielded a  marginally smaller RMSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Overall, this supplementary analysis confirmed that Model 6  appeared to be a robust choice for correcting the OPCs in a mobile setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  This analysis represents one of the first attempts to provide guidance on correcting LCS for  use in mobile deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We found that although more complex machine learning  algorithms appeared to perform best at the stationary collocation site, simpler models  appeared to transfer better to the mobile setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This is likely because more complex  models are overfit to the stationary site, even after appropriate cross-validation, due to  differences in conditions such as meteorology or aerosol type at the stationary site  compared to during the mobile deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Importantly, we also observed that corrections  developed using high-resolution measurements at the stationary co-location site, that were  similar to the frequency of data collection during the mobile deployment, also transferred  best to the mobile setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We note however, that the best correction model only yielded a  low to moderate correlation (R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='29 when comparind the DustTrak with OPC 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='59  when comparing the DustTrak data with OPC 2) between the corrected OPC and DustTrak  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Table 1: Comparing corrected OPC measurements using different correction models derived  from minute- and hourly-averaged collocated data with that of the corresponding (from the  correction model developed using 1 minute-averaged collocation data) DustTrak data in our  mobile deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In Column Model: s1-s15 and b are empirically derived from regression  analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Comparisons using OPC data corrected with Model 6 and the DustTrake data in  the mobile setting are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Mobile deployment  ID  Name  Model  Correction derived from minute-averaged OPC  data  Correction derived from hourly-averaged OPC  data  DustTrak and  OPC 1  DustTrak and  OPC 2  OPC 1 and  OPC2  DustTrak and  OPC 1  DustTrak and  OPC 2  OPC 1 and  OPC 2  R  RMSE  R  RMSE  R  RMSE  R  RMSE  R  RMSE  R  RMSE  Raw OPC  measurements  0  Raw  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='7  Multivariate Regression  1  Linear  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='4  2  +RH  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5  T  s5  ×  ×  ×  ×  + RH  T  s6 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 +  ×  RH  s2 + D  s3 +  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  ×  ×  RH  s4 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  D  ×  ×  ×  ×  s5 + RH  D  s6 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='23  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='45  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  15  PM x T  x D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 +  ×  T  s2 + D  s3 +  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  ×  ×  ×  T  s4 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  D  s5 + T  ×  ×  ×  D  s6 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  T  D  ×  ×  ×  ×  s7 +  b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='17  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='55  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='19  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='27  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='54  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  16  PM x  RH x T  x D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 +  ×  RH  s2 + T  s3 +  D  s4  ×  ×  ×  + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  RH  s5 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  ×  ×  T x s6 + T  RH  s7 +  ×  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  D  s8 + D  RH  ×  ×  ×  ×  s9 + D  T  s10 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  ×  ×  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='47  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='11  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='19  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='60  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  10  RH  T  s11 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  RH  ×  ×  ×  D  s12 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  D  T  ×  ×  ×  ×  s13 + D  RH  T  s14 +  ×  ×  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  RH  T  D  s15  ×  ×  ×  ×  + b  Machine Learning (10-fold CV)  17  Rando  m  Forest  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = f(PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, T,  RH)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2 Evaluating factors that best explained differences between  corrected OPC and DustTrak and OPC 1 and OPC 2  measurements  The results from our ANOVA analyses are displayed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The bulk of the variation in  the absolute difference/difference between the uncorrected OPC measurements remains  unexplained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' However, when evaluating factors that contributed to the variation in  differences between corrected OPC measurements, we see T explains a moderate amount  of variation (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='01% from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='09% when looking at the absolute difference between corrected  versus uncorrected OPC measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='25 % from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='11%, when looking at the  difference between corrected versus uncorrected OPC measurements), followed by  time-of-day (~3% from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Model 6 corrects OPC measurements using RH and D  (but not explicitly T, although D and T are related).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Thus, this result could arise from our  choice of correction model (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  When evaluating factors that contribute to the differences/ absolute differences between  corrected OPC and DustTrak measurements, we found that for OPC 1 >90% of the variation  was unexplained by the factors considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' For OPC 2, T explained >20% of the variation  (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This could be because Model 6 did not adequately correct OPC 2 measurements  for T as T during the collocation was not representative of T during the mobile deployment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  whereas for OPC 1, as the mobile deployment occurred over more days, the T experienced  was more representative (Figure S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' It is thus important to ensure that the environmental  conditions during collocation are representative of conditions experienced during the mobile  deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Another potential explanation is that degradation in the performance of OPC 1  due to aging was responsible for the largest fraction of the difference in OPC 1 and DustTrak  measurements, rendering the impact of T small in comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Date contributed next to the variation in the absolute difference/difference between corrected  OPC 1 and DustTrak measurements, corresponding to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1% and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' For  OPC 2, hour of day contributed marginally more to variation than day, accounting for ~4-5%  of the variation in the difference and absolute difference of the corrected OPC 2 and  DustTrak measurements (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Aerosol composition varies over the course of a day, and  between days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Our results suggest that this variation does not greatly impact the difference  between corrected-OPC and DustTrak results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  SVF, speed and orientation, even when they significantly contributed to variation of the  differences between OPC and DustTrak measurements, or coincident OPC measurements,  11  only accounted for < 1% of total variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In supplementary analyses, we also considered  road class and found that it also significantly contributed to a small proportion (< 5%) of total  variation in these outcomes (Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Our analysis thus suggests, although vehicle speed, orientation of the OPC-N3, and sky  view factor of the road the measurement was made on (a proxy for the street canyon effect)  impacted the variation of the mobile corrected OPC-N3 measurements, their impact was  small and do not greatly impact our mobile monitoring results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The large unexplained  variation (> 80%) in the difference between the corrected measurements of the two OPCs is  due to the difference in age of the two instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The performance of the older OPC which  was in operation for several years prior to the experiment has likely degraded over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  More work is needed to understand how aging impacts the performance of LCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The  majority of variation (> 80%) in the difference between the corrected measurements from  OPC 1 and the DustTrak is also unexplained, likely due to the same reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Table 2: Sum of Squares (Explained variation (%) in the difference and absolute difference  between uncorrected and corrected OPC measurements, and corrected OPC and DustTrak  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurements) from an ANOVA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The T and RH used in ANOVA analyses  comparing the OPC measurements came from City Scanner corresponding to OPC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  When comparing OPC 1 and OPC 2 measurements with DustTrak data, we used T and RH  data corresponding to the City Scanner corresponding to OPC 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  OPC  No  Orientation  RH  T  Speed  Sky view  factor  Day  Hour  Residuals  Total Sum  of Squares  Absolute  difference  between the  uncorrected  OPC PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  measurements  1 33*  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='17%)  18*  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='09%)  1  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='01%)  6  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='03%)  174*  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='90%)  61*  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='31%)  19,087  (98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='49%)  19,380  Absolute  difference  between the  corrected OPC  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  measurements  using Model 6  (derived from  minute-average  d collocated  measurements)  2 55*  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='51%)  1,194*  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='01%)  0  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='00%)  8*  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='07%)  80*  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='86%)  10,848  Absolute  difference  between the  corrected  DustTrak  measurements  and the OPC  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='31%)  15,192*  (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='13%)  1  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='00%)  1,352*  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='41%)  1,640*  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='14%)  21,239  (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='64%)  39,596  Note: The road class data we used had the following categories: 1) Motorways, 2) Major  Roads, 3) Other Major Roads, 4) Secondary Roads, 5) Local Connecting Roads, 6) Local  Roads of High Importance, 7) Local Roads, 8) Local Roads of Minor Importance, 9) Other  Roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' SVF is a dimensionless parameter that indicates how much sky is obstructed by  buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' When the SVF is 0, the sky is totally obstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' When the SVF is 1, there are no  obstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The SVF for the sampling route was calculated elsewhere using Google Street  View Imagery28  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3 Evaluating the use of mobile OPCs to detect hotspots  Figure 1 displays hotspots identified using uncorrected OPC 1 and DustTrak measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We note that although there appear to be differences in the hotspots identified by each  instrument, the most persistent hotspots corresponding to clusters that contain more than  five measurements in the top percentile made in the same 100 m radius overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This result  is remarkable despite concerns of degradation of measurements from OPC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This suggests  that mobile OPCs can be used to identify important hotspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  13  Figure 1: Clusters of high PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations observed by OPC 1 and the DustTrak that  include a minimum of 5 PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 observations in the top percentile recorded by each  instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The legend displays the number of measurements in the top percentile  corresponding to each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The size of each cluster is proportional to the number of runs  over which a high measurement was observed in the same 100 m radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4 Evaluating the use of mobile OPCs to produce  generalizable maps of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 along the study route  Figure S15 displays a 50 m  50 m grid drawn across the study area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Grid cells in which  ×  mobile measurements were made are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Figure 2 displays maps of generalizable  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations from mobile OPC 1, OPC 2 and DustTrak measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' R and RMSE  comparing generalizable OPC 1 concentrations and the corresponding DustTrak  measurements were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='39 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9 μg/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' These metrics signify an improvement  from comparing unaggregated corrected mobile OPC 1 and DustTrak measurements where  the R and RMSE were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='29 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6, respectively (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' R and RMSE comparing  generalizable OPC 2 concentrations and the corresponding DustTrak measurements were  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='60 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8 μg/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' These metrics also signified an improvement from  comparing mobile OPC 1 and DustTrak measurements where the R and RMSE were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='59  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9 μg/m3, respectively (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  The R and RMSE of generalizable OPC 1 and DustTrak measurements for only the 813 (out  of 3,824) grid cells where the normalized standard error in the generalizable OPC 1 PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  concentrations calculated due to sampling variability from each instrument were < 20%, and  measurements on the segment were made over more than single run improved to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='38 and  14  Overlap  HotspotDustTrak(#measurements)  5  6  13  84  HotspotOPC1(#measurements)  5  11  25  StamenToner2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8 μg/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The R and RMSE of generalizable OPC 2 and DustTrak  measurements for only the 1,268 (out of 1,622) grid cells where the normalized standard  error in the generalizable PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations calculated due to sampling variability from  each instrument were < 20%, and measurements on the segment were made over more  than single run were improved: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='59 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6 μg/m3, respectively (Figure S16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The reduced  RMSE and similar R suggests that aggregating mobile LCS measurements to produce  generalizable maps can reduce the error in these instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The error is smaller on  segments with stable generalizable PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Figure 2: Generalizable PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations (median background-corrected PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  concentrations displayed at the center of the corresponding 50 m  50 m grid cell across the  ×  study area) produced using A) OPC 1 measurements, B) DustTrak measurements  coincident with OPC 1, C) OPC 2 measurements, D) DustTrak measurements coincident  with OPC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  4 Limitations and Future Directions  15  median background-crrected OPC1PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  0-1  A)  ctedDustTrak (OPC1)  2-4  4-6  4-6  B)  6-8  6-8  8-10  8-10  10-12  10-12  12-220  ：12-220  Stamen Toner  Stamen Toner  OPC2PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  medianbackgroun  correctedDustTrak(OPC2)  1-2  C)  D)  2-4  4-6  4-6  8-9  6-8  8-10  8-10  10-12  10-12  12-40  Stamen Toner  Stamen TonerThis experiment had several limitations:  1) We performed the LCS testing in a single season, during daytime hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The  labor-intensive nature of this experiment precluded conducting long-term tests in  more seasons and hours of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We hope to conduct longer-term experiments in  the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2) We evaluated a single LCS: the Alphasense OPC-N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In future experiments we aim  to test a wider array of LCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We also aim to evaluate LCS that measure other  pollutants in addition to PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  3) Although a diverse set of LCS installation scenarios was tested in this experiment,  there are many configurations (orientation of the LCS relative to the direction of  travel, vehicle speeds etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=') that a user might choose to implement that differ  significantly from those tested in this evaluation protocol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' thus the results of this  experiment may have limited application to other configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In future  experiments, we aim to evaluate other configurations over a broader range of  experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  This experiment represents one of the first attempts to provide guidance on calibrating LCS  for use in mobile settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In general, we recommend that researchers use simpler  calibration models to correct mobile LCS data, as these models transfer better from  stationary to mobile deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This paper also represents one of the first attempts to  evaluate the feasibility of using mobile LCS data for two key-use cases: the identification of  hotspots and the production of generalizable maps of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We find that LCS  can be used to identify persistent hotspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We also find that LCS can be used to develop  reasonably accurate generalizable maps of PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 in cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Finally, by using OPCs of different  ages, our paper represents one of the first efforts to show that the age of the LCS plays an  important role in mobile experiments, and more work is needed to understand the impact of  age on LCS performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  References  (1)  deSouza, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Braun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Parks, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Schwartz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Dominici, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Kioumourtzoglou,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Nationwide Study of Short-Term Exposure to Fine Particulate Matter and  Cardiovascular Hospitalizations Among Medicaid Enrollees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Epidemiology 2020, 32 (1),  6–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1097/EDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0000000000001265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  (2)  deSouza, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Dey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Mwenda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Kim, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Subramanian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Kinney, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Robust Relationship between Ambient Air Pollution and Infant Mortality in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Science of The Total Environment 2022, 815, 152755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='scitotenv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='152755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  (3)  Boing, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' deSouza, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Boing, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Kim, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Subramanian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Air Pollution,  Socioeconomic Status, and Age-Specific Mortality Risk in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' JAMA  Network Open 2022, 5 (5), e2213540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1001/jamanetworkopen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='13540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  (4)  deSouza, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Boing, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Kim, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Subramanian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Associations between Ambient  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 – Components and Age-Specific Mortality Risk in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Environmental Advances 2022, 9, 100289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='envadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='  (5)  deSouza, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Hammer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Anthamatten, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Kinney, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Kim, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Subramanian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Impact of Air Pollution on Stunting among Children in  16  Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Environ Health 2022, 21 (1), 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' Braun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' Nethery, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Estimating a Causal Exposure  Response Function with a Continuous Error-Prone Exposure: A Study of Fine  Particulate Matter and All-Cause Mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='2573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Anjomshoaa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' Saha, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Robinson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' Presto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Subramanian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Fine Particle Mass Monitoring with Low-Cost Sensors: Corrections  and Long-Term Performance Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Aerosol Science and Technology 2020, 54 (2),  160–174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1623863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  (32) Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Du, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Calibration of a Low-Cost PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 Monitor Using a  Random Forest Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Environment International 2019, 133, 105161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='envint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='105161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  (33) Considine, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Reid, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Ogletree, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content=' Dye, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Improving Accuracy of Air  Pollution Exposure Measurements: Statistical Correction of a Municipal Low-Cost  Airborne Particulate Matter Sensor Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Environmental Pollution 2021, 268,  115833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='115833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  (34) Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Che, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Lau, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Fung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Lu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' A Feasible Experimental  Framework for Field Calibration of Portable Light-Scattering Aerosol Monitors: Case of  TSI DustTrak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Environmental Pollution 2019, 255, 113136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='envpol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='113136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  (35) Hudda, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Simon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Patton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Durant, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Reductions in Traffic-Related Black  Carbon and Ultrafine Particle Number Concentrations in an Urban Neighborhood  during the COVID-19 Pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Science of The Total Environment 2020, 742, 140931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='scitotenv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='140931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  18  Supplementary Information  S1: Performance of the OPC-N3  The Air Quality Sensor Performance Evaluation Center (AQ-SPEC) of the South Coast Air  Quality Management District (SCAQMD) reported that in a laboratory evaluation23 of three  OPC-N3 sensors, 5-min average  PM1, PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 and PM10 measurements were highly  correlated (R2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='99) with measurements from a reference monitor, and the accuracy of the  sensors was consistent over the range of PM1 (accuracy was 11-14%), PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 (accuracy was  17-24%), and PM10 (accuracy was 4-5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In a field-setting, the correlations between the  sensors and the reference monitor (5-min averages) were lower: the R2 was ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='80, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='60,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='50 for PM1, PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, and PM10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  S2: Description of Random Forest  Random forest (RF): RF is a decision-tree-based machine learning algorithm that  has been shown to perform well in air quality predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Briefly, to generate a  random forest model, the user specifies the maximum number of trees that make up  the forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Each tree is constructed using a bootstrapped random sample from the  training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The origin node of the decision tree is split into sub-nodes by  considering a random subset of the possible explanatory variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Trees are split  based on which of the explanatory variables in each subset is the strongest predictor  of the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This process of node splitting is repeated until a terminal node is  reached1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' For our random forest models, the terminal node was specified using a  minimum node size of five data points per node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  S3 10-fold Cross-Validation  Our 10-fold cross-validation (CV) exercise occurred in the following manner: The model  fitting dataset was randomly split into 10 groups, with each group containing about 10% of  the grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In each cross-validation iteration, we select nine groups of the grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' All of  the data from these grid cells are used to fit the model and make predictions of the  remaining group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We repeated this process 10 times, until every group was predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  The CV process was used to tune two major RF hyperparameters: the number of decision  trees (ntree) and the number of predictors randomly tried at each split (mtry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Briefly, the  optimal mtry was determined by optimizing the “pseudo R2”: the fraction of variance of the  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5c measurements explained by the “out of bag” (OOB) predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' An OOB prediction is  based only on trees in the random-forest ensemble that were not trained using the PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5c  observations being predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Each mtry value was used to train 10 random-forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The  average pseudo R2 was used to select the best mtry value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  19  S4: Calibrating the DustTrak  We compared the results from using three widely-used calibration methods, including the  simple linear regression model in Equation (1), a widely-used the RH-adjusted empirical  equation proposed by Laulainen (1993)2 in Equation (2), and the RH-adjusted linear  regressio model in Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='.(1)  𝑃𝑀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 =  𝑠1 +  𝑠2 × 𝑃𝑀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, 𝐷𝑢𝑠𝑡𝑇𝑟𝑎𝑘  s1, and s2 are empirically determined parameters determine from a regression analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, DustTrak and PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5,reference are the coincident 1-minute averaged PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations  reported by the DustTrak and reference site respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The R and RMSE obtained were  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='81 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0 μg/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In supplementary analyses, we also re-ran these  regressions using hourly-averaged data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The R and RMSE obtained were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='77 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  μg/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' (2)  𝑂𝑅 =  𝑠1 +  𝑠2 ×  (𝑅𝐻/100)  2  (1 − 𝑅𝐻/100)  In Equation (2) OR is the overestimation ratio of the DustTrak PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 relative to the reference  monitor and is  We used RH measurements from the reference station for this  𝑃𝑀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, 𝐷𝑢𝑠𝑡𝑇𝑟𝑎𝑘  𝑃𝑀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒  correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' s1 and s2 are empirically determined parameters from a regression analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Many studies have reported  s1 =1 and  s2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Using minute-averaged data, we obtained  an s1 = 1, and an s2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='13, and an R and RMSE corresponding to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='81 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8 μg/m3,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In supplementary analyses, we also re-ran these regressions using  hourly-averaged data, where obtained an R and RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='78 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8 μg/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' (3)  𝑂𝑅 =  𝑠1 +  𝑠2 × 𝑅𝐻  The R and RMSE obtained from Equation 3 were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='79 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='7 μg/m3, while that obtained  from using hourly-averaged data were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='80 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6 μg/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  S5 Developing maps of generalizable PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations for  the time-period of the mobile deployment in the sampling area  S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1 Background Correction  We estimated background contribution for the uncorrected and corrected data for each OPC  and the DustTrak using two different methods:  1) We adopted the lowest 10th percentile of the PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations measured by the  instrument under consideration for a given hour during the run as the fixed  background value for that run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  2) We used a time-series, spline-of-minimums approach, presented by Brantley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  (2013), to estimate background PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 for each sampling run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We did this by: (a)  20  Applying a rolling 30-second mean to smooth the measurements, (b) Dividing the  time  series  into  discrete  10-minute  segments  and  locating  the  minimum  concentration in each segment, and (c) Fitting a smooth, thin-plate regression spline  through the minimum concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We made the assumption that the background is temporally varying but spatially uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  After estimating the background PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 for each instrument, we performed a background  time-of-day correction or standardization using Equations 4 and 5 to account for the period  during which the vehicle operated:  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5c,i= PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5,OPC i - PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, bkg,i + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, bkg,median      ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='…….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='.……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='…….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' (4)  where PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5,OPC,i is the calibrated measurement for event i, PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5,bkg,i is the contemporaneous  background value of pollution over the entire region, and PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, bkg,median is the median of the  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5,bkg values for the sampling run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' By subtracting the time-of-day-run-resolved regional  background  from  the  pollution  measurement,  we  can  now  compare  background-corrected/local PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations measured during different runs over space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  When the background PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurement value was estimated to be more than that of the  concentration measured, we applied a multiplicative background-correction factor as  displayed in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5c,i= PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5,OPC, i x PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, bkg,median / PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5,bkg,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='. (5)  Unlike Equation 4, this assures non-negative PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5c,i values, based on considering the  background value assessed during the measurement hour as a fraction of the day’s median  background, thus reducing the contribution of possible measurement error due to treating  the hourly value as an absolute quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  We found that both methods to calculate the background-corrected PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 for each instrument  produced similar background-corrected values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The median differences between the two  background-corrected concentrations for the uncorrected and corrected OPC 1, OPC 2 and  the DustTrak measurements were < 5 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Figure S14 displays the background-corrected  OPC 1, OPC 2, and DustTrak data during the mobile deployment and confirms that both  methods to derive background PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations yield similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The median  differences between the raw and each background-corrected concentration for the OPC 1,  OPC 2 and the DustTrak were between 0 - 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Given the minimal differences in median  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5c from the different methods, we chose the splines-of-minimum approach to obtain  background concentrations for both pollutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This is supported by previous research that  found this approach to be an effective way to account for background concentrations for a  range of pollutants over a variety of meteorological conditions and sampling routes (Brantley  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2 Calculating the generalizable PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations across  the sampling route  21  We divided the sampling area into grid cells of 50 m in length, using the ‘Create Grid’ QGIS  Tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' We aggregate the background-corrected PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations made over different runs  and derive the sampling error using the method proposed in deSouza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=', (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  First, we snapped each background-corrected PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurement to the grid cell that it fell  in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' This allows measurements made in the same grid cell to be analyzed as a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Second, we selected the median (which is outlier resistant) of all background-corrected PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  measurements made on a given road segment as the ‘generalizable PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentration’ for  that segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' Third, we used a set of bootstrap resampling procedures to quantify the effect  of sample-to-sample variability and of sampling error on the median concentration for each  road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' As a metric of precision, we used the ratio of the standard error of median  concentration to the median concentration itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  22  Figures  Figure S1: Pair-wise correlations between T and RH recorded by OPC 1, OPC 2, and by the  reference monitoring station  23  T Reference  RHReference  RH OPC 1  RHOPC2  2  T OPC  OPC  TReference  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='52  TOPC1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='58  TOPC2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='94  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='63  RHReference  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='89  RHOPC1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='96  RH OPC 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='96 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  1Figure S2: Pairwise correlations between T, RH and D for A) All mobile measurements  made by the CityScanner platform corresponding to OPC 1, B) All coincident mobile  24  RH OPC 1  TOPC1  DOPC1  A)  RH OPC 1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='77  T OPC 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='37  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='87  D OPC 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='87  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  8°0  RHOPC1  RH OPC 2  DOPC1  DOPC2  TOPC 1  TOPC2  B)  RH OPC 1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='45  RH OPC 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='99  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='43  D OPC 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='44  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='61  D OPC 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='98  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='48  T OPC 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='52  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='98  TOPC2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='98  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8:0measurements made by both CityScanner platforms for the time period where both OPC1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='and OPC2 were operational  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='Figure S3: Comparing the distribution of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='Minute-level A) RH and B) T during the collocation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='of the OPCs in comparison to during the mobile monitoring deployment using violin and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='8  1Figure S10: Reference, OPC 1, corrected OPC 1 measurements using the different  calibration models (1 -17) described in Table 1 at the minute-averaged time resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  31  30  20  10  30  20  10  Ref  30  15,16,17,  20  10  40  20  10  0  20  10  30  20  20  30  20  10  25  10  Apr13  Apr15  Apr 17  Apr19  Apr21  Apr 23  Apr 25  Apr 27Figure S11: Reference, OPC 1, corrected OPC 1 measurements using the different  calibration models (1 -17) described in Table 1 at the hourly-averaged time resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  Apr13  Apr15  Apr17  Apr19  Apr 21  Apr23  Apr25  Apr 27Figure S12: Reference, OPC 2, corrected OPC 2 measurements using the different  calibration models (1 -17) described in Table 1 at the minute-averaged time resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  33  10  15  10  15  10  15  10  15  10  15  10  Ret  11  822505  4320  5250  Apr13  Apr 15  Apr 17  Apr19  Apr21  Apr 23  Apr 25  Apr27Figure S13: Reference, OPC 2, corrected OPC 2 measurements using the different  calibration models (1 -17) described in Table 1 at the hourly-averaged time resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  34  Re  Apr 13  Apr 15  Apr17  Apr19  Apr 21  Apr 23  Apr 25  Apr 27Figure S14: Comparing background PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations calculated for each instrument  during the course of the mobile deployment  Figure S15: 50 m x 50 m grid cells drawn over the sampling area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The grid cells in which  mobile PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurements were made are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  35  A)OPC 1  B)OPC 2  Methods  Methods  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0  Spline of minimums  Spline of minimums  LowesttenthPercentile  Lowest tenth Percentile  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='0 -  0  10000  20000  30000  0  2000  4000  6000  8000  10-  C)DustTrakco-locatedwithOPC1  D)DustTrakco-locatedwithOPC2  8:  Methods  Methods  6  Spline of minimums  6  Splineofminimums  LowesttenthPercentile  LowesttenthPercentile  4  2  10000  20000  30000  0  2000  4000  6000  8000-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='44  MA1A  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='42  Nahant  1132  Ariingtor  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='40  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='38  Waitham  Watertoy  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='36  Newton  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='34  Newtol  ChestnutHil  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='32  wtonHighands  lewto  Upp  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='30  HamHeight  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='28e  WestRoxbu  grid on mobile route  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  grid  OpenStreetMap  Dedham  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='24  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  5 km36  median background-corrected OPC 1 (stable)  1-2  2-4  A)  4-6  6-8  8-10  10-12  12-20  StamenToner  median background-corrected OPC 2 (stable)  1-2  2-4  B)  4-6  6-8  8- 10  10 -12  Stamen TonerFigure S16: Generalizable PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations (median background-corrected calibrated  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentrations in 50 m x 50 m grid cells across the study area) produced using A)  OPC 1 measurements, B) OPC 2 measurements for grid cells where the normalized error in  the median PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 concentration was < 20% and measurements in that grid cell were made  over > 1 day  Tables  Table S1: Performance of the correction models using hourly-averaged and  minute-averaged data as captured using root mean square error (RMSE), and Pearson  correlation (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' A 10-fold CV was used to prevent overfitting in the machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Separate correction models were developed for the two OPCs used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' In Column  Model: s1-s15 and b are empirically derived from regression analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  ID  Name  Model  Minute-averaged  Hourly-averaged  OPC 1  OPC 2  OPC 1  OPC 2  R  RMSE  (μg/m3)  R  RMSE  (μg/m3)  R  RMSE  (μg/m3)  R  RMSE  (μg/m3)  Raw OPC measurements  0  Raw  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='9  Multivariate Regression  1  Linear  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='31  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='6  2  +RH  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 +  ×  RH  s2 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='6  3  +T  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 + T  ×  s2 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='6  4  +D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 + D  ×  s2 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='6  5  +RH x T  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 +  ×  RH  s2 + D  s3 + RH  D  ×  ×  ×  s4 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5  s1 + D  ×  s2 + T  s3 + D  T  s4 +  ×  ×  ×  ×  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 +  ×  RH  s2 + T  s3 + D  s4 +  ×  ×  ×  RH  T  s5 + RH  D x s6 +  ×  ×  ×  T  D  s7 + RH  T  D  ×  ×  ×  ×  ×  s8 + b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5  s3 +  ×  ×  ×  b  10  PM x D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 + D  ×  s2 + D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 + T  ×  s2 + T  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 +  ×  s2 +  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  𝑅𝐻  2  (1−𝑅𝐻) ×  𝑅𝐻  2  (1−𝑅𝐻) ×  s3 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 +  ×  RH  s2 + T  s3 +  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 +  ×  RH  s2 + D  s3 +  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  ×  ×  ×  RH  s4 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  D  s5 +  ×  ×  ×  RH  D  s6 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 + T  ×  s2 + D  s3 +  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  T  ×  ×  ×  ×  s4 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  D  s5 + T  D  ×  ×  ×  s6 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='2  16  PM x RH x  T x D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s1 +  ×  RH  s2 + T  s3 +  D  s4 +  ×  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  RH  s5 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  T  ×  ×  ×  x s6 + T  RH  s7 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5  D  T  s13 + D  ×  ×  ×  ×  RH  T  s14 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  RH  ×  ×  ×  T  D  s15 + b  ×  ×  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='2  Machine Learning (10-fold CV)  17  Random  Forest  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = f(PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, T, RH)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='69  Table S2: Mobile monitoring summary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  Date  Sampl  ing  time  (Numb  er of  measu  remen  ts)  Sampling  area  Vehicle  Mean  Speed  (m/s)  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  Mean (Median)  μg/m3  (  10)  ×  RH  Mean (Median)  (%)  Temperature  Mean (Median)  (0C)  OPC position vis a vis vehicle  direction  R (RMSE  (μg/m3))  comparing  corrected  5-second  DustTrak and  uncorrected  OPC  measurements  OPC1  OPC2  OPC1  OPC2  OPC1  OPC2  OPC 1  OPC 2  02/11/202  09:18-  “South  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6 OPC 1: flow rate parallel to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='31 38  2  13:59  (3,351)  route”: I-93  tunnels,  Downtown  Boston,  South  Boston,  Dorchester,  Roxbury,  North  Quincy  (3)  (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5)  direction of vehicle  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5)  2/16/2022  13:10-  16:35  (2,458)  “North  route”:  Everett,  Revere,  Chelsea,  Winthrop,  East Boston  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5  21  (14) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='8) OPC 1: flow rate parallel to  direction of vehicle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='4) 2/23/2022  08:50-  12:58  (2,976)  North route  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='7  14  (12) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5) OPC 1: flow rate parallel to  direction of vehicle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='6) OPC 1: flow rate parallel to  direction of vehicle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='3) 3/2/2022  13:33-  18:26  (3,440)  South route  + extra run  through  1-93  tunnels  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  5  (2) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='9) OPC 1: flow rate parallel to  direction of vehicle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='2) OPC 1: flow rate parallel to  direction of vehicle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='9) OPC 1: flow rate parallel to  direction of vehicle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='6) OPC 1: flow rate perpendicular to  direction of vehicle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='6  2  (1)  5  (3)  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='8)  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='6)  OPC 1: flow rate perpendicular to  direction of vehicle  OPC 2: flow rate parallel to the  direction of the vehicle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='15  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='45  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6)  3/30/2022  12:06 -  16:02  (2,828)  South route  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  2  (1)  9  (5)  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='7)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8)  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6)  OPC 1: flow rate perpendicular to  direction of vehicle  OPC 2: flow rate parallel to the  direction of the vehicle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='33  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='7)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='44  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1)  4/05/2022  10:27 -  14:24  (2,845)  South route  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  4  (3)  7  (4)  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4)  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1)  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8)  OPC 1: flow rate perpendicular to  direction of vehicle  OPC 2: flow rate parallel to the  direction of the vehicle  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='33  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='24  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8)  Note that due to the low uncorrected OPC PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 measurements, we report PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  concentrations  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  ×  Table S3: Comparing corrected OPC measurements using different calibration models  derived from minute- and hourly-averaged collocated data with that of the corresponding  corrected (from the calibration model developed using hourly-averaged collocation data)  DustTrak data in our mobile deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content=' The R and RMSE when comparing the corrected  DustTrak measurements using minute-averaged and hourly-averaged data during the  collocation were 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='69 μg/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='  ID  Name  Model  Calibration derived from minute-averaged  data  Calibration derived from hourly-averaged  data  DustTrak and OPC 1  DustTrak and OPC 2  DustTrak and OPC 1  DustTrak and OPC 2  R  RMSE  (μg/m3)  R  RMSE  (μg/m3)  R  RMSE  (μg/m3)  R  RMSE  (μg/m3)  Raw OPC measurements  0  Raw  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  Multivariate Regression  1  Linear  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0  2  +RH  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + RH  s2 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0  3  +T  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + T  s2 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='31  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='31  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  40  4  +D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + D  s2 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='42  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='27  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='39  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  5  +RH x T  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + RH  s2 + T  s3  ×  ×  + RH  T  s4 + b  ×  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='27  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='33  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='31  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  6  +RH x D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + RH  s2 + D  s3  ×  ×  + RH  D  s4 + b  ×  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='59  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='27  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='56  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  7  +D x T  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + D  s2 + T  s3 +  ×  ×  D  T  s4 + b  ×  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='28  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='48  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='27  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='45  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  8  +RH x T  x D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + RH  s2 + T  s3  ×  ×  + D  s4 + RH  T  ×  ×  ×  s5 + RH  D x s6 + T  ×  D  s7 + RH  T  ×  ×  ×  ×  D  s8 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='08  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  9  PM x  RH  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + RH  s2 + RH  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s3 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='28  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='31  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  10  PM x D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + D  s2 + D  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s3 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='17  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='28  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='19  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='27  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='4  11  PM x T  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + T  s2 + T  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s3 + b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='25  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='26  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='24  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  12  PM x  nonlinea  r RH  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 +  s2 +  𝑅𝐻  2  (1−𝑅𝐻) ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  s3 +  𝑅𝐻  2  (1−𝑅𝐻) ×  ×  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='22  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='23  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  13  PM x  RH x T  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + RH  s2 + T  s3  ×  ×  +  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  RH  s4 +  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  T  s5 + RH  ×  ×  T  s6 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  ×  ×  ×  RH  T  s7 +  b  ×  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='18  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='27  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='25  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  14  PM x  RH x D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + RH  s2 + D  s3  ×  ×  +  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  RH  s4 +  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  D  s5 + RH  ×  ×  D  s6 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  ×  ×  ×  RH  D  s7 +  b  ×  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='21  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='28  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='21  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='23  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  15  PM x T  x D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + T  s2 + D  s3 +  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  T  s4 +  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  D  s5 + T  ×  ×  ×  D  s6 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  T  ×  ×  ×  D  s7 +  b  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='17  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='19  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='27  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='0  41  16  PM x  RH x T x  D  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5 ×  s1 + RH  s2 + T  s3  ×  ×  +  D  s4 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  ×  ×  RH  s5 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  T x  ×  ×  s6 + T  RH  s7 +  ×  ×  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  D  s8 + D  ×  ×  ×  RH  s9 + D  T  s10  ×  ×  ×  + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  RH  T  ×  ×  ×  s11 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  RH  D  ×  ×  s12 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  D  T  ×  ×  ×  s13 + D  RH  T  ×  ×  ×  ×  s14 + PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5  RH  T  ×  ×  D  s15 + b  ×  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='20  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='11  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='19  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='9  Machine Learning (10-fold CV)  17  Random  Forest  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
+page_content='5, corrected = f(PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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+page_content='5  Table S4: Sum of Squares (Explained variation (%) in the difference and absolute difference  between uncorrected and corrected OPC measurements, and corrected OPC and DustTrak  PM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQfXwdZ/content/2301.03847v1.pdf'}
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diff --git a/WNE2T4oBgHgl3EQfYAd6/content/tmp_files/2301.03850v1.pdf.txt b/WNE2T4oBgHgl3EQfYAd6/content/tmp_files/2301.03850v1.pdf.txt
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@@ -0,0 +1,1353 @@
+Effect of temperature-dependent thermophysical properties on
+turbulent forced convection under constant heat flux boundary
+condition
+Himani Garga, Lei Wanga,∗
+aDepartment of Energy Sciences, Lund University, Ole R¨omers v¨ag, Lund, 22100, Sweden
+Abstract
+In this study, we performed highly resolved large-eddy simulations (LES) to investigate the in-
+fluence of variable properties on the forced turbulent convection in a channel. The constant heat
+flux boundary condition permits wall temperature fluctuations and thus induces variations of fluid
+properties. Since the effect of viscosity on the flow exhibits Re−1
+τ
+scaling, we only considered
+Reτ = 180 in the present study. Compared to the flow with constant properties, results indicate
+that the variable properties have trivial effects on the mean velocity and temperature profiles,
+Reynolds shear stress, wall-normal heat flux, as well as the small-scale turbulence characteris-
+tics. However, we also observed that the turbulence intensities, low-speed streaks, burst motions,
+and budgets for temperature variance and wall-normal heat flux are modified by the variable
+properties in a perceptible way. In addition, we showed that the classic wall scaling is a good
+choice for flow with small and moderate variations of fluid properties.
+Keywords: wall-resolved LES, turbulent convection, variable flow properties
+1. Introduction
+For forced convection in a turbulent pipe or channel flow, the Nusselt number can be ob-
+tained, for example, by using the Gnielinski correlation on the basis of constant thermophysical
+properties [1]. However, for fluids under intensive heating, such as in the concentrating solar
+receivers, supercritical water-cooled reactors, heat exchangers in chemical engineering, etc., the
+fluid properties are temperature-dependent, which, in turn, alters the flow and temperature fields
+and cause a departure of heat transfer rate from the constant-property correlations. To account
+for the departure, Petukhov [2] presented a theoretical analysis of heat transfer and friction to flu-
+ids with variable thermophysical properties. In this work, the Nusselt number and friction factor,
+which are calculated by the constant-property equations, are modified by the viscosity ratios eval-
+uated at bulk and wall temperatures, or Nu/Nuconst = (µbulk/µwall)n and f/ fconst = (µbulk/µwall)−m,
+where n and m depend on the specific flow conditions. Since then, further experimental and
+theoretical investigations have been carried out to propose different correlations which take the
+∗Corresponding author
+Email address: lei.wang@energy.lth.se (Lei Wang )
+Preprint submitted to Journal of heat and mass transfer
+January 11, 2023
+arXiv:2301.03850v1  [physics.flu-dyn]  10 Jan 2023
+
+variable properties into account [3, 4, 5, 6]. These works, however, mainly focused on the deter-
+mination of coefficients n and m, rather than on the characterization of the flow behavior from a
+physical point of view.
+Among analytical and numerical studies dealing with variable properties problems, Pinarbasi
+et al. [7] studied the influence of temperature-dependent viscosity and thermal conductivity on
+the velocity and temperature fields in a two-dimensional channel flow with isothermal boundary
+conditions. It was found that variable properties have some impacts on the temperature field
+but very scarce effects on the velocity profiles. Zonta et al. [8] performed a DNS study of the
+forced convection of water in a heated channel in which the viscosity of the fluid is temperature-
+dependent. Compared with the isothermal flow with constant viscosity, they observed that turbu-
+lence is promoted on the cold side of the channel where viscosity is higher; while turbulence is
+dampened on the hot side where viscosity is lower. In addition, they identified the connection be-
+tween the variable viscosity and the near-wall turbulence structures that sustain the regeneration
+cycle, i.e., low-speed streaks, bursting events, and quasi-streamwise vortices. Similarly, using
+water as a working fluid, Lee et al. [9] conducted DNS studies of turbulent boundary layers over
+isothermally heated walls in which the effect of viscosity stratification on turbulence and skin
+friction was studied. They demonstrated that the turbulence intensity and the Reynolds shear
+stress were decreased in the heated walls. In line with the attenuated turbulence transport, the
+skin-friction factor was reduced by up to 26%. Subsequently, the same author, Lee et al. [10],
+found that the Stanton number and the Reynolds analogy factor are augmented by 10% and 44%,
+respectively, in the variable viscosity flow due to the modification of the near-wall turbulence.
+In strongly anisothermal wall-bounded air flows, the momentum and energy exchange are also
+under the influence of variations in temperature. Contrary to the high-speed compressible flow,
+acoustic waves have a negligible impact in anisothermal flows with low Mach numbers. Serra
+et al. [11] performed thermal large-eddy simulations (LES) in a channel flow asymmetrically
+heated from both sides. They showed that the impact of temperature-dependent viscosity and
+thermal conductivity is less important than the impact of variable density which is caused by
+thermal expansion. Toutant and Bataille [12] implemented DNS studies of forced convection in
+an anisothermal channel flow with highly variable fluid properties. Their data illustrated that the
+turbulence intensity and Reynolds shear stress are enhanced on the cold side; while on the hot
+side, the increase of viscosity and thermal expansion are in competition to affect the turbulent
+motions. To assess the isolated influence produced by the temperature-dependent viscosity and
+thermal conductivity, Wang et al. [13] performed LES to study the turbulent heat transfer in an
+anisothermal channel flow. Their data showed that variation in viscosity and thermal conductiv-
+ity leads to the modification of the mean velocity and temperature fields. However, the molecular
+momentum and heat transport across the channel remain unaffected. Consistent with the results
+in [12], they also observed that the turbulence is promoted at the cold side even though variable
+properties have negligible effect in the near-wall region y+ < 20.
+More recently, there has been an increased interest in studying turbulent heat transfer to fluids
+at supercritical pressures [14, 15, 16]. These fluids exhibit strong thermophysical property vari-
+ations as the pseudo-critical temperature is crossed. Yoo [14] pointed out that the two dominant
+mechanisms in modulating turbulence and scalar transport in supercritical fluids are buoyancy
+and large variations of properties. Peeters et al. [15] conducted DNS of supercritical fluid in an
+annular channel with a hot inner wall and a cold outer wall. It was shown that the turbulence was
+attenuated at the hot wall but amplified at the cold wall. It is worth noting that this trend is similar
+to the sub-critical fluid in [8]. However, by analyzing a transport equation for the coherent streak
+flank strength, it was found that thermophysical property fluctuations significantly affect streak
+2
+
+evolution and the self-regeneration process in near-wall turbulence. Nemati et al. [16] studied
+the effect of thermal boundary conditions on turbulent heat transfer to fluids at supercritical pres-
+sures. Two different thermal wall boundary conditions were considered: the first corresponds to a
+Neumann boundary condition that permits wall temperature fluctuations; the second corresponds
+to a Dirichlet boundary condition that does not allow wall temperature fluctuations. The results
+showed an enhancement of 7% in Nusselt number under the Neumann boundary condition be-
+cause large property variations are induced that consequently cause an increase of wall-normal
+velocity fluctuations and, thus, an increased overall heat flux.
+In this work, we aim to investigate the influence of temperature-dependent viscosity and ther-
+mal conductivity on turbulent forced convection under the constant heat flux boundary condition
+(or Neumann boundary condition). To this end, we performed highly-resolved LES studies in a
+heated channel flow where the air is treated as an incompressible fluid and isolated from gravity.
+The variable viscosity and thermal diffusivity are connected by the Prandtl number, which is kept
+constant at 0.71 in our study. To highlight the viscous effect, we focused on Reτ = 180 because
+the viscous term exhibits a Re−1
+τ
+scaling in our methodological framework. Under the constant
+heat flux boundary condition, enthalpy (or, the bulk-mean temperature) usually undergoes a lin-
+ear increase from the inlet to the exit. This makes it hard to implement the periodic boundary
+condition in the computational domain. Therefore, a modification of temperature is introduced
+into the energy equation to accommodate this change. This part is described in Section 2. Results
+and discussions are presented in Section 3. Finally, the conclusions are given in Section 4.
+2. Numerical methodology
+In this study, wall-resolved LES of thermal flow in a Poiseuille turbulent channel has been
+conducted within a computational box of Lx = 2πδ, Ly = 2δ, and Lz = πδ, with spanwise and
+streamwise periodicity, shown in Fig. 1. The flow is considered fully incompressible, Newto-
+nian, and heated with a uniform wall heat flux qw. The streamwise, wall-normal, and spanwise
+coordinates are x, y, and z, respectively and the corresponding velocity components are u, v,
+and w. No-slip boundary condition is imposed on the walls. The temperature is denoted by T.
+Statistically, time-averaged quantities are denoted by angular brackets, ⟨...⟩, whereas fluctuating
+quantities are denoted by prime, (...)′. In LES, the governing equations for the conservation of
+mass, momentum, and energy are averaged by a spatial filtering procedure, which is denoted by
+the symbol˜. After introducing the subgrid-scale (SGS) model, the filtered LES equations read
+∂˜ui
+∂xi
+= 0
+(1)
+∂˜ui
+∂t + ∂˜ui˜u j
+∂x j
+= − ∂ ˜p
+∂xi
++ ∂
+∂xj
+�
+(ν + νsgs)
+� ∂˜ui
+∂xj
++ ∂˜uj
+∂xi
+− 2
+3δi j
+∂˜uk
+∂xk
+��
++ Fδi1,
+(2)
+∂ ˜T
+∂t +
+∂
+� ˜T ˜ui
+�
+∂xi
+= ∂
+∂xi
+��
+α + αsgs
+� ∂ ˜T
+∂xi
+�
+(3)
+where ui is the ith component of the velocity vector, T is the temperature, p is the fluctuat-
+ing kinematic pressure, F is the driving pressure gradient, ν and α are kinematic viscosity and
+thermal diffusivity, respectively. Here, α is connected with ν by Prandtl number which is kept
+constant at 0.71. Under the constant heat flux boundary condition, the bulk mean temperature
+3
+
+Figure 1: Schematic of channel flow configuration.
+grows linearly with respect to the streamwise direction x, which makes it hard to implement peri-
+odic boundary conditions in the computational domain. In this context, we follow the suggestion
+of Kasagi et al. [17] by introducing a transformed temperature Θ = T − Tw in the energy equa-
+tion. The great advantage of this method is that Θ is periodic in the streamwise direction [18, 19].
+Under the fully developed flow, the energy equation is changed to a new energy equation with
+an internal heat source which depends on the wall heat flux qw and the local streamwise velocity
+u1, as shown in Eq. 4,
+∂ ˜Θ
+∂t +
+∂
+� ˜Θ˜ui
+�
+∂xi
+= ∂
+∂xi
+��
+α + αsgs
+� ∂ ˜Θ
+∂xi
+�
++ qwuτ
+ρcpνw
+˜u1
+⟨U1⟩
+(4)
+where density ρ, heat capacity cp, viscosity at the wall νw, are considered to be constant. The
+boundary condition for the transformed temperature is simply Θ = 0 at y = 0 and 2δ.
+To account for the temperature-dependent viscosity, we use a power relation to represent the
+dependency of viscosity on the transformed temperature Θ, as shown in Eq. 5,
+ν = ν (Θ0)
+�Θ + Θ0
+Θ0
+�n
+(5)
+where the reference viscosity ν(Θ0) = 2 × 10−5 m2/s, and the exponent n = 0.2. For the case of
+the constant properties (CP), the viscosity is uniform throughout the domain (ν = 2×10−5 m2/s).
+For the case of the variable properties (VP), the viscosity at the wall is kept at the reference value
+at 2 × 10−5 m2/s, while the bulk-mean viscosity νb in the domain is about 2.4 × 10−5 m2/s. More
+detailed simulation parameters are shown in Table 1. For both considered cases, Reτ = 180.
+However, due to the different bulk velocity and viscosity, the bulk Reynolds numbers Reb for the
+CP and VP cases are 5967 and 4912, respectively. Nevertheless, they are considered to be fully
+turbulent flows.
+In the simulations, the subgrid viscosity νsgs is calculated by means of the Smagorinsky model
+[20]. However, the standard Smagorinsky model results in the non-zero value of νsgs at the solid
+4
+
+qw = const
+28
+Yi
+ZTable 1: Summary of simulation parameters.
+Case
+Ub (m/s)
+uτ (m/s)
+νb (m2/s)
+Reb
+Reτ
+CP
+0.0597
+0.0036
+2.0 × 10−5
+5967
+180
+VP
+0.0587
+0.0036
+2.4 × 10−5
+4912
+180
+boundary. This defect can be avoided by introducing a van-Driest damping function [21] into
+the length scale. The turbulent heat flux is modeled with a gradient transport model [22], with
+αsgs = νsgs/Prt, where the turbulent Prandtl number is equal to Prt = 0.85.
+We carried out wall-resolved LES with an open-source OpenFOAM solver. Herein we limit
+ourselves to a summary of the numerical method. The governing equations are discretized on
+a collocated grid system. To avoid spurious pressure-velocity coupling, the Rhie-Chow inter-
+polation is adopted [23]. A second-order implicit scheme implements time integration. The
+convective terms for the momentum and energy equations are discretized with a second-order
+Gauss limited linear scheme [24]. The Poisson equation is solved for the pressure following both
+the predictor and corrector steps to satisfy the continuity equation at each step. Once the velocity
+field is known, the temperature field is computed as a solution of the energy equation, and finally,
+viscosity is updated using Eq. (5).
+The computational domain contains 102×129×128 grid points. The mesh is considered non-
+uniform in the wall-normal direction and uniform in the streamwise and spanwise directions. The
+grid points in the wall-normal direction are determined by a hyperbolic tangent transformation
+[25].
+yk = H
+�
+1 + 1
+a tanh
+�
+−1 + 2k − 1
+Ny
+�
+tanh−1(a)
+�
+, k ∈
+�
+1,Ny
+�
+,
+(6)
+where Ny = 129 is the number of grid points on the y-axis, and a is an adjustable parameter
+depending on the mesh dilation. In the present case, a = 0.96135. The dimensionless mesh
+sizes are ∆+
+x = 11.5, ∆+
+z = 4.5, and ∆+
+y = 0.2 at the wall to 5.7 at the center. These values
+are substantially lower than the recommendations by Piomelli & Chasnov [26] for wall-resolved
+LES.
+For the characterization of present simulations, classical wall scaling is used. In particular,
+the friction velocity, uτ, is obtained by solving the near-wall mean velocity gradient,
+uτ =
+�
+νw
+�∂U
+∂y
+�
+w
+,
+(7)
+where the subscript w refers to the wall quantities. The friction temperature, Tτ is defined as:
+Tτ =
+qw
+ρcpuτ
+,
+(8)
+where qw is the wall heat flux. Moreover, y+ is defined as:
+y+ = yuτ
+νw
+.
+(9)
+In the present study, the flow remains turbulent for all the considered cases, and the turbulent
+Reynolds number, Reτ, is based on the uτ and νw:
+Reτ = uτδ
+νw
+(10)
+5
+
+Prior to collecting data, each case has been run for approximately 100 diffusion time units
+(δ/uτ) to reach a statistically steady state which is identified by the convergence of the root-
+mean-square velocity (u, v, w) and temperature (θ) fluctuations. Subsequently, data are collected
+over roughly 40 diffusion time units. Finally, the mean scalar quantities and various statistical
+moments are computed by integrating in time and in the homogeneous x and z directions (i.e.,
+the Reynolds average).
+3. Results and Discussions
+3.1. Validation
+Before detailing the effect of temperature-dependent viscosity on the flow properties, we per-
+formed a detailed validation against the DNS database available for the channel flow at Reτ = 180
+where the fluid properties are constant. Figure 2(a) shows the profiles for the mean streamwise
+velocity U+ and its fluctuations U+
+rms, plotted in semi-logarithmic coordinates. The DNS results
+of Moser et al. [27] are also plotted for the comparison’s sake. Our simulation results in an
+overestimation of the mean velocity in the log region. General agreement for root-mean-square
+fluctuations between present simulation results and DNS is reasonably good; however, the peak
+value is slightly shifted outwards. In Fig. 2(b), the profiles of mean temperature and its variance
+are also plotted. The results of Seki et al. [28] are presented for comparison. To keep consistent
+with [28], we use T +
+rms instead of θ+
+rms to represent the temperature fluctuations. Similarly to the
+mean flow profiles, a slight overestimation of the results compared to DNS is observed. Like
+U+
+rms, the peak value of T +
+rms is properly predicted, but the peak position is also shifted outwards.
+It is noted that even finer meshes would reduce the disagreement, but they cannot completely
+remove it. This relates to the Smagorinsky model, which reproduces the action of small scales
+by introducing a subgrid viscosity, but it does not model the backward of the energy cascade
+[29]. Nevertheless, compared to the reference DNS data, the essential features of the flow and
+scalar quantities are captured, especially close to the wall, and the maximum discrepancies are
+within 10%.
+(a)
+(b)
+Figure 2: Comparison of (a) mean velocity and turbulent velocity fluctuations and (b) mean temperature and temperature
+fluctuations, with DNS [27, 28]. The results are extracted for the constant properties case.
+6
+
+4
+22
+LES,presentstudy
+20
+DNS,Moseretal.(1999)
+18
+3
+16
+14
+su.
+12
++
+2
+U
+10
+8
+6
+4
+2
+0
+0
+10°
+101
+102
++4
+20
+LES,presentstudy
+DNS,Sekietal.(2008)
+3
+15
+sul.
+10
+5
+0
+0
+10°
+101
+1023.2. Mean flow variables
+The profiles of the mean streamwise velocity and temperature plotted against the wall units
+are presented in Fig. 3, normalized by uτ and Tτ, respectively. In Fig. 3(a), the impact of
+variable viscosity on the profile of the mean streamwise velocity is trivial, and only a slight
+difference is seen in the log region; In Fig 3(b), the mean temperature profiles for the two cases
+agree reasonably well. This is in contrast with the results of Lee et al. [10], showing an upward
+shift of mean velocity profile and a downward shift of temperature profile in the log region for the
+cases of variable viscosity compared to the constant reference. The difference between [10] and
+our studies can be attributed to the different thermal boundary conditions. In [10], the isothermal
+wall heating with different temperatures (or Dirichlet boundary conditions) was used. In their
+simulations, the higher wall temperature leads to lower viscosity of water and, thus, lower skin
+friction, which is associated with the upshift of the mean velocity profiles. In the present studies,
+the Neumann boundary condition results in the same viscosity at the wall for the two concerned
+cases. In particular, for the VP case, the increased viscosity in the fluid does reduce the bulk
+Reynolds number in the channel, as shown in Table 1, but it does not change the skin friction
+on the wall. In addition, we observe that the variable viscosity and thermal conductivity do not
+modify the mean temperature profile. This is also inconsistent with the results in [7], showing that
+temperature-dependent properties have some impact on the temperature field. However, in [7],
+they also used the isothermal boundary condition. Despite the influence of variable properties,
+the good agreement of the mean velocity and temperature profiles between CP and VP cases
+indicates that the classic wall units are a good choice for scaling. Of course, this conclusion
+may only apply to the fluids having small to moderate dependence on temperature. For fluids
+undergoing significant variations of physical properties like superfluids, however, scaling that
+takes into account the temperature-dependent properties is suggested [30, 31].
+y
+U
+10
+0
+10
+1
+10
+2
+5
+10
+15
+20
+constant properties
+variable properties
++
++
+(a)
+y
+T
+10
+0
+10
+1
+10
+2
+5
+10
+15
+20
+constant properties
+variable properties
++
++
+(b)
+Figure 3: Comparison of the wall-normal distribution of (a) mean streamwise velocity and (b) temperature in the wall
+coordinates. The profiles of streamwise velocity and temperature are normalized by uτ and Tτ respectively, for each case.
+3.3. Statistical properties
+Apart from the mean velocity and temperature profiles, it is of interest to investigate the
+influence of variable properties on turbulent statistical properties. In Fig. 4(a), we plot the
+7
+
+profiles of the Reynolds shear stress, ⟨−u′v′⟩, which are normalized by u2
+τ, against the wall unit
+y+. In general, the two profiles collapse very well in the region of y+ > 100. Close to the wall,
+a small deviation of the profile with variable properties from the benchmark case is observed. In
+particular, the property variations slightly shift the peak position outwards but without modifying
+its peak value. A very similar trend is found in the profiles of wall-normal heat flux ⟨−v′θ′⟩,
+which are normalized by uτTτ and shown in Fig. 4(b). The similarity between the wall-normal
+heat flux and the Reynolds shear stress displayed in the present study is in line with the DNS
+results in [32], indicating a strong correlation between u and θ. This feature will be visited later.
+Fig. 4(c) illustrates the profiles of streamwise turbulent heat flux, ⟨u′θ′⟩, normalized by uτTτ. It
+is observed that the profiles collapse very well in the wall region, y+ < 20. Further away from
+the wall, the variable properties increase its magnitudes and shift the profile upward from the
+benchmark case. It is worth noting here in wall-bounded turbulent flow, the momentum and heat
+are mainly transported via Reynolds shear stress and wall-normal heat flux, which are almost
+unaffected by the property variations, as shown in the present study. Thereby, we can reach a
+conclusion that the variable properties have a trivial influence on the turbulent transport across
+the channel.
+To further inspect the impact of fluid properties on the turbulence intensities, the distributions
+of root-mean-square streamwise, wall-normal, and spanwise velocity, as well as temperature fluc-
+tuations, are presented in Fig. 5. Here, we observe the striking similarity between the streamwise
+velocity fluctuations u+
+rms and temperature fluctuations θ+
+rms. In the near-wall region y+ < 20, the
+variable properties have no effect on the streamwise velocity and temperature fluctuations, and
+the profiles collapse very well for the CP and VP cases. In the outer region y+ > 20, the increased
+viscosity and thermal conductivity tend to enhance the magnitudes of u+
+rms and θ+
+rms, respectively.
+This is also in contrast with the results of Lee et al. [10]. In [10], the wall heating results in
+the diminished turbulence intensity; however, if normalized by the friction velocity uτ, a better
+agreement was achieved for various u+
+rms profiles. Further inspection of Fig. 5 shows that the
+variable properties have some visible impacts on the quantities of v+
+rms and w+
+rms. In particular,
+close to the wall region, both vrms and wrms levels are suppressed compared to the benchmark
+case, and further away from the wall, the fluctuation level enhances. In other words, v+
+rms and
+w+
+rms can feel the influence of the variable properties throughout the channel, while u+
+rms and θ+
+rms
+are modified only in the outer region y+ > 20.
+3.4. Probability density functions
+Probability density functions (PDFs) of velocity and scalar quantities such as temperature
+are fundamental in studying and modeling statistical characteristics of flow and heat transfer
+in wall-bounded turbulent flows. The PDFs of streamwise, wall-normal, and spanwise velocity
+components and of temperature θ are depicted in Fig. 6 for both constant properties (see Fig.
+6(a)) and variable properties cases 6(b). To highlight the dynamic structures in the near-wall
+region, the PDF profiles are plotted only at the position y+ = 5.5. By inspecting Fig. 6(a), the first
+impression conveyed to us is that PDF profiles of u and θ are in good agreement, indicating that
+the two quantities are strongly coupled; Second, the negatively skewed peak of u profile suggests
+that low-speed streaks are dominant structures in the near-wall sublayer. From a heat transfer
+point of view, the low-speed streaks mainly contain the colder fluid. In addition, both PDF
+profiles for u and θ show a long tail in the positive region, indicating that the flow in the vicinity
+of the wall is of intermittent nature. These predictions are in accordance with the experimental
+results of Antonia et al. [33] and the DNS results of Redjem-Saad et al. [34]. Under the
+variable properties, as shown in Fig. 6(b), the strong coupling between u and θ is not affected,
+8
+
+y
+< -u’v’>
+0
+50
+100
+150
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+constant properties
+variable properties
++
++
+(a)
+y
+< -v’ ’>
+0
+50
+100
+150
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+constant properties
+variable properties
++
++
+θ
+(b)
+y
+< u’ ’>
+0
+50
+100
+150
+0
+1
+2
+3
+4
+5
+6
+7
+8
+constant properties
+variable properties
++
++
+θ
+(c)
+Figure 4: Comparison of normalized profiles of (a) turbulent shear stress, (b) wall-normal heat flux, and (c) streamwise
+heat flux in the wall coordinates.
+however, the negatively skewed peak is replaced by a plateau which keeps being negatively
+inclined. Regarding the other two velocity components, under constant properties, the PDF
+profile of w displays a quasi-Gaussian distribution, while v profile exhibits a higher peak than
+w. This is due to the fact that approaching towards the wall, v is proportional to y3 while w ∝
+y2 [35]. Under the variable properties, as shown in Fig. 6(b), the PDF profiles still keep their
+symmetrical shape; however, the peak value is seen to increase, especially for the w component.
+This indicates that the distribution of the spanwise velocity loses its Gaussian nature under the
+influence of variable properties.
+3.5. Joint probability distribution function
+The joint PDFs (JPDFs) are very valuable in describing the physics of turbulent flows. At
+y+ = 5.5, the JPDFs of the streamwise and wall-normal velocity component fluctuations u′ and
+v′ are presented in Fig. 7(a) and Fig. 7(b), respectively. Here, the fluctuating values used
+for JPDFs are scaled with respect to the rms value. Under the constant properties, Fig. 7(a)
+9
+
+y
+U
+0
+50
+100
+150
+0
+0.5
+1
+1.5
+2
+2.5
+3
+constant properties
+variable properties
++
++
+rms
+(a)
+y
+V
+0
+50
+100
+150
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+constant properties
+variable properties
++
++
+rms
+(b)
+y
+W
+0
+50
+100
+150
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+constant properties
+variable properties
++
++
+rms
+(c)
+y
+T
+0
+50
+100
+150
+0
+0.5
+1
+1.5
+2
+2.5
+3
+constant properties
+variable properties
++
++
+rms
+(d)
+Figure 5: Comparison of normalized profiles of the root-mean-square of the (a) axial velocity fluctuations, (b) wall-
+normal velocity, (c) spanwise velocity, and (d) root-mean-square of the temperature fluctuations, in the inner scale units.
+demonstrates that the second quadrant (u′ < 0 and v′ > 0, ejections) and fourth quadrant (u′ > 0
+and v′ < 0, sweeps) have larger contributions to the Reynolds shear stress and thus to turbulent
+energy production. This is in agreement with most existing studies. For variable properties, Fig.
+7(b) shows that the ejection events are slightly less vigorous. Similarly, we observe that ejections
+(v′ > 0, θ′ < 0) and sweeps (v′ < 0, θ′ > 0) are dominant motions in contributing to the wall-
+normal heat flux, as shown in Fig. 7(c). This corroborates the DNS results by Kong et al. [32].
+Under the influence of variable properties, as shown in Fig. 7(d), this trend is almost unaffected.
+As we discussed earlier, the similarity between the Reynolds shear stress < −u′v′ > and wall-
+normal heat flux < −v′θ′ > indicates the strong coupling between streamwise velocity u′ and
+temperature θ′. Fig. 7(e) does demonstrate such a strong correlation and further supports this
+conclusion. Under the influence of variable properties, Fig. 7(f) illustrates that the peak values
+of JPDF are somewhat attenuated. Nevertheless, the magnitude of < u′θ′ > at this position is
+found to be unaffected, as shown in Fig. 4(c).
+10
+
+i/i
+pdf(i/i
+)
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+u
+v
+w
+θ
+constant properties
+rms
+rms
+(a)
+i/i
+pdf(i/i
+)
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+u
+v
+w
+variable properties
+rms
+rms
+θ
+(b)
+Figure 6: Probability density function at y+ = 5.5.
+3.6. Distribution of velocity and temperature increments
+In order to further verify the strong coupling between streamwise velocity and temperature,
+we calculate the PDFs of the increments of u and θ, respectively. Here, the increments are
+denoted as ∆u = u(x + r) − u(x), and ∆θ = θ(x + r) − θ(x), where r represents the distance
+between two given points. Usually, the separation r is located in the inertial sub-scale, i.e.,
+η < r < δ, where δ is the half-height of the channel, and η is the Kolmogorov scale which is
+estimated by the relation η = δ/Re3/4
+τ . The PDFs of both increments at the position y+ = 5.5
+are plotted in Fig. 8, where the separation r = 3η and 15η, respectively. It is clear for both
+constant and variable properties, the distribution of ∆u and ∆θ are in good agreement. For smaller
+separation at r = 3η, however, the PDF profiles are characterized by wider tails. This indicates
+the small-scale intermittency which has been observed in various turbulent flows [36, 37]. A
+proper understanding of small-scale turbulence is especially important for practical calculations
+such as LES and RANS models. At a larger scale r = 15η, it is noted that the wider tails become
+narrower, and the curves approach the Gaussian distribution. If r = δ, for example, we noticed
+that the PDFs of increments are essentially indistinguishable from a Gaussian (not shown here).
+3.7. Budget terms
+Concerning thermal flows, the equation for the budgets of the turbulent heat fluxes, ⟨uiθ⟩,
+and temperature variance, kθ =
+�
+θ2�
+, are referred to Kawamura et al. [38]. These equations are
+transcribed here to facilitate the reading of the paper.
+The budget equation for the turbulent heat fluxes, ⟨uiθ⟩, is given by:
+D
+�
+u′
+i
++θ′+�
+Dt+
+=
+Piθ
+����
+Production
+−
+ϵiθ
+����
+Dissipation
++
+Tiθ
+����
+Turbulent diffusion
++
+Diθ
+����
+Molecular diffusion
++
+Πiθ
+����
+Pressure-temperature gradient correlation
+(11)
+where D/Dt is the material derivative. The different terms on the right-hand side respectively are
+defined as follows:
+Piθ = −
+�
+u′
+k
++θ′+� ∂ �Ui+�
+∂x+
+k
+−
+�
+u′
+i
++u′
+k
++� ∂ ⟨Θ+⟩
+∂x+
+k
++
+�
+u′
+1
++u′
+i
++� ∂ ⟨Tw⟩
+∂x+
+1
+(12)
+11
+
+u’
+v’
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+constant properties
+(a)
+u’
+v’
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+variable properties
+(b)
+v’
+θ
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+’
+constant variables
+(c)
+v’
+θ
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+’
+variable variables
+(d)
+u’
+θ
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+’
+constant properties
+(e)
+u’
+θ
+-3
+-2
+-1
+0
+1
+2
+3
+-3
+-2
+-1
+0
+1
+2
+3
+’
+variable properties
+(f)
+Figure 7: Joint probability distribution function at y+ = 5.5.
+12
+
+i/ i
+pdf( i/ i
+)
+-15
+-10
+-5
+0
+5
+10
+15
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+u
+Gaussian
+constant properties
+rms
+θ
+∆
+∆
+∆ rms
+∆
+∆
+∆
+(a)
+i/ i
+pdf( i/ i
+)
+-15
+-10
+-5
+0
+5
+10
+15
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+u
+Gaussian
+variable properties
+rms
+θ
+∆
+∆
+∆ rms
+∆
+∆
+∆
+(b)
+i/ i
+pdf( i/ i
+)
+-15
+-10
+-5
+0
+5
+10
+15
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+u
+Gaussian
+constant properties
+rms
+θ
+∆
+∆
+∆ rms
+∆
+∆
+∆
+(c)
+i/ i
+pdf( i/ i
+)
+-15
+-10
+-5
+0
+5
+10
+15
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+u
+Gaussian
+variable properties
+rms
+θ
+∆
+∆
+∆ rms
+∆
+∆
+∆
+(d)
+Figure 8: Probability distribution of second order structure function for (a), (b) r ≈ 3η; (c), (d) r = 15η. Here r is the
+distance between two points and η is the Kolmogorov scale.
+ϵiθ =
+�
+1 + 1
+Pr
+� �∂θ′+
+∂x+
+k
+∂u′+
+i
+∂x+
+k
+�
+(13)
+Tiθ = − ∂
+∂x+
+k
+�
+u′
+i
++u′
+k
++θ′+�
+(14)
+Πiθ = −
+�
+θ′ ∂p′+
+∂x+
+i
+�
+(15)
+Diθ =
+∂
+∂x+
+k
+��
+θ′+ ∂u′
+i
++
+∂x+
+k
+�
++ 1
+Pr
+�
+u′
+i
++ ∂θ′+
+∂x+
+k
+��
+.
+(16)
+13
+
+The budget terms of the temperature variance are expressed as follows:
+Dk+
+θ
+Dt+ =
+Pθ
+����
+Production
++
+Tθ
+����
+Turbulent diffusion
++
+Dθ
+����
+Molecular diffusion
+−
+ϵθ
+����
+Dissipation
+(17)
+The different terms on the right-hand side are defined as follows:
+Pθ = −
+�
+u′
+k
++θ′+� ∂ ⟨Θ+⟩
+∂x+
+k
++
+�
+u′
+1
++θ′+� ∂ ⟨Tw⟩
+∂x+
+1
+(18)
+Tθ = −1
+2
+∂
+∂x+
+k
+�
+u′
+k
++θ′+2�
+(19)
+Dθ =
+1
+2Pr
+�∂2θ′+2
+∂x+
+k
+2
+�
+.
+(20)
+ϵθ = 1
+Pr
+�∂θ′+
+∂x+
+k
+�2
+(21)
+The effect of the variable properties in the budget terms of the temperature variance and the wall-
+normal heat flux is given in Fig. 9 and Fig. 9(b), respectively. Compared to the DNS study of
+Kawamura et al. [38], the essential features of various budget terms are captured. In particular,
+for the production term of temperature variance and wall-normal heat flux, the peak position is
+appropriately predicted while the peak value is underestimated. In terms of the influence of the
+variable properties on the budgets, Fig. 9(a) and Fig. 9(b) show that only the turbulent diffusion
+term is slightly modified, while the other terms almost remain unaffected, as different budgets
+for variable properties collapse well with the benchmark case.
+14
+
+(a)
+(b)
+Figure 9: Budget terms for (a) temperature variance and (b) wall-normal heat flux.
+4. Conclusions
+In this paper, the influence of temperature-dependent viscosity and thermal conductivity on
+the forced convection in a channel is investigated by the wall-resolved LES. Different from the
+isothermal boundary condition, the present isoflux boundary condition permits the wall tem-
+perature fluctuations, thus induces the variations of fluid property. To highlight the influence of
+variable properties on turbulent flow, which is inversely proportional to the Reynolds number, we
+select Reτ = 180 in this study. Although the variable viscosity results in a lower bulk Reynolds
+number than the benchmark case with constant properties, the two cases are considered to be
+fully turbulent flow.
+15
+
+0.15
+Production
+Gain
+Dissipation
+Turb.Diffusion
+Mole.Diffusion
+LoSS
+Lines: CP
+Symbols: VP
+-0.15
+50
+100
+1500.05
+Production
+口
+Dissipation
+Loss
+Temp. Press-Grad.Corr.
+△
+Turb.Diffusion
+Mole.Diffusion
+Gain
+Lines: CP
+Symbols: VP
+-0.05
+50
+100
+150Using the classic wall scaling, we observed that the variable properties have a trivial ef-
+fect on the mean velocity and temperature profiles. Meanwhile, the Reynolds shear stress and
+wall-normal heat flux, which play a fundamental role in transporting momentum and heat in
+wall-bounded turbulence, remain almost unaffected by the variations of property. However, we
+also found that the turbulence intensities, which are represented by the rms values of velocity
+fluctuations and temperature, are modulated in a perceptible way. Consistent with [32], we also
+noticed the strong correlation between streamwise velocity u and temperature θ, which results in
+the remarkable similarity between ⟨−u′v′⟩ and ⟨−v′θ′⟩ as well as the similarity between u+
+rms and
+θ+
+rms. Our study shows that this conclusion can be extended to the flow with variable properties.
+The PDF profiles of u and θ show a good agreement, however, further inspection of the figure
+reveals that the negatively skewed peak is replaced by a plateau, indicating that the structures
+of low-speed streaks are modified by the variable properties. In addition, the effect of variable
+properties on the burst motions, such as ejections and sweeps, is illustrated. In terms of small-
+scale turbulence and intermittency, we cannot detect any obvious influence of variable properties.
+Finally, we calculated the budgets for the temperature variance and wall-normal heat flux, it is
+seen that only the turbulent diffusion term is slightly affected by the variable properties, whereas
+other terms remain unaffected.
+Acknowledgement
+The authors greatly appreciate the financial support provided by Vinnova under project num-
+ber ”2020-04 529”. Computer time was provided by the Swedish National Infrastructure for
+Computing (SNIC), partially funded by the Swedish Research Council through grant agreement
+no. 2018-05973.
+16
+
+References
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+//www.rs.tus.ac.jp/$\sim$t2lab/db/index.html, 2008.
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+18
+
diff --git a/WNE2T4oBgHgl3EQfYAd6/content/tmp_files/load_file.txt b/WNE2T4oBgHgl3EQfYAd6/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..da6ab33f57f57b65437b94804187b52fafc59c42
--- /dev/null
+++ b/WNE2T4oBgHgl3EQfYAd6/content/tmp_files/load_file.txt
@@ -0,0 +1,947 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf,len=946
+page_content='Effect of temperature-dependent thermophysical properties on  turbulent forced convection under constant heat flux boundary  condition  Himani Garga, Lei Wanga,∗  aDepartment of Energy Sciences, Lund University, Ole R¨omers v¨ag, Lund, 22100, Sweden  Abstract  In this study, we performed highly resolved large-eddy simulations (LES) to investigate the in-  fluence of variable properties on the forced turbulent convection in a channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The constant heat  flux boundary condition permits wall temperature fluctuations and thus induces variations of fluid  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Since the effect of viscosity on the flow exhibits Re−1  τ  scaling, we only considered  Reτ = 180 in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Compared to the flow with constant properties, results indicate  that the variable properties have trivial effects on the mean velocity and temperature profiles,  Reynolds shear stress, wall-normal heat flux, as well as the small-scale turbulence characteris-  tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' However, we also observed that the turbulence intensities, low-speed streaks, burst motions,  and budgets for temperature variance and wall-normal heat flux are modified by the variable  properties in a perceptible way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In addition, we showed that the classic wall scaling is a good  choice for flow with small and moderate variations of fluid properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  Keywords: wall-resolved LES, turbulent convection, variable flow properties  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Introduction  For forced convection in a turbulent pipe or channel flow, the Nusselt number can be ob-  tained, for example, by using the Gnielinski correlation on the basis of constant thermophysical  properties [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' However, for fluids under intensive heating, such as in the concentrating solar  receivers, supercritical water-cooled reactors, heat exchangers in chemical engineering, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=', the  fluid properties are temperature-dependent, which, in turn, alters the flow and temperature fields  and cause a departure of heat transfer rate from the constant-property correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' To account  for the departure, Petukhov [2] presented a theoretical analysis of heat transfer and friction to flu-  ids with variable thermophysical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In this work, the Nusselt number and friction factor,  which are calculated by the constant-property equations, are modified by the viscosity ratios eval-  uated at bulk and wall temperatures, or Nu/Nuconst = (µbulk/µwall)n and f/ fconst = (µbulk/µwall)−m,  where n and m depend on the specific flow conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Since then, further experimental and  theoretical investigations have been carried out to propose different correlations which take the  ∗Corresponding author  Email address: lei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='wang@energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='lth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='se (Lei Wang )  Preprint submitted to Journal of heat and mass transfer  January 11, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='03850v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='flu-dyn]  10 Jan 2023  variable properties into account [3, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' These works, however, mainly focused on the deter-  mination of coefficients n and m, rather than on the characterization of the flow behavior from a  physical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  Among analytical and numerical studies dealing with variable properties problems, Pinarbasi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [7] studied the influence of temperature-dependent viscosity and thermal conductivity on  the velocity and temperature fields in a two-dimensional channel flow with isothermal boundary  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' It was found that variable properties have some impacts on the temperature field  but very scarce effects on the velocity profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Zonta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [8] performed a DNS study of the  forced convection of water in a heated channel in which the viscosity of the fluid is temperature-  dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Compared with the isothermal flow with constant viscosity, they observed that turbu-  lence is promoted on the cold side of the channel where viscosity is higher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' while turbulence is  dampened on the hot side where viscosity is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In addition, they identified the connection be-  tween the variable viscosity and the near-wall turbulence structures that sustain the regeneration  cycle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=', low-speed streaks, bursting events, and quasi-streamwise vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Similarly, using  water as a working fluid, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [9] conducted DNS studies of turbulent boundary layers over  isothermally heated walls in which the effect of viscosity stratification on turbulence and skin  friction was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' They demonstrated that the turbulence intensity and the Reynolds shear  stress were decreased in the heated walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In line with the attenuated turbulence transport, the  skin-friction factor was reduced by up to 26%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Subsequently, the same author, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [10],  found that the Stanton number and the Reynolds analogy factor are augmented by 10% and 44%,  respectively, in the variable viscosity flow due to the modification of the near-wall turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  In strongly anisothermal wall-bounded air flows, the momentum and energy exchange are also  under the influence of variations in temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Contrary to the high-speed compressible flow,  acoustic waves have a negligible impact in anisothermal flows with low Mach numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Serra  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [11] performed thermal large-eddy simulations (LES) in a channel flow asymmetrically  heated from both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' They showed that the impact of temperature-dependent viscosity and  thermal conductivity is less important than the impact of variable density which is caused by  thermal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Toutant and Bataille [12] implemented DNS studies of forced convection in  an anisothermal channel flow with highly variable fluid properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Their data illustrated that the  turbulence intensity and Reynolds shear stress are enhanced on the cold side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' while on the hot  side, the increase of viscosity and thermal expansion are in competition to affect the turbulent  motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' To assess the isolated influence produced by the temperature-dependent viscosity and  thermal conductivity, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [13] performed LES to study the turbulent heat transfer in an  anisothermal channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Their data showed that variation in viscosity and thermal conductiv-  ity leads to the modification of the mean velocity and temperature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' However, the molecular  momentum and heat transport across the channel remain unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Consistent with the results  in [12], they also observed that the turbulence is promoted at the cold side even though variable  properties have negligible effect in the near-wall region y+ < 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  More recently, there has been an increased interest in studying turbulent heat transfer to fluids  at supercritical pressures [14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' These fluids exhibit strong thermophysical property vari-  ations as the pseudo-critical temperature is crossed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Yoo [14] pointed out that the two dominant  mechanisms in modulating turbulence and scalar transport in supercritical fluids are buoyancy  and large variations of properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Peeters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [15] conducted DNS of supercritical fluid in an  annular channel with a hot inner wall and a cold outer wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' It was shown that the turbulence was  attenuated at the hot wall but amplified at the cold wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' It is worth noting that this trend is similar  to the sub-critical fluid in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' However, by analyzing a transport equation for the coherent streak  flank strength, it was found that thermophysical property fluctuations significantly affect streak  2  evolution and the self-regeneration process in near-wall turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Nemati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [16] studied  the effect of thermal boundary conditions on turbulent heat transfer to fluids at supercritical pres-  sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Two different thermal wall boundary conditions were considered: the first corresponds to a  Neumann boundary condition that permits wall temperature fluctuations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' the second corresponds  to a Dirichlet boundary condition that does not allow wall temperature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The results  showed an enhancement of 7% in Nusselt number under the Neumann boundary condition be-  cause large property variations are induced that consequently cause an increase of wall-normal  velocity fluctuations and, thus, an increased overall heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  In this work, we aim to investigate the influence of temperature-dependent viscosity and ther-  mal conductivity on turbulent forced convection under the constant heat flux boundary condition  (or Neumann boundary condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' To this end, we performed highly-resolved LES studies in a  heated channel flow where the air is treated as an incompressible fluid and isolated from gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  The variable viscosity and thermal diffusivity are connected by the Prandtl number, which is kept  constant at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='71 in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' To highlight the viscous effect, we focused on Reτ = 180 because  the viscous term exhibits a Re−1  τ  scaling in our methodological framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Under the constant  heat flux boundary condition, enthalpy (or, the bulk-mean temperature) usually undergoes a lin-  ear increase from the inlet to the exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This makes it hard to implement the periodic boundary  condition in the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Therefore, a modification of temperature is introduced  into the energy equation to accommodate this change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This part is described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Results  and discussions are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Finally, the conclusions are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Numerical methodology  In this study, wall-resolved LES of thermal flow in a Poiseuille turbulent channel has been  conducted within a computational box of Lx = 2πδ, Ly = 2δ, and Lz = πδ, with spanwise and  streamwise periodicity, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The flow is considered fully incompressible, Newto-  nian, and heated with a uniform wall heat flux qw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The streamwise, wall-normal, and spanwise  coordinates are x, y, and z, respectively and the corresponding velocity components are u, v,  and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' No-slip boundary condition is imposed on the walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The temperature is denoted by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  Statistically, time-averaged quantities are denoted by angular brackets, ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='⟩, whereas fluctuating  quantities are denoted by prime, (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=')′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In LES, the governing equations for the conservation of  mass, momentum, and energy are averaged by a spatial filtering procedure, which is denoted by  the symbol˜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' After introducing the subgrid-scale (SGS) model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' the filtered LES equations read  ∂˜ui  ∂xi  = 0  (1)  ∂˜ui  ∂t + ∂˜ui˜u j  ∂x j  = − ∂ ˜p  ∂xi  + ∂  ∂xj  �  (ν + νsgs)  � ∂˜ui  ∂xj  + ∂˜uj  ∂xi  − 2  3δi j  ∂˜uk  ∂xk  ��  + Fδi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  (2)  ∂ ˜T  ∂t +  ∂  � ˜T ˜ui  �  ∂xi  = ∂  ∂xi  ��  α + αsgs  � ∂ ˜T  ∂xi  �  (3)  where ui is the ith component of the velocity vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' T is the temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' p is the fluctuat-  ing kinematic pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' F is the driving pressure gradient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' ν and α are kinematic viscosity and  thermal diffusivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Here, α is connected with ν by Prandtl number which is kept  constant at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Under the constant heat flux boundary condition, the bulk mean temperature  3  Figure 1: Schematic of channel flow configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  grows linearly with respect to the streamwise direction x, which makes it hard to implement peri-  odic boundary conditions in the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In this context, we follow the suggestion  of Kasagi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [17] by introducing a transformed temperature Θ = T − Tw in the energy equa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The great advantage of this method is that Θ is periodic in the streamwise direction [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  Under the fully developed flow, the energy equation is changed to a new energy equation with  an internal heat source which depends on the wall heat flux qw and the local streamwise velocity  u1, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 4,  ∂ ˜Θ  ∂t +  ∂  � ˜Θ˜ui  �  ∂xi  = ∂  ∂xi  ��  α + αsgs  � ∂ ˜Θ  ∂xi  �  + qwuτ  ρcpνw  ˜u1  ⟨U1⟩  (4)  where density ρ, heat capacity cp, viscosity at the wall νw, are considered to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The  boundary condition for the transformed temperature is simply Θ = 0 at y = 0 and 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  To account for the temperature-dependent viscosity, we use a power relation to represent the  dependency of viscosity on the transformed temperature Θ, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 5,  ν = ν (Θ0)  �Θ + Θ0  Θ0  �n  (5)  where the reference viscosity ν(Θ0) = 2 × 10−5 m2/s, and the exponent n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' For the case of  the constant properties (CP), the viscosity is uniform throughout the domain (ν = 2×10−5 m2/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  For the case of the variable properties (VP), the viscosity at the wall is kept at the reference value  at 2 × 10−5 m2/s, while the bulk-mean viscosity νb in the domain is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='4 × 10−5 m2/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' More  detailed simulation parameters are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' For both considered cases, Reτ = 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  However, due to the different bulk velocity and viscosity, the bulk Reynolds numbers Reb for the  CP and VP cases are 5967 and 4912, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Nevertheless, they are considered to be fully  turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  In the simulations, the subgrid viscosity νsgs is calculated by means of the Smagorinsky model  [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' However, the standard Smagorinsky model results in the non-zero value of νsgs at the solid  4  qw = const  28  Yi  ZTable 1: Summary of simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  Case  Ub (m/s)  uτ (m/s)  νb (m2/s)  Reb  Reτ  CP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='0597  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='0036  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='0 × 10−5  5967  180  VP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='0587  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='0036  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='4 × 10−5  4912  180  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This defect can be avoided by introducing a van-Driest damping function [21] into  the length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The turbulent heat flux is modeled with a gradient transport model [22], with  αsgs = νsgs/Prt, where the turbulent Prandtl number is equal to Prt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  We carried out wall-resolved LES with an open-source OpenFOAM solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Herein we limit  ourselves to a summary of the numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The governing equations are discretized on  a collocated grid system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' To avoid spurious pressure-velocity coupling, the Rhie-Chow inter-  polation is adopted [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' A second-order implicit scheme implements time integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The  convective terms for the momentum and energy equations are discretized with a second-order  Gauss limited linear scheme [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The Poisson equation is solved for the pressure following both  the predictor and corrector steps to satisfy the continuity equation at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Once the velocity  field is known, the temperature field is computed as a solution of the energy equation, and finally,  viscosity is updated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  The computational domain contains 102×129×128 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The mesh is considered non-  uniform in the wall-normal direction and uniform in the streamwise and spanwise directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The  grid points in the wall-normal direction are determined by a hyperbolic tangent transformation  [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  yk = H  �  1 + 1  a tanh  �  −1 + 2k − 1  Ny  �  tanh−1(a)  �  , k ∈  �  1,Ny  �  ,  (6)  where Ny = 129 is the number of grid points on the y-axis, and a is an adjustable parameter  depending on the mesh dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In the present case, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='96135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The dimensionless mesh  sizes are ∆+  x = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5, ∆+  z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5, and ∆+  y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='2 at the wall to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='7 at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' These values  are substantially lower than the recommendations by Piomelli & Chasnov [26] for wall-resolved  LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  For the characterization of present simulations, classical wall scaling is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In particular,  the friction velocity, uτ, is obtained by solving the near-wall mean velocity gradient,  uτ =  �  νw  �∂U  ∂y  �  w  ,  (7)  where the subscript w refers to the wall quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The friction temperature, Tτ is defined as:  Tτ =  qw  ρcpuτ  ,  (8)  where qw is the wall heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Moreover, y+ is defined as:  y+ = yuτ  νw  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  (9)  In the present study, the flow remains turbulent for all the considered cases, and the turbulent  Reynolds number, Reτ, is based on the uτ and νw:  Reτ = uτδ  νw  (10)  5  Prior to collecting data, each case has been run for approximately 100 diffusion time units  (δ/uτ) to reach a statistically steady state which is identified by the convergence of the root-  mean-square velocity (u, v, w) and temperature (θ) fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Subsequently, data are collected  over roughly 40 diffusion time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Finally, the mean scalar quantities and various statistical  moments are computed by integrating in time and in the homogeneous x and z directions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=',  the Reynolds average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Results and Discussions  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Validation  Before detailing the effect of temperature-dependent viscosity on the flow properties, we per-  formed a detailed validation against the DNS database available for the channel flow at Reτ = 180  where the fluid properties are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Figure 2(a) shows the profiles for the mean streamwise  velocity U+ and its fluctuations U+  rms, plotted in semi-logarithmic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The DNS results  of Moser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [27] are also plotted for the comparison’s sake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Our simulation results in an  overestimation of the mean velocity in the log region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' General agreement for root-mean-square  fluctuations between present simulation results and DNS is reasonably good;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' however, the peak  value is slightly shifted outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 2(b), the profiles of mean temperature and its variance  are also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The results of Seki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [28] are presented for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' To keep consistent  with [28], we use T +  rms instead of θ+  rms to represent the temperature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Similarly to the  mean flow profiles, a slight overestimation of the results compared to DNS is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Like  U+  rms, the peak value of T +  rms is properly predicted, but the peak position is also shifted outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  It is noted that even finer meshes would reduce the disagreement, but they cannot completely  remove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This relates to the Smagorinsky model, which reproduces the action of small scales  by introducing a subgrid viscosity, but it does not model the backward of the energy cascade  [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Nevertheless, compared to the reference DNS data, the essential features of the flow and  scalar quantities are captured, especially close to the wall, and the maximum discrepancies are  within 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  (a)  (b)  Figure 2: Comparison of (a) mean velocity and turbulent velocity fluctuations and (b) mean temperature and temperature  fluctuations, with DNS [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The results are extracted for the constant properties case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  6  4  22  LES,presentstudy  20  DNS,Moseretal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' (1999)  18  3  16  14  su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  12  +  2  U  10  8  6  4  2  0  0  10°  101  102  +4  20  LES,presentstudy  DNS,Sekietal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' (2008)  3  15  sul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  10  5  0  0  10°  101  1023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Mean flow variables  The profiles of the mean streamwise velocity and temperature plotted against the wall units  are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 3, normalized by uτ and Tτ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 3(a), the impact of  variable viscosity on the profile of the mean streamwise velocity is trivial, and only a slight  difference is seen in the log region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In Fig 3(b), the mean temperature profiles for the two cases  agree reasonably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This is in contrast with the results of Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [10], showing an upward  shift of mean velocity profile and a downward shift of temperature profile in the log region for the  cases of variable viscosity compared to the constant reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The difference between [10] and  our studies can be attributed to the different thermal boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In [10], the isothermal  wall heating with different temperatures (or Dirichlet boundary conditions) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In their  simulations, the higher wall temperature leads to lower viscosity of water and, thus, lower skin  friction, which is associated with the upshift of the mean velocity profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In the present studies,  the Neumann boundary condition results in the same viscosity at the wall for the two concerned  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In particular, for the VP case, the increased viscosity in the fluid does reduce the bulk  Reynolds number in the channel, as shown in Table 1, but it does not change the skin friction  on the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In addition, we observe that the variable viscosity and thermal conductivity do not  modify the mean temperature profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This is also inconsistent with the results in [7], showing that  temperature-dependent properties have some impact on the temperature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' However, in [7],  they also used the isothermal boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Despite the influence of variable properties,  the good agreement of the mean velocity and temperature profiles between CP and VP cases  indicates that the classic wall units are a good choice for scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Of course, this conclusion  may only apply to the fluids having small to moderate dependence on temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' For fluids  undergoing significant variations of physical properties like superfluids, however, scaling that  takes into account the temperature-dependent properties is suggested [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  y  U  10  0  10  1  10  2  5  10  15  20  constant properties  variable properties  +  +  (a)  y  T  10  0  10  1  10  2  5  10  15  20  constant properties  variable properties  +  +  (b)  Figure 3: Comparison of the wall-normal distribution of (a) mean streamwise velocity and (b) temperature in the wall  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The profiles of streamwise velocity and temperature are normalized by uτ and Tτ respectively, for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Statistical properties  Apart from the mean velocity and temperature profiles, it is of interest to investigate the  influence of variable properties on turbulent statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 4(a), we plot the  7  profiles of the Reynolds shear stress, ⟨−u′v′⟩, which are normalized by u2  τ, against the wall unit  y+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In general, the two profiles collapse very well in the region of y+ > 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Close to the wall,  a small deviation of the profile with variable properties from the benchmark case is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In  particular, the property variations slightly shift the peak position outwards but without modifying  its peak value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' A very similar trend is found in the profiles of wall-normal heat flux ⟨−v′θ′⟩,  which are normalized by uτTτ and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The similarity between the wall-normal  heat flux and the Reynolds shear stress displayed in the present study is in line with the DNS  results in [32], indicating a strong correlation between u and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This feature will be visited later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 4(c) illustrates the profiles of streamwise turbulent heat flux, ⟨u′θ′⟩, normalized by uτTτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' It  is observed that the profiles collapse very well in the wall region, y+ < 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Further away from  the wall, the variable properties increase its magnitudes and shift the profile upward from the  benchmark case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' It is worth noting here in wall-bounded turbulent flow, the momentum and heat  are mainly transported via Reynolds shear stress and wall-normal heat flux, which are almost  unaffected by the property variations, as shown in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Thereby, we can reach a  conclusion that the variable properties have a trivial influence on the turbulent transport across  the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  To further inspect the impact of fluid properties on the turbulence intensities, the distributions  of root-mean-square streamwise, wall-normal, and spanwise velocity, as well as temperature fluc-  tuations, are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Here, we observe the striking similarity between the streamwise  velocity fluctuations u+  rms and temperature fluctuations θ+  rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In the near-wall region y+ < 20, the  variable properties have no effect on the streamwise velocity and temperature fluctuations, and  the profiles collapse very well for the CP and VP cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In the outer region y+ > 20, the increased  viscosity and thermal conductivity tend to enhance the magnitudes of u+  rms and θ+  rms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  This is also in contrast with the results of Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In [10], the wall heating results in  the diminished turbulence intensity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' however, if normalized by the friction velocity uτ, a better  agreement was achieved for various u+  rms profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Further inspection of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 5 shows that the  variable properties have some visible impacts on the quantities of v+  rms and w+  rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In particular,  close to the wall region, both vrms and wrms levels are suppressed compared to the benchmark  case, and further away from the wall, the fluctuation level enhances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In other words, v+  rms and  w+  rms can feel the influence of the variable properties throughout the channel, while u+  rms and θ+  rms  are modified only in the outer region y+ > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Probability density functions  Probability density functions (PDFs) of velocity and scalar quantities such as temperature  are fundamental in studying and modeling statistical characteristics of flow and heat transfer  in wall-bounded turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The PDFs of streamwise, wall-normal, and spanwise velocity  components and of temperature θ are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 6 for both constant properties (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  6(a)) and variable properties cases 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' To highlight the dynamic structures in the near-wall  region, the PDF profiles are plotted only at the position y+ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' By inspecting Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 6(a), the first  impression conveyed to us is that PDF profiles of u and θ are in good agreement, indicating that  the two quantities are strongly coupled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Second, the negatively skewed peak of u profile suggests  that low-speed streaks are dominant structures in the near-wall sublayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' From a heat transfer  point of view, the low-speed streaks mainly contain the colder fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In addition, both PDF  profiles for u and θ show a long tail in the positive region, indicating that the flow in the vicinity  of the wall is of intermittent nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' These predictions are in accordance with the experimental  results of Antonia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [33] and the DNS results of Redjem-Saad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Under the  variable properties, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 6(b), the strong coupling between u and θ is not affected,  8  y  < -u’v’>  0  50  100  150  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='9  1  constant properties  variable properties  +  +  (a)  y  < -v’ ’>  0  50  100  150  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='9  1  constant properties  variable properties  +  +  θ  (b)  y  < u’ ’>  0  50  100  150  0  1  2  3  4  5  6  7  8  constant properties  variable properties  +  +  θ  (c)  Figure 4: Comparison of normalized profiles of (a) turbulent shear stress, (b) wall-normal heat flux, and (c) streamwise  heat flux in the wall coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  however, the negatively skewed peak is replaced by a plateau which keeps being negatively  inclined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Regarding the other two velocity components, under constant properties, the PDF  profile of w displays a quasi-Gaussian distribution, while v profile exhibits a higher peak than  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This is due to the fact that approaching towards the wall, v is proportional to y3 while w ∝  y2 [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Under the variable properties, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 6(b), the PDF profiles still keep their  symmetrical shape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' however, the peak value is seen to increase, especially for the w component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  This indicates that the distribution of the spanwise velocity loses its Gaussian nature under the  influence of variable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Joint probability distribution function  The joint PDFs (JPDFs) are very valuable in describing the physics of turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' At  y+ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5, the JPDFs of the streamwise and wall-normal velocity component fluctuations u′ and  v′ are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 7(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 7(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Here, the fluctuating values used  for JPDFs are scaled with respect to the rms value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Under the constant properties, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 7(a)  9  y  U  0  50  100  150  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  3  constant properties  variable properties  +  +  rms  (a)  y  V  0  50  100  150  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='9  1  constant properties  variable properties  +  +  rms  (b)  y  W  0  50  100  150  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='2  constant properties  variable properties  +  +  rms  (c)  y  T  0  50  100  150  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  3  constant properties  variable properties  +  +  rms  (d)  Figure 5: Comparison of normalized profiles of the root-mean-square of the (a) axial velocity fluctuations, (b) wall-  normal velocity, (c) spanwise velocity, and (d) root-mean-square of the temperature fluctuations, in the inner scale units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  demonstrates that the second quadrant (u′ < 0 and v′ > 0, ejections) and fourth quadrant (u′ > 0  and v′ < 0, sweeps) have larger contributions to the Reynolds shear stress and thus to turbulent  energy production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This is in agreement with most existing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' For variable properties, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  7(b) shows that the ejection events are slightly less vigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Similarly, we observe that ejections  (v′ > 0, θ′ < 0) and sweeps (v′ < 0, θ′ > 0) are dominant motions in contributing to the wall-  normal heat flux, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 7(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This corroborates the DNS results by Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  Under the influence of variable properties, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 7(d), this trend is almost unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  As we discussed earlier, the similarity between the Reynolds shear stress < −u′v′ > and wall-  normal heat flux < −v′θ′ > indicates the strong coupling between streamwise velocity u′ and  temperature θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 7(e) does demonstrate such a strong correlation and further supports this  conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Under the influence of variable properties, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 7(f) illustrates that the peak values  of JPDF are somewhat attenuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Nevertheless, the magnitude of < u′θ′ > at this position is  found to be unaffected, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  10  i/i  pdf(i/i  )  4  3  2  1  0  1  2  3  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='9  1  u  v  w  θ  constant properties  rms  rms  (a)  i/i  pdf(i/i  )  4  3  2  1  0  1  2  3  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='9  1  u  v  w  variable properties  rms  rms  θ  (b)  Figure 6: Probability density function at y+ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Distribution of velocity and temperature increments  In order to further verify the strong coupling between streamwise velocity and temperature,  we calculate the PDFs of the increments of u and θ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Here, the increments are  denoted as ∆u = u(x + r) − u(x), and ∆θ = θ(x + r) − θ(x), where r represents the distance  between two given points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Usually, the separation r is located in the inertial sub-scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=',  η < r < δ, where δ is the half-height of the channel, and η is the Kolmogorov scale which is  estimated by the relation η = δ/Re3/4  τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The PDFs of both increments at the position y+ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='5  are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 8, where the separation r = 3η and 15η, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' It is clear for both  constant and variable properties, the distribution of ∆u and ∆θ are in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' For smaller  separation at r = 3η, however, the PDF profiles are characterized by wider tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' This indicates  the small-scale intermittency which has been observed in various turbulent flows [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' A  proper understanding of small-scale turbulence is especially important for practical calculations  such as LES and RANS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' At a larger scale r = 15η, it is noted that the wider tails become  narrower, and the curves approach the Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' If r = δ, for example, we noticed  that the PDFs of increments are essentially indistinguishable from a Gaussian (not shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Budget terms  Concerning thermal flows, the equation for the budgets of the turbulent heat fluxes, ⟨uiθ⟩,  and temperature variance, kθ =  �  θ2�  , are referred to Kawamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' These equations are  transcribed here to facilitate the reading of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  The budget equation for the turbulent heat fluxes, ⟨uiθ⟩, is given by:  D  �  u′  i  +θ′+�  Dt+  =  Piθ  ����  Production  −  ϵiθ  ����  Dissipation  +  Tiθ  ����  Turbulent diffusion  +  Diθ  ����  Molecular diffusion  +  Πiθ  ����  Pressure-temperature gradient correlation  (11)  where D/Dt is the material derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' The different terms on the right-hand side respectively are  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='defined as follows:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='Piθ = −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='u′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='+θ′+� ∂ �Ui+�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='∂x+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='u′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='+u′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
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+page_content='+� ∂ ⟨Θ+⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='∂x+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
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+page_content='u′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='+u′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
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+page_content='+� ∂ ⟨Tw⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='∂x+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
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+page_content='∆ rms  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='∆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
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+page_content='(d)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='Figure 8: Probability distribution of second order structure function for (a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' (b) r ≈ 3η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' (c), (d) r = 15η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Here r is the  distance between two points and η is the Kolmogorov scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  ϵiθ =  �  1 + 1  Pr  � �∂θ′+  ∂x+  k  ∂u′+  i  ∂x+  k  �  (13)  Tiθ = − ∂  ∂x+  k  �  u′  i  +u′  k  +θ′+�  (14)  Πiθ = −  �  θ′ ∂p′+  ∂x+  i  �  (15)  Diθ =  ∂  ∂x+  k  ��  θ′+ ∂u′  i  +  ∂x+  k  �  + 1  Pr  �  u′  i  + ∂θ′+  ∂x+  k  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  (16)  13  The budget terms of the temperature variance are expressed as follows:  Dk+  θ  Dt+ =  Pθ  ����  Production  +  Tθ  ����  Turbulent diffusion  +  Dθ  ����  Molecular diffusion  −  ϵθ  ����  Dissipation  (17)  The different terms on the right-hand side are defined as follows:  Pθ = −  �  u′  k  +θ′+� ∂ ⟨Θ+⟩  ∂x+  k  +  �  u′  1  +θ′+� ∂ ⟨Tw⟩  ∂x+  1  (18)  Tθ = −1  2  ∂  ∂x+  k  �  u′  k  +θ′+2�  (19)  Dθ =  1  2Pr  �∂2θ′+2  ∂x+  k  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  (20)  ϵθ = 1  Pr  �∂θ′+  ∂x+  k  �2  (21)  The effect of the variable properties in the budget terms of the temperature variance and the wall-  normal heat flux is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 9(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Compared to the DNS study of  Kawamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' [38], the essential features of various budget terms are captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In particular,  for the production term of temperature variance and wall-normal heat flux, the peak position is  appropriately predicted while the peak value is underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In terms of the influence of the  variable properties on the budgets, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 9(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' 9(b) show that only the turbulent diffusion  term is slightly modified, while the other terms almost remain unaffected, as different budgets  for variable properties collapse well with the benchmark case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  14  (a)  (b)  Figure 9: Budget terms for (a) temperature variance and (b) wall-normal heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Conclusions  In this paper, the influence of temperature-dependent viscosity and thermal conductivity on  the forced convection in a channel is investigated by the wall-resolved LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Different from the  isothermal boundary condition, the present isoflux boundary condition permits the wall tem-  perature fluctuations, thus induces the variations of fluid property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' To highlight the influence of  variable properties on turbulent flow, which is inversely proportional to the Reynolds number, we  select Reτ = 180 in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Although the variable viscosity results in a lower bulk Reynolds  number than the benchmark case with constant properties, the two cases are considered to be  fully turbulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='15  Production  Gain  Dissipation  Turb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='Diffusion  Mole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='Diffusion  LoSS  Lines: CP  Symbols: VP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='15  50  100  1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='05  Production  口  Dissipation  Loss  Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Press-Grad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  △  Turb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='Diffusion  Mole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='Diffusion  Gain  Lines: CP  Symbols: VP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='05  50  100  150Using the classic wall scaling, we observed that the variable properties have a trivial ef-  fect on the mean velocity and temperature profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Meanwhile, the Reynolds shear stress and  wall-normal heat flux, which play a fundamental role in transporting momentum and heat in  wall-bounded turbulence, remain almost unaffected by the variations of property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' However, we  also found that the turbulence intensities, which are represented by the rms values of velocity  fluctuations and temperature, are modulated in a perceptible way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Consistent with [32], we also  noticed the strong correlation between streamwise velocity u and temperature θ, which results in  the remarkable similarity between ⟨−u′v′⟩ and ⟨−v′θ′⟩ as well as the similarity between u+  rms and  θ+  rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Our study shows that this conclusion can be extended to the flow with variable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  The PDF profiles of u and θ show a good agreement, however, further inspection of the figure  reveals that the negatively skewed peak is replaced by a plateau, indicating that the structures  of low-speed streaks are modified by the variable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In addition, the effect of variable  properties on the burst motions, such as ejections and sweeps, is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' In terms of small-  scale turbulence and intermittency, we cannot detect any obvious influence of variable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  Finally, we calculated the budgets for the temperature variance and wall-normal heat flux, it is  seen that only the turbulent diffusion term is slightly affected by the variable properties, whereas  other terms remain unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content='  Acknowledgement  The authors greatly appreciate the financial support provided by Vinnova under project num-  ber ”2020-04 529”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
+page_content=' Computer time was provided by the Swedish National Infrastructure for  Computing (SNIC), partially funded by the Swedish Research Council through grant agreement  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfYAd6/content/2301.03850v1.pdf'}
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+The E-Posterior
+Peter D. Gr¨unwald∗
+January 5, 2023
+Abstract
+We develop a representation of a decision maker’s uncertainty based on e-variables.
+Like the Bayesian posterior, this e-posterior allows for making predictions against arbi-
+trary loss functions that may not be specified ex ante. Unlike the Bayesian posterior,
+it provides risk bounds that have frequentist validity irrespective of prior adequacy: if
+the e-collection (which plays a role analogous to the Bayesian prior) is chosen badly, the
+bounds get loose rather than wrong, making e-posterior minimax decision rules safer than
+Bayesian ones. The resulting quasi-conditional paradigm is illustrated by re-interpreting
+a previous influential partial Bayes-frequentist unification, Kiefer-Berger-Brown-Wolpert
+conditional frequentist tests, in terms of e-posteriors.
+Accepted for publication in a special issue of the
+Philosophical Transactions of the Royal Society A, doi: 10.1098/rsta.2022.146.
+1
+Introduction and Overview
+It is one of the main attractions of Bayesian inference: one may assess the posterior-expected
+performance of arbitrary decision rules in arbitrary decision tasks, and, as a consequence,
+perform posterior-optimal decisions for any such task. An important special case arises if we
+are given a statistical model P = {Pθ ∶ θ ∈ Θ} equipped with a prior distribution W on Θ,
+a space of actions A and a loss function L ∶ Θ × A → R, and we observe data Y = y, with
+Y ∼ Pθ for some θ ∈ Θ, taking values in some set Y. We call such a combination of P and L a
+decision problem. A decision rule δ is then any function mapping data y ∈ Y to action a ∈ A;
+we write δ(y) = a. We then assess the expected performance of δ given data Y = y as
+Eθ∼W∣y[L(θ,δ(y))],
+(1)
+with W ∣ y denoting the Bayes posterior distribution based on prior W and data Y = y.
+In their classic Bayesian theory (1994), Jos´e Bernardo and Adrian Smith adopt, as they
+make clear already on page 2, a wholehearted subjectivist position on the interpretation of
+probability. Such a subjective Bayesian stance implies that if the prior distribution truly
+describes one’s initial degrees of belief, then the assessment (1) is not just correct for any
+decision rule δ, it also implies optimality of the Bayes decision rule, defined to be any rule
+δBayes such that, for all y ∈ Y,
+¯Rbayes(W ∣ y) ∶= Eθ∼W∣y[L(θ,δBayes(y))] = min
+a∈A Eθ∼W∣y[L(θ,a)].
+(2)
+∗CWI, Amsterdam, and Mathematical Institute, Leiden University. CWI is the National Resarch Institute
+for Mathematics and Computer Science in the Netherlands.
+1
+arXiv:2301.01335v1  [math.ST]  3 Jan 2023
+
+Unfortunately, in practice, for lack of time, computational power or imagination, one often —
+in fact, nearly always — uses pragmatic priors in combination with potentially ‘misspecified’
+(wrong) models. Then the priors do not truly represent one’s beliefs, and as a consequence (1)
+may give a highly misleading impression of δ’s quality in the real world, and using δBayes may
+lead to disastrous results. Much of my own research in the past has concerned misspecified
+models (Gr¨unwald, 1999, Gr¨unwald and van Ommen, 2017). In the present paper, I assume
+the model is correct and concentrate on the case in which ”good” priors (i.e. sufficiently
+sophisticated to justify using (2) in practice) are hard to come by. The pragmatic priors
+that are then used often lead to overconfident posteriors, examples of which abound in the
+literature: decisions based on W ∣ y are then substantially worse in practice than predicted
+by W ∣ y itself via (1) (Oelrich et al., 2020), especially in nonparametrics (Szab´o et al.,
+2015); naturally such assessments can lead to (very) bad decisions (Balch et al., 2019). In
+this paper, we provide the e-posterior as an alternative for such cases: while remaining close
+to the Bayesian ideal, it has a frequentist justification that avoids overconfidence and other
+misleading assessments. As an additional advantage, in contrast to Bayesian methods, e-
+posteriors are readily used in highly nonparametric situations in which a likelihood is not
+easy to define — we defer discussion of this bonus until the discussion Section 5.
+The E-Posterior
+The e-posterior is an analogue to the Bayes posterior for which fre-
+quentist analogues ((5) and (6) below) of Bayesian assessments ((1) and (2) above) hold
+simultaneously under all θ ∈ Θ. It is based on e-values, a far-reaching generalization of like-
+lihood ratios. Crucially, just as with the Bayesian posterior, the same e-posterior can be
+combined with arbitrary decision problems and loss functions. That is, if a given prior W
+truly represents beliefs, the Bayesian (1) and (2) are appropriate simultaneously for arbi-
+trary sets of loss functions {Lb ∶ b ∈ B}, substantiating the common Bayesian claim that the
+Bayesian posterior summarizes all one’s uncertainty about the parameter given the data —
+since all conceivable decisions can be based on it. The same holds for the e-posterior ana-
+logues below: they are simultaneously valid for all loss functions, so that, analogously to the
+Bayesian case, the e-posterior can also be thought of as a single function summarizing one’s
+uncertainty. As discussed in Section 2, this is what sets out the e-posterior from previous
+attempts to provide frequentist guarantees for Bayesian posteriors or frequentist analogues
+thereof such as confidence distributions. There we also explain why we regard validity under
+post-hoc choice of loss function as crucial in statistical testing and estimation in the applied
+sciences.
+Both the Bayesian and the e-representation of uncertainty are relative: whereas the
+Bayesian posterior is defined relative to a prior defined upon a statistical model P, the
+e-posterior is defined relative to a collection S of e-variables, which is itself defined relative
+to P. This ‘e-collection’ S in turn may itself be implicitly based on a prior (Example 2) or
+a set of priors (Section 3.1). Because it gives rise to stochastic upper bounds on risks rather
+than precise risks, we are able to obtain the result that the e-posterior is valid in a frequentist
+sense irrespective of the collection S taken; but the bounds may get better or worse depend-
+ing on S. This makes it considerably more robust than the Bayesian risk assessment (1),
+which, at least under a subjective Bayes interpretation, one can only trust if one truly fully
+believes one’s prior W and one’s model P. It puts the e-posterior approach in the luckiness-
+framework (Gr¨unwald, 2007, Shawe-Taylor and Williamson, 1997, Herbrich and Williamson,
+2002, Gr¨unwald and Mehta, 2019) in which Bayesian belief (one has to believe one’s prior to
+2
+
+trust one’s assessment) is replaced by guarantees and hope that the guarantees are good: if
+the prior is well-aligned with the data, one is lucky and gets tight bounds; if not, the bounds
+one gets are loose but still valid. It goes substantially further though than standard ‘lucki-
+ness’ approaches (such as PAC-Bayesian methods (Gr¨unwald and Mehta, 2019)), since these
+invariably incorporate an a-priori given loss function into the likelihood (e.g. via a Gibbs
+posterior) so that the same (e.g. Gibbs) posterior can only be used for a single loss function.
+Our e-posterior being combinable with arbitrary loss functions, we chose to call the resulting
+type of inference quasi-conditional, a name we explain further in Section 2. Before we start
+with the details, we should point out that, of course, not all can be rosy: the specific type
+of stochastic upper bound that provides our sense of quasi-conditional frequentist validity,
+as given in Proposition 1 and interpreted underneath in terms of a De Finetti-type of game
+between a decision-maker and a bookie, while holding universally, may in some instances
+provide a weaker inference than one would really like to see — we consider it as the main
+open problem of this work to further investigate this novel type of validity.
+Formal Definitions
+Formally, we associate P = {Pθ ∶ θ ∈ Θ} with some corresponding
+collection S of e-variables S = {Sθ ∶ θ ∈ Θ}. We focus on what we call the simple setting, in
+which all parameters in parameter vector θ ∈ Θ ⊆ Rk are of interest. An e-variable Sθ relative
+to any θ ∈ Θ is then simply any nonnegative statistic (i.e. a random variable that can be
+written as a function of the data, Sθ = Sθ(Y )) that satisfies the inequality:
+EY ∼Pθ[Sθ] ≤ 1.
+(3)
+We defer the case of nonparametrics and nuisance parameters, requiring a slightly extended
+definition, to Section 5.
+Over the last three years, interest in e-variables has exploded
+(Gr¨unwald et al., 2019, Shafer, 2021, Vovk and Wang, 2021, Henzi and Ziegel, 2021, Ramdas
+et al., 2022), as tools to extend traditional Neyman-Pearson tests to situations with optional
+stopping and continuation while keeping Type-I error guarantees — see Section 5 for details.
+Here we use them in a novel way: we define the e-posterior based on S and Y to be the
+reciprocal of Sθ:
+¯P(θ ∣ y) ∶=
+1
+Sθ(y),
+(4)
+where by convention we set 1/0 to be ∞; and we define the e-posterior risk assessment of
+decision rule δ based on S by
+e-posterior risk bound: ¯R(δ) ∶= ¯R(Y,δ(Y )) with ¯R(y,a) ∶= sup
+θ∈Θ
+¯P(θ ∣ y) ⋅ L(θ,a),
+(5)
+where we use the convention that ∞ ⋅ a ∶= 0 if a = 0 and ∞ if a > 0.
+¯R(δ) is the random
+variable that maps Y to the corresponding risk bound (5), which is to be interpreted as a
+specific type of stochastic upper bound on the risk that holds under all θ ∈ Θ simultaneously
+— Proposition 1 below makes this precise. It holds for all loss functions L ∶ Θ × A → R
+satisfying the condition that supθ∈Θ L(θ,a) ≥ 0 for each a. We call this the no-sure-gain
+condition. It is satisfied if L ≥ 0, but in general it also allows negative losses (as will be useful
+in Example 6). In contrast to the upper bound-nature of (5), the Bayesian (1) does not just
+bound but precisely gives the conditional risk. Next, in analogy to (2), we suggest, once S
+3
+
+has been fixed, the S-based e-posterior-minimax decision rule as any rule δmm satisfying, for
+all y ∈ Y, any loss function L,
+e-posterior minimax rule: sup
+θ∈Θ
+¯P(θ ∣ y) ⋅ L(θ,δmm(y)) = min
+a∈A sup
+θ∈Θ
+¯P(θ ∣ y) ⋅ L(θ,a).
+(6)
+Contents of this Paper
+Section 2 provides Proposition 1 which establishes the frequentist
+sense in which ¯R(δ) as defined by (5) is valid. We discuss this quasi-conditional validity at
+length and provide extended examples. The paper continues in Section 3 by listing various
+types of e-collections, and investigating the risk assessment and the corresponding e-posterior
+minimax rules that ensue from them when instantiated to simple yet important situations. In
+particular, Section 3.1 considers the Gaussian location family and a weighted squared error
+loss, with data-dependent weights. In this situation (extended to 1-dimensional exponential
+families in the Supplementary Material), Bayesian risk assessment based on standard objec-
+tive Bayes posteriors can fail spectacularly if the weights strongly vary with the data, whereas
+our type of risk assessment via (5) is always valid; moreover, if the weights are constant,
+the assessment for e-posterior minimax rules can match minimax optimal frequentist and
+Bayesian rates up to a small constant factor, thereby alleviating concerns that the sup in (5)
+may make assessments overly pessimistic. In Section 3.2, we consider a Bayesian-frequentist
+unification for hypothesis testing with a simple null that was introduced in an influential
+paper by Berger, Brown and Wolpert (1994) (BBW from now on), building on earlier work
+by Kiefer (1976, 1977), Brownie and Kiefer (1977). BBW’s approach was later extended with
+various collaborators (Berger et al., 1994, Wolpert, 1996, Berger and Guglielmi, 2001, Dass
+and Berger, 2003), culminating in Berger’s (2003) IMS Lecture paper Could Fisher, Neyman
+and Jeffreys have agreed on testing? We show how to re-interpret the BBW conditional error
+probabilities, as well as Kiefer’s earlier conditional confidence estimators (1976, 1977) and
+Vovk’s (1993) extension of it, in terms of (quite special) e-posteriors. Our re-interpretation
+suggests that in practice, we will usually want to use other e-variable collections than the
+BBW one — which is not at all to criticize their pioneering and highly original work, that,
+together with Vovk’s (1993), has served as the main inspiration for this paper. Then, in Sec-
+tion 4, we describe a variation of the e-posterior risk assessment (5) in which the e-posterior
+is replaced by its capped version (see Figure 1) and we investigate when this arguably more
+intuitive form of the e-posterior gives rise to the same assessment. This turns out to be the
+case under Condition Zero, a fundamental condition on the decision rules employed, which
+may or may not be enforceable in any given decision problem. Finally, in Section 5, we give
+more background on the traditional use and interpretation of e-variables and generalize to
+testing with composite nulls, estimating with nuisance parameters and nonparametrics, and
+we provide some concluding discussion. But first, we introduce several running examples of
+e-posteriors, highlighting via (9) the connection with the Bayes posterior.
+1.1
+Examples of E-Collections
+Example 1 [A Simple E-Posterior: Savage-Dickey] A simple way to design e-variables
+is to start with a prior distribution W on Θ and setting pW (y) ∶= ∫ pθ(y)dW(θ) to be the
+Bayes marginal based on W. Here, as in the rest of the paper, we assume that for all θ ∈ Θ,
+Pθ has density pθ relative to some common underlying measure µ. One then defines, for each
+4
+
+θ ∈ Θ,
+Ssd
+θ = pW (y)
+pθ(y)
+;
+¯P sd(θ ∣ y) = pθ(y)
+pW (y) = w(θ ∣ y)
+w(θ)
+(7)
+where the latter equality holds if W has density w and we denote prior/posterior densities
+by w(θ) and w(θ ∣ y), respectively.
+Ssd
+θ
+is an e-variable, since it satisfies EY ∼Pθ[Ssd
+θ ] =
+∫Y pW (y)dy = 1. Pawel et al. (2022) study this e-variable, focusing on an ‘evidential’ in-
+terpretation; it also figures prominently in (Gr¨unwald et al., 2019) and, under the name
+‘prior-to-posterior e-variable’, in (Neiswanger and Ramdas, 2021). We chose our alternative
+name because, in the Bayesian literature, the quantity w(θ ∣ y)/w(θ) is generally known as
+the Savage-Dickey density ratio after the prominent Bayesians L. Savage and J. Dickey. The
+Savage-Dickey e-variables are by no means the only reasonable ones to base an e-posterior
+on, but they are educationally insightful because they give a first interpretation of risk assess-
+ment (5). Namely, using Bayes’ theorem we can trivially but unusually rewrite the standard
+Bayesian uncertainty assessment (1) as follows:
+Eθ∼W∣y[L(θ,δ(y))] = ∫
+pθ(y)
+pW (y) ⋅ L(θ,δ(y))dW(θ) = Eθ∼W [ ¯P sd(θ ∣ y) ⋅ L(θ,δ(y))],
+(8)
+whereas with Y = y, we have
+¯R(δ) =
+max
+θ
+¯P sd(θ ∣ y) ⋅ L(θ,δ(y)).
+(9)
+Replacing prior expectation (8) by maximum (9), the Savage-Dickey e-posterior is always
+more pessimistic than the Bayesian posterior based on the same prior, and might thus not
+look so outlandish to Bayesians any more.
+Example 2 [Savage-Dickey for the Normal Location Family] Let {Pθ ∶ θ ∈ R} repre-
+sent the normal location family, where Pθ with density pθ has mean θ and variance 1. We
+take as prior a normal distribution with mean θ0 and variance ρ2 > 0, and define the precision
+λ ∶= ρ−2. Suppose we observe Y = xn = (x1,...,xn). By standard calculations, the standard
+Bayesian posterior is given by a normal distribution with mean ˘θ = (n/(n+λ))ˆθ+(λ/(n+λ))θ0,
+with ˆθ = ˆθ(y) the MLE (∑n
+i=1 xi)/n, and posterior variance 1/(n + λ), i.e. with density
+w(θ ∣ xn) =
+√
+n + λ
+2π
+⋅ e− (n+λ)(θ−˘θ)2
+2
+(10)
+so that, by (7),
+¯P sd(θ ∣ y) =
+√
+n + λ
+λ
+⋅ e− n+λ
+2 (θ−˘θ)2+ λ
+2 ⋅(θ−θ0)2 =
+√
+n + λ
+λ
+⋅ e− n
+2 (θ−ˆθ)2+ 1
+2 ⋅ nλ
+n+λ⋅(ˆθ−θ0)2
+(11)
+where the latter equality follows by simple calculus when θ0 = 0 and reducing the general
+case to this case by considering translated data x1 − θ0,...,xn − θ0; see Figure 1.
+Example 3 [Two-Point Prior-Based E-Posterior for Normal Location] Fix some
+C > 0 and set n∗ equal to the sample size n (in Example 8 we generalize to and explain cases
+with n∗ ≠ n) and, for each θ, define θ− < θ and θ+ > θ such that
+1
+2(θ − θ+)2 = 1
+2(θ − θ−)2 = C
+n∗ .
+(12)
+5
+
+Then Sθ(y) = (1/2)pθ−(y)+(1/2)pθ+(y)
+pθ(y)
+is an e-variable (just plug it into the definition to check).
+It is not of Savage-Dickey form, since θ− and θ+ depend on θ. In fact it coincides with the
+two-sided version of Johnson’s (2013) uniformly most powerful Bayes factor at level α if we
+were to set C ∶= −log α, which might, according to Johnson, motivate its use for testing
+H0 = {Pθ} against H1 = {Pθ′ ∶ θ′ ∈ Θ,θ′ ≠ θ}. Here we will use it for estimating and not just
+testing, i.e. for general θ ∈ Θ; then its Bayesian interpretation is not so clear, but it will tend
+to give better risk assessments (5) than the Savage-Dickey e-variable above, as we explain in
+Section 3. Both Example 2 and 3 are extended to general regular (Barndorff-Nielsen, 1978)
+1-dimensional exponential families in the Supplementary Material.
+Example 4 [Simple Hypothesis Testing with LR E-Variables] Consider testing a
+simple H0 = {P0} vs. a simple H1 = {P1} (we remark on the composite case in Section 5).
+We can embed this in our setting by taking P = H0 ∪ H1 and as our loss function a Wald-
+Neyman-Pearson-type loss given by A = A = {0,1}, L ∶ {0,1}2 → R+
+0 nonnegative, and
+L(0,1) > 0, L(1,0) > 0.
+We further say that the loss function satisfies Condition Zero,
+henceforth abbreviated to C0, if L(0,0) = L(1,1) = 0. The interpretation is that action 0
+stands for ‘accept the null’ leading to loss 0 if the true state of nature is P0; and 1 stands for
+‘reject the null’, leading to loss 0 if the alternative P1 is true. The use of such loss functions is
+implicit in the Neyman-Pearson testing paradigm, as noted in 1939 by Wald (1939) who stated
+that one may ‘of course’ set L(0,0) = L(1,1) = 0, i.e. impose C0, and because it simplifies
+treatment we will do so in most testing examples in this paper — but in Example 6 we will
+deviate from it, and see that this in a sense makes for the most interesting applications of
+our risk assessments. We associate both P0 and P1 with an e-variable. A natural choice is to
+take the likelihood ratio, S1 = p0(Y )/p1(Y ) and S0 = p1(Y )/p0(Y ), seen to be e-variables by
+noting EY ∼Pθ(pθ′(Y )/pθ(Y )) = ∫ pθ′(y)dy = 1. In fact, they are the ‘prototypical’ e-variables
+for simple-vs.-simple testing (Gr¨unwald et al., 2019). The risk assessment (5) now becomes
+¯R(δ) = max{p1(Y )
+p0(Y )L(1,δ(Y )), p0(Y )
+p1(Y )L(0,δ(Y ))} under C0
+=
+p1(Y )
+p0(Y )L(1,δ(Y ))+p0(Y )
+p1(Y )L(0,δ(Y )).
+(13)
+Example 5 [Simple Testing with BBW E-Variables] Several other e-variables may
+be considered for simple hypothesis testing as well, including the Savage-Dickey e-variables;
+we treat these in Section 3. There we also show that BBW’s conditional error probabilities
+(Berger et al., 1994, Wolpert, 1996, Berger, 2003) can be interpreted as e-posteriors. Formally,
+we obtain a parameterized set of e-posteriors { ¯P bbw
+w
+(θ ∣ y)} for 0 < w < 1. We consider the
+special case w = 1/2 here, deferring the general case and definitions to Section 3.2. As we will
+see there, this e-posterior is quite special: for the e-posterior minimax decision rule δmm we
+get, with ¯R defined relative to the BBW e-posterior, under C0,
+¯R(δmm)
+=
+p1(Y )
+p0(Y ) + p1(Y )L(1,δmm(Y )) +
+p0(Y )
+p0(Y ) + p1(Y )L(0,δmm(Y )).
+(14)
+Thus, it can again be rewritten as a sum rather than a supremum and we see that it is formally
+identical to the Bayes assessment (1) when P0 and P1 are equipped with the uniform prior.
+We thus have a frequentist risk assessment, not requiring a prior assumption on P, that is
+as strong as a Bayesian one, which is optimal under a specific prior assumption. This seems
+almost too good to be true, especially since it is slightly, but uniformly, better than the
+6
+
+Figure 1:
+(left and middle) The brown solid curve is the pure Savage-Dickey e-posterior
+¯P(θ ∣ y) for the normal location family based on sample y = (x1,...,x100) with ˆθ(y) = 1 and
+prior W with λ = 1 and θ0 = 0 as in (11) (it reaches a maximum of about 17). The green curve
+represents the 1/2-dampened version (see (35)) of this e-posterior, by design bounded by 2.
+The dashed brown curve is the capped version ¯P cap(θ ∣ y) (Definition 1) of ¯P(θ ∣ y), equal
+to it whenever the latter is ≤ 1. The blue curve represents the generalized Savage-Dickey
+two-point prior-based e-posterior optimized for n∗ = n = 100 as in Example 3 and 8. The
+red curve is twice the tail area of the objective Bayes posterior w○(θ ∣ y) of Example 7, i.e.
+given by f(θ) = 2W ○(¯θ ≥ θ ∣ y) = 2∫
+∞
+θ
+w○(θ ∣ y)dθ. Viz. the remark underneath Definition 1,
+the set {θ ∶ ¯P(θ ∣ y) ≥ α} is a conservative 2-sided (1 − α)-confidence interval, whereas, viz.
+Example 7, the set {θ ∶ f(θ) ≥ α} is an exact nonconservative 2-sided (1 − α)-confidence
+interval, so that plotting f(θ) provides a better comparison to ¯P(θ ∣ y) than directly plotting
+w○(θ ∣ y). The middle display zooms in on the 0.05 right tail.
+(right) The value of (θ − ˆθ)2 ¯P(θ ∣ y) is depicted for the same four e-posteriors, using the
+same colors. The two-point prior-based e-posterior is optimized to give good assessments
+relative to squared error, its maximum is considerably smaller than for the others; for the
+pure Savage-Dickey prior, the maximum is excessively high.
+alternative (13) — and as we will see in Section 3.2, there is indeed no free lunch: (14) only
+holds for δmm, which for the BBW e-variable can sometimes be a rather bad decision rule,
+whereas (13) holds for arbitrary rules; there are some other prices to pay as well when using
+BBW’s e-variable. In fact, a major motivation for the author’s involvement in e-variables was
+my realization, some fifteen years ago, that when deriving frequentist Type-I error bounds
+for decisions based on Bayes factors/likelihood ratios in simple testing, one ends up with
+the expression p1/p0 as in (13) if one follows the reasoning of classic Bayesian texts such as
+(Edwards et al., 1963, Good, 1991) or Royall’s universal bound (Royall, 1997); whereas using
+BBW’s ideas, one obtains p1/(p1 +p0) as in (14): the small but essential discrepancy between
+the two assessments tripped me up for about 15 years until I finally developed the necessary
+tool to understand it — the much more general e-posterior of which both approaches are
+simply a special case, each with its own (dis)advantages, as discussed in Section 3.2.
+2
+Quasi-Conditional Risk Assessment
+We want risk bounds that are valid, in some precise sense, simultaneously for all loss functions
+of the form L ∶ Θ × A → R that satisfy the no-sure-gain condition (see underneath (5)).
+7
+
+45
+3-
+2
+1-
+0.6
+0.8
+1.0
+12
+1.4
+80.05
+0.04
+0.03
+0.02
+0.01-
+12
+1.3
+1.4
+1.50.15
+0.10
+0.05
+0
+0.6
+0.7
+0.8
+60
+1.0Proposition 1 below shows that, for any choice of e-variable collection S = {Sθ ∶ θ ∈ Θ} and
+any given decision rule δ, we have that ¯R(δ) as in (5) satisfies this purpose.
+Proposition 1 [Validity of E-Risk Assessments] Consider a decision problem as defined
+in the introduction relative to any model {Pθ ∶ θ ∈ Θ} for random variable Y ∈ Y and loss
+function L as above that satisfies the no-sure-gain condition. Let ¯R(δ) be as in (5) relative
+to some e-variable collection S = {Sθ ∶ θ ∈ Θ}. For all θ ∈ Θ we have:
+EY ∼Pθ [L(θ,δ(Y ))
+¯R(δ)
+] ≤ 1,
+(15)
+where we adopt the convention 0/0 ∶= 0. In particular, this implies, for every function b ∶
+Y × U × Θ × A → R+, and for any random variable U taking values in some set U and any
+distribution P ′
+θ for (Y, U) whose marginal on Y coincides with Pθ, that
+E(Y,U)∼P ′
+θ [b(Y,U,θ, δ(Y )) ⋅ L(θ,δ(Y ))] ≤
+sup
+(y,u,θ)∈Y×U×Θ
+b(y,u,θ,δ(y)) ⋅ ¯R(y,δ(y)).
+(16)
+Moreover, for any e-collection S, the risk bound (5) with any minimax decision rule δmm
+as given by (6) dominates (5) when applied with any other decision rule δ: for all y ∈ Y:
+¯R(y,δmm(y)) ≤ ¯R(y,δ(y)).
+Proof: Fix any θ ∈ Θ. By definition of ¯R(y,a), we have for all y ∈ Y that ¯R(y,δ(y)) ≥ ¯P(θ ∣
+y)L(θ,δ(y)) so that
+EY ∼Pθ [L(θ,δ(Y ))
+¯R(δ)
+] ≤ EY ∼Pθ [
+L(θ,δ(Y ))
+S−1
+θ (Y )L(θ,δ(Y ))] = EY ∼Pθ [Sθ] ≤ 1,
+(17)
+noting that the derivation is valid as long as the denominator ¯R(δ) inside the first expectation
+cannot evaluate to 0. In case it can, we split up the expectand into two terms, 1 ¯R(δ)=0 ⋅
+L(θ,δ(Y ))/ ¯R(δ) + 1 ¯R(δ)>0 ⋅ L(θ,δ(Y ))/ ¯R(δ), and use that the first fraction is 0/0 which by
+our convention evaluates to 0; we proceed with the second term as above. This proves (15).
+(16) and domination of δmm is then immediate. 2
+The bounds (15) and (16) (we mostly use the latter) are unusual and for them to be practically
+interesting, they should satisfy several desiderata. In particular:
+1. They should have a clear interpretation. We show that this is the case directly below.
+2. They should exceed standard risk bounds obtainable for standard estimators (actions)
+in standard decision problems by only a small factor, at least if we choose a reasonable
+e-collection S. This is indeed the case in our running examples (see Example 8, for the
+Gaussian location family with standard squared error loss, Example 10 for an extension
+to exponential families, and Example 9, underneath (28) for simple-vs.-simple testing).
+3. There should exist settings in which, just like Bayesian risk assessments, they allow
+one to make more risky decisions conditional on more extreme data. This happens in
+Example 6 below.
+4. There should exist situations in which they give useful bounds whereas a standard
+Bayesian assessment, based on a standard convenience prior, is really highly misleading.
+We give such a case in Example 7.
+8
+
+Bookie Interpretation
+The following interpretation of (16) has a De Finettian flavour to
+it. Suppose a bookie offers decision-maker (DM) to play the following game, really a meta-
+decision problem: DM first specifies a decision-rule δ for the original decision-problem. DM
+then gets paid some amount ℓ > 0; then θ and Y are revealed to both DM and bookie. Then
+bookie presents a collection of numbers {Bθ,a ∶ θ ∈ Θ,a ∈ A}, all assumed > 0. Bookie may
+choose Bθ,a as functions of the data Y and external data, here encoded as U; hence we have
+Bθ,a = b(Y,U, θ,a) for some function b. DM then has to pay back Bθ,δ(Y ) ⋅ L(θ,δ(Y )). If the
+rules are such that (i.e. bookie a priori guarantees that), no matter what Y,U are observed,
+bookie will provide a B such that Bθ,δ(Y ) ≤ ℓ/ ¯R(δ) then DM should accept to play this game,
+for her expected gain is ≥ 0; if this cannot be guaranteed, then DM has no guarantee that the
+expected gain is nonnegative. This directly follows from (16); note there is no need for DM to
+know the details of how B = b(Y,U,θ, δ(Y )) is defined, as long as the promise Bθ,δ(Y ) ≤ ¯R−1(δ)
+always holds.
+In all examples to follow, we consider a simple bookie, by which we mean that the B =
+b(y,u) presented is a single number that can be written as a function of Y and U only, and
+does not vary with θ or a. The effect of this is that, both for a Bayesian DM and for an
+e-posterior minimax DM, knowledge of B does not give any incentive to change the action
+taken: for any given prior W, a decision rule δ is Bayes relative to loss function L iff it is
+Bayes relative to loss function L′(θ,a) ∶= b(y,u) ⋅ L(θ,a); and the same holds for e-posterior
+minimaxity. Bookies who provide a Bθ,a depending on θ and a are certainly worth looking
+at as well, but the analysis is substantially more difficult and will be left for future work.
+Example 6 [Policy Makers vs. the Decision Maker Interpretation] While the bookie
+interpretation may seem rather abstract, here we show that it can translate into practically
+relevant settings in which risk bound (16) has actual decision-theoretic consequences, allowing
+one to reliably make more extreme decisions if the data are more extreme.
+In standard
+hypothesis testing problems, this cannot happen, since there are only two actions, ‘reject’
+and ‘accept’, and then ‘more extreme actions’ do not exist. But in practical testing problems
+there is often an additional factor, measuring the importance of the decision in light of the
+observed data. The following real-world example (in essence taken from Gr¨unwald (2022))
+illustrates: a study about vaccine efficacy (ve) in a pandemic has been set up as a test between
+null hypothesis ve ≤ 30% and alternative ve ≥ 50%. The original plan was to vaccinate all
+people above 60 years of age if the null is rejected.
+But suppose the null actually gets
+rejected with unexpectedly strong evidence for the alternative — there is substantially more
+evidence than policy makers had hoped for — and at the same time the virus’ reproduction
+rate may be much higher than anticipated. Based on the observed data and the changed
+circumstances, the policy makers might now contemplate an alternative action: vaccinate
+every adult, instead of everyone above 60. This would change the losses of all decisions. We
+can model this (in an admittedly highly stylized way) by assuming that the loss incurred is
+really B ⋅ L(θ,a) where B is some positive number depending on how the decision ‘δ(y) = a’
+really gets translated into practical measures. For example, δ(y) = 1 (vaccine effective) may
+translate into ‘vaccinate everyone over the age of V ’. If L(1,1) and L(0,0) are both negative
+(as we allow), then the lower V , the higher the value of B will be. As indicated above, we
+simplify things by assuming that the B = b(y,u) ultimately chosen does not depend on θ or
+the action taken; all losses are multiplied by the same amount. In practice, the employed
+weight B might arise from a government committee (say the ‘policy makers’) that analyzes
+the vaccine study results independently from DM, potentially in a very different way (e.g.
+9
+
+fully Bayesian, or just informally by eyeballing graphs). The more convincing they determine
+the evidence to be, the higher the B they may propose; similarly, B may depend on external
+U — if the question is more pressing then one might want to take more drastic measures so
+the B’s go up. The role of the DM (who should now perhaps be called the ‘final risk checker’)
+is simply to formulate an absolute upper bound ℓ for the risk and tell the policy makers: as
+long as you guarantee policies so that B ⋅ ¯R(Y,δ(Y )) ≤ ℓ, I can guarantee you a risk of no
+more than ℓ (here ℓ should be determined before the data become available; in contrast, the
+details of how B arises need not be known and can be arrived at by ‘vague’ means, as long
+as B is conditionally (given the data) independence of θ and a).
+The Quasi-Conditional Stance
+By stressing the importance of (16), we adopt a philos-
+ophy closely related to Neyman’s (1950) influential inductive behaviour view of statistics —
+we cannot say anything about our performance for an individual study, but we can make
+claims about our expected performance (or, by the law of large numbers and concentration
+inequalities, with high probability, about our average performance in several runs). One can
+say less than under strict Bayesian assumptions, i.e the assumption that one fully believes
+one’s prior: in that case, the assessments one makes will be conditionally correct, and not just
+in the average sense of (16). But there is still a ‘conditional’ aspect to (16), in the informal
+sense that more extreme data (small ¯R(δ)) leads to a decision rule being acceptable at higher
+‘stakes’ (value of B). More formally, upon observing ¯R(δ) = r, a Bayesian who insists on
+risk bounded by ℓ would condition on this event and hence accept importance weight B iff
+B⋅r ≤ ℓ — a simple bookie can present any B he likes and the Bayesian can decide based on r
+whether to accept or not. Our DM can safely accept B if the subtly stronger property holds
+that it is known a priori that the bookie must present a B with B ⋅ ¯R(δ) ≤ ℓ, no matter the
+value ¯R(δ) takes, or equivalently, DM can tell, after observing ¯R(δ) = r, bookie that he must
+present a B ≤ ℓ/r and bookie is guaranteed to obey. Thus, our DM can safely accept higher
+stakes if better bounds are observed and is thus more flexible than a DM in the unconditional
+frequentist Neyman-Pearson setting; but only a subset of the games that are acceptable to
+the conditioning Bayesian is acceptable to the DM — in this sense, she is ‘quasi-conditioning’.
+Such substantial consequences ensuing from subtle differences in the protocol between bookie
+and DM are also found, in a different context, in (Gr¨unwald and Halpern, 2011), and are
+a subject matter of the Vovk-Shafer theory of game-theoretic probability (Shafer and Vovk,
+2019). They motivate the term quasi-conditional, emphasizing that the DM may behave in
+some but not all senses as if conditioning on the data; and that her final performance is
+evaluated in expectation over all possible data, and not conditionally on any given data.
+To our knowledge, this is a new paradigm.
+It is certainly different from the main exist-
+ing attempts to unify Bayesian ideas (represent posterior uncertainty by a data-dependent
+distribution over parameters) and frequentist guarantees — here we think of (a) objective
+Bayes methods with matching priors chosen to get frequentist coverage (Berger et al., 2022);
+(b) confidence and the related fiducial distributions (Schweder and Hjort, 2016); and (c)
+conditionalist frequentist methods such as BBW’s. The example below illustrates this for (a)
+and (b), focusing on the normal location family, for which matching objective Bayes priors,
+standard objective Bayes priors, confidence distribution and fiducial distribution all agree.
+It shows that — despite the matching! — if one were to use them as if they were standard
+posteriors in decision problems with an incentive to play more extreme actions with more ex-
+treme data, they fail to give proper frequentist error bounds. (c) is more subtle: in Section 3.2
+10
+
+we show that BBW and other conditionalist frequentist approaches can be re-interpreted in
+terms of e-posteriors — so they can be used against arbitrary loss functions even though
+there were not designed as such. However, these e-posteriors only give meaningful (< ∞)
+bounds for a small subset of all decision rules (in the BBW case, effectively only for a single
+one (!)), making them highly restricted in practice — whereas other e-collections such as the
+Savage-Dickey and LR ones give meaningful bounds for essentially any decision rule.
+Example 7 [Weighted Squared Error] Take the Gaussian location family with variance
+1 as in Example 2 and 3, where we allow data-dependent stopping times such as ‘τ is the
+smallest n at which xn > 5’), defined in the standard manner (Williams, 1991, Hendriksen
+et al., 2021). Hence, data Y takes the form Y = (τ,Xτ) with τ a stopping time and Xτ a
+vector of τ outcomes. Suppose we aim to find the estimator with smallest mean square error,
+where the importance of the problem at hand can depend on the data via some function
+B = b(Y ) (we will ignore external circumstances U); it is known to DM that B is some
+deterministic function of Y . We thus set
+L(θ,a) = (θ − a)2,Ly(θ,a) ∶= b(y)L(θ,a)
+(18)
+where Ly is the loss function we are interested in in the end, a squared error loss with
+importance weighted by some function b(y). In general, B = b(Y ) may again be determined
+by policy makers who may reconsider, using potentially vague ways, the importance of the
+problem in light of the observed data as in Example 6 above. We can think of any estimator
+˘θ for θ as a decision rule, that, upon observing Y = (n,Xn), outputs ˘θ ∶= ˘θ(n,Xn).
+We first consider how one would go about this problem in an ‘objective Bayes’ man-
+ner. The standard (uniform, improper) ‘objective Bayes’ prior for this family and data xn
+corresponds to the limit for λ ↓ 0 in (10).
+The ensuing posterior W ○ ∣ xn has a normal
+density w○(θ ∣ xn) with mean and median equal to the maximum likelihood estimator (MLE)
+ˆθ(xn) = n−1 ∑n
+i=1 xi and variance 1/n, and it is ‘matching’ (Berger et al., 2022) in the sense that
+1 − α credible posterior intervals (taken symmetrically around the posterior mean) coincide
+exactly, for each fixed n, with 1−α confidence intervals. In this case the objective Bayes pos-
+terior also coincides with the fiducial and the confidence distribution (Schweder and Hjort,
+2016) based on xn. The name ‘confidence distribution’ already warns that this posterior
+should perhaps only be used for ‘confidence’ (statements about confidence intervals) but in
+the fiducial and objective Bayes world, it seems to have been advocated more generally. That
+this is problematic was already exhibited in different settings by (Gr¨unwald, 2018, Example 7)
+and Balch et al. (2019). As is well-known, Bayesian posteriors do not depend on the definition
+of the stopping time (Berger and Wolpert, 1988, Hendriksen et al., 2021); also, they condition
+on Y = (n,Xn) so further conditioning on B = b(Y ) also does not affect them, since it is com-
+pletely determined by Y . Thus we have w○(θ ∣ Y = (n,Xn),B = b0) = w○(θ ∣ xn) (Hendriksen
+et al., 2021) so that the Bayes optimal action based on (18) and W ○(θ ∣ Y = (n,xn),B = b0)
+is simply the MLE, ˆθ(xn) irrespective of the definition of τ, the definition of b and the value
+of b(y). Now, based on this posterior, the loss we think we make upon observing Y = (n,xn)
+is given, as in (2), by
+¯Rbayes(W ○ ∣ n,xn) = E¯θ∼W ○∣xn[Ly(¯θ, ˆθ)] = b(n,xn)
+n
+,
+(19)
+where we used that W ○ ∣ Y is a normal with mean ˆθ and variance 1/n, independently of the
+definition of the stopping time random variable τ. Therefore, if data are in fact sampled from
+11
+
+θ∗, then the average of the losses we expect to make, in several studies, is given by
+E(τ,Xτ)∼Pθ∗[E¯θ∼W ○∣Xτ [LY (¯θ, ˆθ(Xτ))]] = E(τ,Xτ)∼Pθ∗ [b(τ,Xτ)
+τ
+],
+(20)
+whereas the loss we actually make on average is
+¯R∗ ∶= E(τ,Xτ)∼Pθ∗[LY (θ∗, ˆθ(Xτ))] = E(τ,Xτ)∼Pθ∗[b(τ,Xτ) ⋅ (θ∗ − ˆθ(Xτ))2]
+(21)
+We now show that the difference between believed risk (20) and actual (21) can be unbounded,
+i.e. we can get it to be at least k∗, for arbitrarily large k∗ > 0. To this end, simply set
+b(τ,Xτ) ∶= τ and set the stopping time τ to be the smallest n such that (ˆθ(xn)−θ∗)2/n ≥ k∗.
+By the law of the iterated logarithm (Williams, 1991) we know that τ is finite, Pθ∗-almost
+surely, under any θ∗ ∈ Θ. With this definition of b, the believed risk (20) is 1 yet the actual
+risk (21) is at least k∗. By Proposition 1 such a discrepancy is impossible if we base our risk
+assessment on e-posteriors via (5) instead. This will be made concrete in Example 8 and 11
+below.
+3
+A Small Taxonomy of E-Specifications
+Existing papers on e-variables such as Turner et al. (2021), P´erez-Ortiz et al. (2022), Wasser-
+man et al. (2020) consider Pθ that define a random process X1,X2,..., and then they provide
+constructions for an e-variable for sample size n, for each n.
+Thus, they really provide
+e-specifications: a sequence of e-variables S[1]
+θ ,S[2]
+θ ,..., for each θ ∈ Θ, where S[n]
+θ
+is an
+e-variable for data Y = (X1,...,Xn).
+Straightforwardly, upon defining q(Xn) = pθ(Xn)S[n]
+θ
+(Xn), we have ∫ q(xn)dµ(xn) ≤ 1
+where µ is the underlying measure, so q can be thought of as a probability density (if ∫ qdµ < 1,
+we may think of the missing mass being put on an outcome that, under Pθ, has probability
+0). S[n]
+θ
+can then be written as a likelihood ratio q(Xn)/pθ(Xn). Thus, in the simple (no-
+nuisance) parameter case, we can rewrite every e-variable as a likelihood ratio. Importantly
+though, q is allowed to depend on n and θ; the specification S[1]
+θ ,S[2]
+θ ,... may deliver, for
+different n and θ, entirely different q’s. To emphasize this possibility, we will from now on
+make the potential dependence on n and θ explicit, and henceforth write
+S[n]
+θ
+= q[n]
+�θ(Xn)
+pθ(Xn) .
+(22)
+For an important subclass of e-variable specifications, the corresponding q[n]
+�θ = q�θ does not
+depend on n after all (it may still depend on θ). In that case q�θ defines a random process
+for X1,X2,... and the specification (S[n]
+θ
+)n is called an e-process (Ramdas et al., 2022). E-
+processes have a major advantage: for every e-process, for every stopping time τ (e.g. ‘stop
+at the smallest n at which S[n] ≥ 1/α, and set τ to this n’), we have (Ramdas et al., 2022):
+EXτ∼Pθ [S[τ](Xτ)] = EXτ∼Pθ [q�θ(Xτ)
+pθ(Xτ) ] ≤ 1.
+As a result, with e-processes we can engage in optional stopping — under any stopping time
+τ — even the aggressive stopping rule mentioned above or some externally imposed rule,
+12
+
+the details of the definition of which we do not need to know — S[τ]
+θ
+is still an e-variable,
+and decision making based on (5) as above can proceed. With e-specifications that do not
+provide e-processes, one cannot engage in such optional stopping (see the extended discussion
+by (Gr¨unwald et al., 2019, Section 5)). Indeed, in Example 7, which involved a non-constant
+stopping time τ, one only gets valid assessments with e-processes rather than mere e-variables,
+and we shall use these in Example 8 and 11 to provide risk assessments for that setting.
+We will now reconsider the e-collections of all our running examples — as we will see,
+some provide e-processes, others do not.
+3.1
+Generalized Savage-Dickey E-Processes
+We have already seen two instances of e-processes: the (specification, for each n, of) Savage-
+Dickey e-variables Ssd and the likelihood ratio e-variables Slr. Both are instances of what
+we may call generalized Savage-Dickey e-processes (S[n])θ, with
+S[n]
+θ
+= pW�θ(Xn)
+pθ(Xn)
+(23)
+where pW (Xn) = ∫ pθ(Xn)dW(θ) is a Bayes marginal distribution on n. Thus, the e-variable
+numerator is a Bayes marginal, but the prior W one uses is allowed to depend on the param-
+eter θ one is comparing to.
+Example 8 [Gaussian Location, Example 3 Continued] Fix some anticipated sample
+size n∗, representing our best a priori guess of the actual sample size n that we will observe.
+Our aim is to obtain an e-process that is valid for all n, including n ≠ n∗, but is optimized to
+give the best possible risk bound if we happen to observe y = xn with n equal or close to n∗. To
+this end, for each θ ∈ R we define θ− < θ and θ+ > θ to satisfy (12) with C = 1. The construction
+below gives a valid e-process for every C > 0, but C = 1 gives the best bounds for the squared
+error, at the cost of potentially somewhat worse yet of course still valid, bounds for other
+loss functions. Now define, for each n, the e-variable S[n]
+θ
+(y) = (1/2)pθ−(y)+(1/2)pθ+(y)
+pθ(y)
+. Clearly
+the S[n]
+θ
+form a generalized Savage-Dickey e-process, with W � θ the uniform distribution
+on {θ−,θ+}. In the Supplementary Material (Section B, Proposition 3) we show that for the
+corresponding e-posterior ¯P(θ ∣ y), we have that
+min
+a max
+θ∈R (θ − a)2 ¯P(θ ∣ y)
+is achieved by setting a = ˆθ (independently of n and n∗), so that δmm(y) = ˆθ, and we also
+show that (Section A) (5) evaluates to
+max
+θ∈R (θ − ˆθ)2 ¯P(θ ∣ y) = 1
+n ⋅ g(n,n∗) ⋅ 1.45... if n∗=n
+=
+1
+n ⋅ 1.45...
+(24)
+where g(n,n∗) = (n∗
+n ) ⋅ en/n∗−1. Note that g(⋅,n∗) has a minimum at n = n∗ with g(n,n∗) = 1
+and increases very slowly for n/n∗ close to 1. The maximum over θ is, for n = n∗, achieved
+approximately at θ = ˆθ ± 1.46/√n at which ¯P(θ ∣ y) ≈ 0.68, independently of the value of
+n = n∗. S[n]
+θ
+gives an e-process for any n∗ we choose in (12). In the Supplementary Material,
+Section A, we show that (24) still holds up to lower order factors, for general 1-dimensional
+families, as long as they are restricted to a compact subinterval of the parameter space.
+13
+
+Incidentally, this shows that for standard decision problems (e.g.
+n fixed in advance,
+uniform weight function), the e-posterior based risk bounds (5) are not much worse than the
+Bayesian risk assessment and minimax risk bounds for the same model, as long as sufficiently
+‘clever’ e-collections are used: the factor is a modest 1.45 if n can be assumed known in
+advance, and a log n-factor otherwise.
+Example 9 [Hypothesis Testing with Generalized Savage-Dickey and LR] Consider
+a simple-vs.-simple test as in Example 4. From (23) we see that the generalized Savage-Dickey
+e-variable at fixed sample size n, with Y = Xn, and based on priors W � θ with mass function
+wθ, for θ ∈ Θ = {0,1}, is given by
+Sθ(y) = wθ(0)p0(y) + wθ(1)p1(y)
+pθ(y)
+with wθ(1) = 1 − wθ(0).
+(25)
+For degenerate priors with w0(0) = w1(1) = 0 this reduces to the LR e-process of (13), and
+for ‘pure’ (i.e. nongeneralized) Savage-Dickey e-posteriors, we have w0 = w1. Consider Wald-
+Neyman-Pearson loss functions under Condition Zero (C0) as in Example 4. Generalizing
+(13), the e-posterior minimax decision rule δmm upon observing Y = y is then given by
+δmm(y) = a, for a achieving
+min
+a∈{0,1}max{L(0,a) ⋅
+p0(y)
+w0(0)p0(y) + w0(1)p1(y),L(1,a) ⋅
+p1(y)
+w1(0)p0(y) + w1(1)p1(y)}
+C0
+=
+min
+a∈{0,1}(L(0,a) ⋅
+p0(y)
+w0(0)p0(y) + w0(1)p1(y) + L(1,a) ⋅
+p1(y)
+w1(0)p0(y) + w1(1)p1(y)).
+(26)
+It follows that under C0, δmm selects hypothesis 0 (a = 0) iff f (p0(y)
+p1(y)) ≥ L(1,0)
+L(0,1) where f(r) =
+r ⋅ (1−w1)r+w1
+(1−w0)r+w0 with wj = wj(1). Straightforward calculus gives that, irrespective of the choice
+for w0 and w1, f(r) is strictly increasing in r, so that there is a unique point r∗ depending
+on w1 and w0, such that for all r ≥ r∗, δmm(y) = 0 and for all r < r∗, δmm(y) = 1. Further, for
+any pure Savage-Dickey e-collection with w1 = w0, we have f(r) = r, i.e. we select action 1 if
+p1(y) ⋅ (L(1,0))γ ≥ p0(y) ⋅ (L(0,1))γ
+(27)
+for γ = 1. This is equivalent to taking the Bayes decision under a uniform prior. On the other
+hand, for the LR e-collection, δmm(y) selects action 1 if the above holds for γ = 1/2. Finally,
+if the loss is symmetric, L(0,1) = L(1,0), then in fact we select action 1 iff p1(y) > p0(y)
+even for general w1 ≠ w0.
+How about our basic risk assessment (5) for general decision rules, including these min-
+imax ones? While (5) does not give rise to easily interpretable formulae that work for all
+generalized Savage-Dickey e-variables, under C0 we do get intuitive expressions for pure
+Savage-Dickey e-variables with w0 = w1 = 1/2 the uniform prior, and for LR e-collections. For
+the former, we can replace the ‘max’ by a ‘+’ using C0 as in (26) and we get:
+L(0,δ(y)) ⋅
+p0(y)
+1
+2p0(y) + 1
+2p1(y) + L(1,δ(y)) ⋅
+p1(y)
+1
+2p0(y) + 1
+2p1(y) =
+2 ⋅ L(0,δ(y)) ⋅ wu(0 ∣ y) + 2 ⋅ L(1,δ(y)) ⋅ wu(1 ∣ y) = 2 ⋅ Eθ∼Wu∣y [L(θ,δ(y))]
+(28)
+14
+
+where wu(j ∣ y) =
+pj(y)
+p0(y)+p1(y) is the posterior based on uniform prior wu with wu(0) = wu(1) =
+1/2, showing that our bound is always within a factor 2 of the best Bayes-with-uniform-prior
+bound and hence within a factor of less than 2 of the standard frequentist minimax bound
+for this setting, obtainable by a procedure that depends on the specific L of interest. For the
+LR e-variables, we similarly get, refining (13),
+L(0,δ(y)) ⋅ p0(y)
+p1(y) + L(1,δ(y)) ⋅ p1(y)
+p0(y) =
+L(0,δ(y)) ⋅ wu(0 ∣ y)
+wu(1 ∣ y) + L(1,δ(y)) ⋅ wu(1 ∣ y)
+wu(0 ∣ y) = Eθ∼wu∣y [
+1
+wu(δ(y) ∣ y) ⋅ L(θ,δ(y))].
+(29)
+We note that for δ ∶= δmm and the symmetric case L(0,1) = L(1,0), we choose action 1 iff
+p1(y) ≥ p0(y) so that wu(δmm(y) ∣ y) ≥ 1/2: the LR bound (29) dominates the uniform-prior
+Savage-Dickey bound (28), and 1/wu(δmm(y) ∣ y) in (29) is a luckiness factor, bounded by
+2: if we happen to assign high posterior to the selected hypothesis, we are ‘lucky’ and our
+loss assessment will get tighter; still, it is valid irrespective of the value of this posterior. If
+asymmetric losses are observed, 1/wu(δmm(y) ∣ y) can in general be both larger and smaller
+than 2, and the LR risk assessment bound does not dominate the Savage-Dickey one.
+3.2
+Conditionalist-Frequentist E-Specifications — not E-Processes
+We first continue the Berger-Brown-Wolpert (BBW) simple-vs.-simple hypothesis testing
+setting of Example 5, later extending it to more general conditionalist frequentist inference.
+Let V ∶= p1(Y )/p0(Y ). We will assume that (a) V is strictly increasing in Y , which (b)
+has a (Lebesgue) density under P0 and P1. Both assumptions are merely for simplicity; for
+example (a) can be dropped by directly analyzing V instead of Y , i.e. ‘the likelihood ratio of
+the likelihood ratio’.
+Fix some w with 0 < w < 1. Since, for any y∗, the sum of both integrals below is 1, there
+must be a specific y∗ such that
+∫y<y∗((1 − w)p0(y) + wp1(y))dy = w ; ∫y≥y∗((1 − w)p0(y) + wp1(y))dy = 1 − w,
+(30)
+so that, upon setting
+Sbbw(w)
+0
+(y) = 1y≥y∗ (1 − w)p0(y) + wp1(y)
+(1 − w)p0(y)
+;
+Sbbw(w)
+1
+(y) = 1y<y∗ (1 − w)p0(y) + wp1(y)
+wp1(y)
+(31)
+we can verify that EY ∼Pθ[Sbbw(w)
+θ
+] = 1 for θ ∈ {0,1} so that {Sbbw(w)
+θ
+∶ θ ∈ {0,1}} is an e-
+collection. The corresponding e-posteriors only give useful risk assessment bounds (5) in quite
+restricted situations, however. To be sure, the bound (5) holds (as it must by Proposition 1),
+but it must be trivial for all decision rules except one as soon as we assume — as will be the
+case for every Wald-Neyman-Pearson loss, i.e. with L(0,1) > 0,L(1,0) > 0. The reason is that
+¯P bbw(w)(θ = 1 ∣ y) = ∞ if y ≥ y∗ and ¯P bbw(w)(θ = 0 ∣ y) = ∞ if y < y∗: for every loss function
+L and decision rule δ there will be y such that ¯R(y,δ(y)) = ∞, unless L(0,0) ≤ 0,L(1,1) ≤ 0
+and δ is the bbw(w)-decision rule, which satisfies δbbw(w)(y) = 1 iff y ≥ y∗. This is, in fact,
+the only decision rule that BBW themselves study. Note that it must then also be the case
+that the minimax decision rule δmm based on ¯P bbw(w) must coincide with δbbw(w), and thus
+15
+
+does not depend on the loss L of interest. This indicates that, for many loss functions, it
+will not be a very good rule — it is simply the only rule for which the risk assessment (5)
+is nontrivial (and then evaluates to (14)). To be sure, BBW provide a result implying that
+if the loss function L satisfies C0, then we may set w as a function of this loss function so
+that the rule δbbw(w) = δmm is to some extent reasonable after all in the following sense: it
+coincides with the Bayes decision rule δBayes(y) based on prior W with w(1) = w as long as
+y /∈ [y−,y+] for some specific values y− ≤ y+. They call [y−,y+] the no-decision region and
+notice that in practice it is often quite small.
+This support for δbbw(w) notwithstanding, we feel that the pure Savage-Dickey (with
+w = 1/2) and LR e-posteriors should be preferred: at the small price of the slightly larger risk
+assessments (28) and (29), they (a) allow us to use these assessments for general decision rules
+rather than the potentially defective (since independent of b) δbbw(w), and (b) they provide
+e-processes rather than just e-specifications (it is easy to see that the likelihood ratios (22)
+corresponding to BBW e-specifications have q[n]
+θ
+depending on n — BBW e-variables do not
+remain e-variables under optional stopping, generalized Savage-Dickey e-variables do).
+BBW’s Original, Conditionalist-Frequentist Interpretation
+For general p0 and p1
+as above, we may partition the sample space Y into regions {Yα ∶ α ∈ [0,1]} where Yα =
+{y+
+α,y−
+α} are chosen such that P0(Y ≥ y+
+α) = α and P1(Y ≤ y−
+α) = β(α) for some function
+β ∶ [0,1] → [0,1]. We may think of observing Y = y as first observing α so that y ∈ Yα, and
+only then observing Y = y itself. We may then, by construction, after observing α, perform
+a conditional frequentist test with Type-I and Type-II ‘conditional’ error probabilities α and
+β(α), by rejecting θ = 0 if Y = y+
+α and accepting if Y = y−
+α. The BBW approach was originally
+conceived as a specific way to construct such a partition, with the special property that for
+any α we have that y+
+α ≥ y∗,y−
+α ≤ y∗ (so that δmm invariably rejects if y+
+α is observed for some
+α) and
+α =
+(1 − w)p0(y+
+α)
+(1 − w)p0(y+α) + wp1(y+α),
+β(α) =
+wp1(y−
+α)
+(1 − w)p0(y−α) + wp1(y−α).
+These are exactly our BBW e-posteriors ¯P bbw(w)(0 ∣ y+
+α) and ¯P bbw(w)(1 ∣ y−
+α) — BBW in
+fact call them conditional error probabilities, because that is how they derived them, and
+note that they are formally identical to Bayesian posteriors; the interpretation in terms of
+e-posteriors is ours.
+Kiefer and Vovk Confidence Estimators
+Essentially the same analysis as for BBW
+conditional error probabilities also applies to general conditionalist frequentist e-variables,
+based on Kiefer’s and Vovk’s conditional confidence estimators, to which we now turn: every
+such confidence estimator gives a valid e-specification, hence e-posteriors for each n, but in
+general these will not be e-processes, and they can evaluate to infinity, hence only giving
+useful, nontrivial risk assessments for a restricted set of loss functions and decision rules —
+although this set may not be as restricted as in the BBW case, where only a single rule,
+δbbw(w) gave nontrivial risk assessments.
+Consider a partition {Yα ∶ α ∈ [0,1]} of Y together with a function CI(y) that maps
+each y ∈ Y into a subset of Θ. In case Θ is a continuum, we usually take CI(y) to be an
+interval, so that CI(y) = [θL,θR] with endpoints θL and θR determined by y. We say that
+({Yα ∶ α ∈ [0,1]},CI) define a Kiefer-type conditional frequentist confidence set estimator if
+16
+
+for all θ ∈ Θ, all α ∈ [0,1],
+Pθ(θ /∈ CI(Y ) ∣ Y ∈ Yα) ≤ α.
+Such confidence estimators were first considered by Kiefer (1976). If β(α) = α, the BBW
+approach can be reinterpreted as a Kiefer-type confidence estimator: we set Yα as above and
+CI(y) = {δbbw(w)(y)} so that CI(y+
+α) = {1},CI(y−
+α) = {0}.
+(32)
+If β(α) ≠ α then the BBW approach does not give a Kiefer-type confidence estimator. How-
+ever, Vovk (1993) provided a notion of confidence estimator that generalizes both BBW’s
+conditional error probabilities and Kiefer-type confidence estimators: for any function CI(y)
+that maps each y ∈ Y into a subset of Θ combined with any function ˆα(Y ) taking values in
+[0,1], we say that CI and ˆα define a Vovk-type confidence set estimator if for all θ,
+EY ∼Pθ [1θ/∈CI(Y ) ⋅
+1
+ˆα(Y )] ≤ 1.
+(33)
+Vovk (1993) defines confidence set estimators in this partition-free manner, motivated by a
+game-theoretic interpretation. He gives a (straightforward) proof showing that any confidence
+set estimator in the sense of Kiefer is also a confidence set estimator in the sense of Vovk,
+with ˆα(y) set to the α such that y ∈ Yα. Similarly, if we set CI as in (32) and, for each
+α ∈ [0,1], ˆα(y+
+α) ∶= α, ˆα(y−
+α) = β(α), then it is easily seen that (33) hold with equality for
+both θ ∈ {0,1}, i.e. we obtain a Vovk-type confidence estimator.
+Now, if (CI(Y ), ˆα(Y )) is a confidence set estimator in sense of Vovk, it directly follows
+from (33) that for all θ, Sθ = 1θ/∈CI(Y )1/ˆα(Y ) is an e-variable, so ¯P(θ ∣ y) is an e-posterior.
+Just as in the specific BBW scenario, such a posterior may become ∞.
+4
+Condition Zero and Capped Posteriors
+Consider Example 2 again, the normal location family with the squared error loss (18) as
+in Example 7. If we base our risk assessment directly on the Savage-Dickey e-posterior as
+defined there, we get bounds that, while correct, are not always sharp. This is due to the
+fact that, as seen from Figure 1, ¯P(θ ∣ Y ) = S−1
+θ (Y ) can become very high for θ close to ˆθ —
+a milder form of the issues with the confidence-estimator based e-variables of the previous
+section, for which it can become even infinite. There, the issues were avoided if the loss
+function satisfied Condition Zero. It turns out that a weakened version of this condition is
+sufficient to avoid this problem, for decision problems more general than testing. We now
+introduce a new concept, and weaken and generalize Condition Zero with this in mind:
+Definition 1 [Capped E-Posterior; Condition Zero] For any given e-posterior ¯P(θ ∣ y),
+its capped version ¯P cap(θ ∣ y) is defined by setting, for all θ ∈ Θ, y ∈ Y:
+¯P cap(θ ∣ y) ∶=
+min{1, ¯P(θ ∣ y)}. Let δ be a decision rule. We say that δ satisfies C0, i.e. Condition Zero,
+if
+sup
+θ∈Θ
+¯P(θ ∣ y)L(θ,δ(y)) = sup
+θ∈Θ
+¯P cap(θ ∣ y)L(θ,δ(y))
+(34)
+for every y ∈ Y. We say that the loss function L itself satisfies Condition Zero if there exists
+a decision rule δ such that (34) holds.
+17
+
+Our original Condition Zero in simple-vs.-simple testing was defined as a condition on a given
+loss function L; the present condition is strictly weaker as a condition on loss functions and
+generalizes the idea to decision rules δ. Why would we want to impose such a condition?
+First note that setting Sθ = 1 for all θ ∈ Θ gives a trivial, yet valid, e-collection; it provides
+no indication at all about θ based on data y. This suggests that, when using a nontrivial
+collection, based on data y, we cannot make any inferences about θ for which Sθ(y) < 1 (i.e.
+¯P(θ ∣ y) > 1): our e-variable provides less evidence against θ than what can be obtained
+trivially: in a sense, y gives us no information at all for making inferences about θ. It is
+therefore of interest to restrict to decision rules satisfying the new Condition Zero. For such
+problems, our risk assessment will never be based on evaluating ¯P(θ ∣ y) at a value of θ for
+which ¯P(θ ∣ y) > 1, i.e. for which it really does not give any information about this θ. Also,
+it may be argued that the capped e-posterior is a more intuitive tool than the e-posterior:
+for any α ≥ 0, the set {θ ∈ Θ ∶ ¯P(θ ∣ y) ≥ α} is a 1 − α-confidence interval (Gr¨unwald,
+2022) (see Figure 1, middle panel), whereas for sets containing θ with ¯P(θ ∣ y) ≥ 1, no clear
+interpretation seems to exist. Thus, by insisting on C0 one really insists on inferences that
+can be based on the more intuitive ¯P cap while still satisfying (5).
+Example 10 [Two-Point prior for exponential families:
+Condition Zero holds]
+The generalized Savage-Dickey two-point prior-based e-posterior that we employed for the
+Gaussian location family in Example 8 gives rise to e-collections such that the corresponding
+minimax δmm(y) = ˆθ(y), also satisfy Condition Zero whenever the b-weighted squared error
+(18) is used: independently of n, ¯P(θ ∣ y) ≈ 0.68 for the maximizing θ, as was established
+below (24). For exponential families, we show in the Supplementary Material (Section A,
+inside the proof of Theorem 1) that the same holds if the model {Pθ ∶ θ ∈ Θ○} is restricted
+to an (arbitrary) compact subset of the parameter space and n is large enough. We do not
+know if the result can be extended beyond exponential families, since its proof makes heavy
+use of their properties.
+In practice, the DM may be presented with a loss function for which Condition Zero does not
+hold, for any reasonable decision rule, and the high-risk problem looms. What to do in such a
+case? There are two alternative solutions. As to the first (we describe the second underneath
+Example 11), one can (sometimes vastly) improve risk assessments by slightly modifying any
+given e-variable Sθ to another e-variable S′
+θ that provides almost as much evidence against θ
+as Sθ whenever Sθ provides more evidence against y than the trivial 1 (i.e. when Sθ(y) > 1),
+and almost as much as the trivial S = 1 otherwise. The corresponding ¯P ′(θ ∣ y) = S′−1
+θ
+(y) will
+then be similar to ¯P(θ ∣ y) if ¯P(θ ∣ y) ≤ 1 and bounded by a small constant otherwise. Three
+straightforward ways of doing this are provided by setting S′
+θ equal to any of the following
+options:
+S∗
+θ (y) ∶=
+max{Sθ(y),1}
+EY ∼Pθ[max{Sθ(Y ),1}]
+,
+S○
+θ(y) ∶= 1
+2 ⋅ max{1,Sθ(y)}
+S[γ]
+θ (y) ∶= (1 − γ) + γSθ(y) for γ = 1/2.
+(35)
+One immediately verifies that all three options produce e-variables: S[γ] produces one for
+all γ, because mixtures of e-variables are e-variables. Using that the expectation in S∗
+θ is
+bounded by 2 (to see this, upper bound max by a sum), we see that S○ is dominated by both
+S∗
+θ and S[1/2]
+θ
+hence must also be an e-variable and we also see that for all three choices, we
+18
+
+have supθ ¯P ′(θ ∣ y) ≤ 2sup(S○
+θ(y))−1 = 2. S∗
+θ may be the most direct solution, but it can turn
+an e-process into an e-specification that is not an e-process anymore, whereas S[1/2]
+θ
+preserves
+the e-process property. Since S[1/2]
+θ
+dominates S○
+θ, we prefer it; it is depicted in Figure 1.
+Example 11 [Improving Pure Savage-Dickey for Exponential Families] We illus-
+trate the idea above for the case that ¯P(θ ∣ y) = 1/Sθ(y) is the Savage-Dickey e-posterior
+based on an arbitrary prior W on the Gaussian location family. Let ¯P [γ](θ ∣ y) ∶= 1/S[γ]
+θ (y) be
+the corresponding ‘dampened’ e-posterior as defined in (35). In the Supplementary Material,
+Section C we describe this setting in more detail and extend it to 1-dimensional exponential
+families, also showing (in Proposition 3) that for any 0 < γ ≤ 1 the MLE ˆθ is the ¯P [γ]-e-
+posterior minimax estimator irrespective of the weight b in the loss (18). Now for simplicity
+let W be a Gaussian prior with some fixed precision λ as in Example 2 and with prior mean
+θ0 = 0. We also show in the Supplementary Material (Proposition 4), that, for the choice
+γ = 1/2 that we advocated above, the bound (15) and (16) hold with
+¯R(δmm) = ¯R(Y, ˆθ(Y )) = 2
+n (1
+2 log n + λ
+λ
++
+nλ
+n + λ
+ˆθ(Y )2),
+(36)
+losing just a log n factor compared to the e-posterior that achieves the standard frequentist
+minimax bound up to a constant factor, and that we considered in Example 8. Again we
+extend this result to general 1-dimensional families with smooth and strictly positive prior
+densities (and again we do not know if it extends to more general 1-dimensional families). In
+contrast, for the original Savage-Dickey e-posterior ¯P(θ ∣ y) we get a much larger ¯R(δmm) ⪰
+n−1/2, as can be seen by evaluating ⋅ ¯P(θ ∣ y)(θ − ˆθ)2 at θ = ˆθ + n−1/2. By ‘dampening’ the
+posterior we improved the factor of order n−1/2 to a factor of order (log n)n−1.
+As a second solution to the high-risk problem, we can alternatively modify our risk assessment
+(5) for decision problems and rules that do not satisfy C0. We then find that if we replace the
+e-posterior in the risk assessment ¯R(δ) as in (5) by its capped version, the risk assessment itself
+may become invalid (i.e. ¯R(δ) does not satisfy (15)/(16)), but, for arbitrary data-dependent
+decision problems, 2 ¯R(δ) does remain valid:
+Proposition 2 Set, for arbitrary multi-loss decision problem and rule δ, not necessarily
+satisfying C0,
+¯Rcap(Y, A) ∶= sup
+θ∈Θ
+¯P cap(θ ∣ y) ⋅ L(θ,a) ,
+¯Rcap(δ) ∶= ¯Rcap(Y,δ(Y )).
+(37)
+Then (15) and (16) hold with 2 ¯Rcap(δ) in the role of ¯R(δ).
+Proof: Calculating ¯Rcap(y,a) based on the capped posterior is equivalent to calculating
+¯R(y,a) based on the modified posterior ¯P ′(θ ∣ y) = (S○
+θ(y))−1 as defined in (35) and multi-
+plying the answer by 2. Since ¯P ′(θ ∣ y) is a standard e-posterior and no requirements on δ
+are made, we can use Proposition 1 based on ¯P ′(θ ∣ y) to get the required result. 2
+This solution to the high-risk problem of some e-collections shows that, for some collections,
+the original e-posterior in its capped version may not be so bad after all. This solution may in
+fact be preferable, since it does not interfere with the use of e-values in optional continuation:
+we can keep multiplying e-values as explained in the next section, and get a factor of 2 in
+the bound for the final product. In contrast, if we use any of the modifications in (35), we
+must modify the e-variable at each study, making the final (i.e. when we stop adding studies)
+product of e-values potentially much smaller.
+19
+
+5
+Additional Background and Discussion
+The General Definition of E-Values
+The definition of e-collection, e-posterior, risk as-
+sessment (5) and e-posterior minimax decision rule readily generalize to models with nuisance
+parameters, composite hypothesis testing and nonparametric settings. All are covered by the
+following extended definition: let P be a set of distributions for random variable Y and
+consider a parameter (vector) of interest defined by a function δ ∶ P → Θ with Θ ⊂ Rk for
+some k > 0. We now call {Sθ ∶ θ ∈ Θ} an e-collection for P if for all θ ∈ Θ, Sθ = Sθ(Y ) is
+a nonnegative function of Y and for all P ∈ P with δ(P) = θ, we have the corresponding
+analogue of (3), i.e. for all such P:
+EP [Sθ] ≤ 1.
+(38)
+The e-posterior ¯P(θ ∣ y) relative to such a collection is still defined as S−1
+θ (y), and the
+definition of risk assessment (5) and e-posterior minimax rule (6) remains unchanged. It is
+easily checked that the results that validate the e-posterior risk assessment and minimax rule,
+Proposition 1 and Proposition 2, still hold in this more general setting.
+The Original Interpretation of E-Variables
+E-variables, while being implicit in earlier
+work going back at least to Darling and Robbins (1967), were largely unknown until 2019,
+when the concept was given a name and was popularized and extensively developed by papers
+such as (Gr¨unwald et al., 2019, Shafer, 2021, Vovk and Wang, 2021). In the mean time they
+have been studied in a wide variety of contexts (Henzi and Ziegel, 2021, Ren and Barber, 2022,
+Bates et al., 2022) and even had an international workshop devoted to them. Originally, they
+were used as tools to extend traditional Neyman-Pearson tests to situations with optional
+continuation while keeping Type-I error guarantees: suppose one observes a sequence of
+studies concerning the same null hypothesis {P ∈ P ∶ δ(P) = θ0}. One can then multiply the
+e-values derived from the individual studies. Broadly speaking, the product will be a new
+e-variable, irrespective of whether the decision to perform a subsequent study depends on
+previous study outcomes (the ‘continuation rule’), and irrespective of the rule used to decide
+when to stop performing new studies. Since, for any fixed α, the probability, under the null,
+that an e-variable gets larger than 1/α, is bounded by α, one can use e-values, combined by
+multiplication, as an alternative to p-values in a setting in which optional continuation of
+experiments is allowed (note that this is always possible, whereas optional stopping within a
+study is only possible if we deal with e-processes rather than e-variables, as in Section 3).
+Composite Testing
+In light of this original motivation, much initial work on e-values
+focused on hypothesis testing in Neyman-Pearson (NP) style, but under optional continua-
+tion. Just as in the original NP paradigm, Type-I error of decision rules was required to be
+bounded by a pre-given significance level α, and other desiderata such as small Type-II error
+only came second. This is different from the treatment of simple-vs.-simple testing in the
+present paper. Here, as in Bayesian and minimax approaches, we do allow asymmetric losses
+(L(0,1) ≠ L(1,0)), but these potentially different losses appear in the risk assessment (5) in
+a symmetric fashion after all.
+Existing papers on e-values within the original paradigm in fact emphasize composite
+hypotheses: the theoretical papers Gr¨unwald et al. (2019), P´erez-Ortiz et al. (2022) and
+the practical (Turner et al., 2021, Turner and Gr¨unwald, 2022) consider situations in which
+the null has nuisance parameters (such as, for example, the variance in the t-test or the
+20
+
+proportion in 2 × 2 tables).
+For such cases (the null P = {Pλ ∶ λ ∈ Λ} is characterized
+by a (potentially) high-dimensional parameter vector λ ; the parameter of interest δ is of
+(potentially) much smaller dimension, often a scalar), Gr¨unwald et al. (2019) offer a generic
+method to construct e-variables as in (38) for such situations via the reverse information
+projection. This method is employed by all papers mentioned above to construct e-variables,
+with the explicit goal of providing Type-I error bounds under optional continuation in the
+Neyman-Pearson style; but we can equally use them to provide e-posterior risk assessments
+and minimax rules with general loss functions as in this paper — in our examples above,
+we only used ‘simple’ e-collections for ease of illustration. Of particular interest is the paper
+(Gr¨unwald, 2022), which may be viewed as a companion to the present one. It also extends
+the use of e-variables to deal with post-hoc specified loss functions but it sticks to a variation
+of the Neyman-Pearson tenet: there is a distinction between Type-I risks (expected losses),
+on which a hard constraint is imposed (similar to the significance level α as a constraint
+on Type-I error probability in traditional testing), and Type-II risks, which are minimized
+subject to this constraint.
+Nonparametrics
+The same generalization (38) covers nonparametric settings.
+For ex-
+ample, in Cox’s celebrated proportional hazards model, the P ∈ P specify an underlying
+continuous time process with dynamics conditional on some covariate vector ⃗x. While the
+full likelihood of this process is intractable, Ter Schure et al. (2021) show that, for any vector
+θ of covariate coefficients, the ratio between any two partial likelihoods defined relative to the
+same set of covariates constitute an e-process. This implies that all the results on e-posterior
+risk assessment of the present paper also pertain to ratios of Cox’s partial likelihoods. In fact,
+for any P and any θ ∈ Θ, any ratio between conditional, marginal or partial likelihoods that
+has the same distribution under all distributions in P with δ(P) = θ is trivially an e-variable,
+and can be used to define an e-collection. We regard all this as an additional advantage of
+e-posteriors over Bayesian ones: strictly speaking, a Bayesian needs to specify the full likeli-
+hood for each distribution in P in order to get a valid a posterior — as has been argued by
+e.g. Robins and Wasserman (2012), this may be viewed as ‘overkill’ and poses practical (and
+unnecessary) difficulties for Bayesian nonparametric inferences especially when the samples
+are small. In contrast, Waudby-Smith and Ramdas (2022) provides an e-collection which
+can be used to learn the mean of a sequence of bounded random variables based on a small
+sample without any assumptions on the underlying random variables except boundedness
+and existence of a common mean.
+Conclusion and Future Work
+We have presented the idea of representing uncertainty
+by e-posteriors based on e-collections. While e-collections already existed, the idea to use
+them in risk assessments such as (5) and to define e-posterior minimax rules as in (6) is,
+to our knowledge, completely new. There are of course many open questions regarding this
+quasi-conditional methodology. The main one is perhaps: what e-collection should be used in
+practice? If one takes as one’s goal “collecting as much evidence as possible in subsequent
+studies against a single θ ∈ Θ”, and one is able to formulate a prior on Θ representing
+a ‘best guess’ (note that Θ does not necessarily index a set of distributions; rather a set of
+properties) then there is a good case to be made for the GRO (growth-rate optimal) e-variable
+as defined by Gr¨unwald et al. (2019), which is always well-defined; we refer to that paper for
+extensive discussion. For simple nulls, the GRO e-variable is always of pure Savage-Dickey
+form. However, if we are interested not in testing but in estimation and want an e-posterior
+21
+
+over the full range of θ ∈ Θ, then the e-posterior based on the GRO e-variables may not be
+intuitively optimal, as we saw in this paper: the pure Savage-Dickey e-variable based on a
+normal prior W in the Gaussian location family (which is GRO relative to W) gives worse
+risk assessments than the one based on the two-point prior as in Example 3. We do not have
+a general recommendation (in the style of ‘reference priors’ for Bayesians) for e-collections,
+and obviously more research is needed here. This paper is really just a first, exploratory one,
+comparing various possibilities such as two-point, capped- and dampened e-collections. A
+second related important question is: in what situations is the bound (16) sufficiently tight
+to be practically relevant? Essentially, if we are in the setting that (bookie, or policy makers)
+provide B such that B ⋅ ¯R(δ) is constant or varies only very little over y, then the bound
+(15) looks like a standard expected risk bound. If ¯R(δ) can vary highly depending on the
+data, then the bounds (15) and (16) are still valid, but the supremum in (16) may make them
+rather weak. Can we get better bounds in such cases? And what if we let B also vary with θ
+and a? Then, upon learning {Bθ,a ∶ θ ∈ Θ,a ∈ A}, DM might want to re-consider the originally
+planned decision rule δ, significantly complicating the analysis. Related to this is the question
+what loss functions L correspond most tightly to real-world problems at all. In our hypothesis
+testing examples, we considered asymmetric loss functions which seems reasonable, but in our
+estimation example, we used the squared error loss — which is mathematically convenient
+and incentives one to output the posterior mean; but in practical situations one may be more
+interested in very different, hard-to-formalize, problem-dependent losses. All in all, there is
+plenty of room for future work here!
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+
+Supplementary Material
+In this Supplementary Material, in Section A we provide details and proofs concerning the
+two-point prior-based generalized Savage-Dickey e-variables for the Gaussian location family
+and its extension to one-dimensional exponential families. In the final Section C we provide
+details and proofs concerning the ’pure’ Savage-Dickey e-variables for Gaussian location and
+one-dimensional exponential families. In between in Section B we show how the MLE is e-
+posterior minimax both for Gaussian location with two-point prior and for general exponential
+families with pure Savage-Dickey.
+A
+Two-point prior-based E-Collections
+A.1
+A Theorem for One Dimensional Exponential Families
+Here we present Theorem 1, which extends the result (24) for two-point prior-based e-
+collections of Example 8 to 1-dimensional exponential families. More precisely, we let {Pθ ∶
+θ ∈ Θ} be any regular (Barndorff-Nielsen, 1978) 1-dimensional exponential family given in
+any diffeomorphic parameterization (for concreteness, take the mean-value one) and extended
+to n outcomes by independence. We write the KL divergence between two members of the
+family defined on a single outcome as D(θ′∥θ). We denote by ˆθ(y) the MLE based on data
+y = xn, which is known to be unique and equal to the empirical average n−1 ∑n
+i=1 φ(Xi), with
+φ the sufficient statistic, whenever this average lies in Θ, which for regular families is an open
+set. Suppose we observe Y = xn with ˆθ(Y ) ∈ Θ. We now use as our loss function
+L(θ, ˘θ) ∶= 2 ⋅ D(˘θ∥θ),
+which is parameterization independent; but with the mean-value parameterization, in the
+Gaussian location family, 2D(˘θ∥θ) = (θ − ˘θ)2 becomes the squared error loss as in Example 7;
+of course the factor 2 is merely for mathematical convenience in the proofs.
+As we show below (42), for every C > 0, every θ ∈ Θ, there exist θ− < θ < θ+ such that,
+consistent with the Gaussian case,
+D(θ∥θ+) = C
+n∗ , D(θ∥θ−) = C
+n∗ .
+(39)
+As in the Gaussian example, we take C = 1, and we construct, for each n, the e-variables
+S[n]
+θ
+(y) = (1/2)pθ−(y)+(1/2)pθ+(y)
+pθ(y)
+, obtaining again a generalized Savage-Dickey e-process. Based
+on the standard Taylor expansion
+D(θ∗∥θ) = 1
+2I(θ○)(θ∗ − θ)2
+(40)
+where I(⋅) is the Fisher information, which must hold exactly for some θ○ in between θ∗
+and θ, we may expect risk assessments based on this e-variable to behave similarly as in the
+Gaussian case. Indeed, below we first prove the result (24) of the main text for the Gaussian
+case, after which we proceed to prove, heavily using (40), the following extension to regular
+1-dimensional exponential families:
+Theorem 1 Let Θ○ be an arbitrary compact subinterval of Θ. Consider restricted model
+{Pθ ∶ θ ∈ Θ○} and the collection of e-processes {Sθ ∶ θ ∈ Θ○}. Then, for every 0 < c ≤ 1,
+26
+
+there is a constant C∗ such that whenever ˆθ ∈ Θ, and c ≤ n∗/n ≤ 1/c, (24) still holds up to a
+vanishing 1 ± O(n−1/2) factor. That is, there is C∗ such that, for all n, uniformly for all xn
+with ˆθ(xn) ∈ Θ○,
+max
+θ∈Θ○ L(θ, ˆθ(xn)) ⋅ ¯P(θ ∣ xn) = 1
+n ⋅ (n∗
+n ) ⋅ expn/n∗ ⋅fn ⋅ 0.53... if n∗=n
+=
+1
+n ⋅ fn ⋅ 1.45...
+(41)
+where 1 −
+√
+C∗/n ≤ fn ≤ 1 +
+√
+C∗/n.
+Thus, even though now ˆθ may not be precisely equal to the e-posterior minimax estimator
+δmm any more, we may use it as our decision rule anyway, and then still get, for each compact
+sub-model Θ○, risk bounds that, at least for sufficiently large n, are of the same order as
+standard minimax frequentist risk bounds.
+A.2
+Proofs for Example 8, 10 in main text, and Theorem 1 above
+Below we prove all results that were stated in the main text for the Gaussian location family
+and 1-dimensional exponential families. We will freely, without further reference, use standard
+terminology and results concerning exponential families (all to be found in (Barndorff-Nielsen,
+1978)), but we first do highlight two slightly nonstandard results that we will use repeatedly.
+The first (easily proved using steepness of regular exponential families) is that for each fixed
+θ′ ∈ Θ, D(θ′∥θ) is a continuous function of θ ∈ Θ satisfying
+sup
+θ<θ′ D(θ′∥θ) = sup
+θ>θ′ D(θ′∥θ) = ∞.
+(42)
+The second result we need is the KL robustness property (Gr¨unwald, 2007) that holds for all
+regular exponential families: for any fixed y = xn such that ˆθ(y) is well-defined, any θ ∈ Θ
+and any prior W on Θ (W does not need to have a density), we have:
+pθ(y)
+pW (y) = exp(−nD(ˆθ∥θ) + D(P (n)
+ˆθ
+∥P (n)
+W )),
+(43)
+where PW is the Bayes mixture distribution as defined in Example 1, and, for any distribution
+P underlying random process X1,X2,..., we denote the marginal distribution of the first n
+outcomes as P (n).
+Proof of (24) in Example 8 and Claim in Example 10
+Fix C > 0. By (42) above,
+for general regular 1-dimensional exponential families, there exist θ− < θ < θ+ such that (39)
+holds.
+Now define, for i = 1,2,..., xi ∈ Ri, the e-processes Sl
+θ(xi) = pθ−(xi)
+pθ(xi) and Sr
+θ (xi) = pθ+(xi)
+pθ(xi)
+and Sθ ∶= (1/2)Sl
+θ +(1/2)Sr
+θ . While these e-processes provide valid e-variables at each sample
+size i we have that when evaluated at sample size n (on which their definition depends), Sl
+θ
+coincides with the uniformly most powerful Bayes factor for a 1-sided test at sample size n and
+level α∗/2, for any n and α∗ such that C− = −log(α∗/2), for H0 = {Pθ} vs. H1 = {Pθ′ ∶ θ > θ};
+similarly for Slr
+θ
+with C+ and H1 = {Pθ′ ∶ θ′ < θ} respectively (Johnson, 2013).
+We first focus on the Gaussian location family for which D(θ′∥θ) = (1/2)(θ′−θ)2 . ¯P(θ ∣ y)
+can be written by (43) with W a prior putting mass 1/2 on θ+ and 1/2 on θ−. For the Gaussian
+27
+
+location family, (43) then gives:
+¯P(θ ∣ y) =
+pθ(y)
+1
+2pθ−(y) + 1
+2pθ+(y)
+=
+2e−nD(ˆθ∥θ)
+e−nD(ˆθ∥θ−) + e−nD(ˆθ∥θ+)
+=
+2e−(n/2)(ˆθ−θ)2
+e−(n/2)⋅(ˆθ−θ+
+√
+2C/n)2 + e−(n/2)(ˆθ−θ−
+√
+2C/n)2 =
+=
+2eC⋅(n/n∗)
+e−√n⋅(ˆθ−θ)
+√
+2C(n/n∗) + e
+√n⋅(ˆθ−θ)
+√
+2C(n/n∗)
+(44)
+We will assume that, for any observed y, the MLE ˆθ = ˆθ(y) is chosen as action (we will derive
+that this is the e-posterior minimax action later on, in Proposition 3). Thus, to calculate the
+risk assessment bound (5) we need to find the maximum over θ of
+(ˆθ − θ)2 ¯P(θ ∣ y)
+(45)
+We find, using (44), and the fact that the maximum (45) must be achieved at the same θ as
+the maximum of 2Cn(n/n∗)(ˆθ − θ)2 ¯P(θ ∣ y), that the maximizing θn satisfies
+√
+2Cn ⋅ n
+n∗ (θn − ˆθ)2 = x∗ where x∗ maximizes
+x2
+exp(x) + exp(−x),
+(46)
+a function with a unique maximum, found numerically to be achieved at x∗ = ±2.065338138969....
+(45) then becomes,
+1
+n ⋅
+(x∗)2
+2C(n/n∗) ⋅ 2eC(n/n∗)
+e−x∗ + ex∗
+(47)
+We want to find the C giving the best possible bounds for the choice n = n∗, i.e. when the
+anticipated sample size turns out correct. This can be found by minimizing (47) for n = n∗:
+by differentiation we find C = 1, corresponding to the choice in Example 8. We then find
+(still at n = n∗) that θn = ˆθ ± x∗/
+√
+2n ≈ ˆθ ± 1.46/√n. Similarly, ¯P(θ ∣ y) for the maximizing
+θn at n = n∗ is given by 2e1/(ex∗ + e−x∗) = 0.6783206434.... The desired maximum (45),
+at general n, is then given, using (47) with C = 1, by (1/n)(n∗/n)en/n∗x∗2/(ex∗ + e−x∗) =
+(1/n)(n∗/n)en/n∗ ⋅ 0.53222078, which, if n = n∗, evaluates to (1/n) ⋅ 1.446729604... and (24)
+follows.
+Proof of Theorem 1 above and generalized Claim in Example 10
+For simplicity we
+only consider the case with n = n∗ here; the extension to c ≤ n/n∗ ≤ 1/c can then be derived
+analogously to the Gaussian case. We now let Θ represent the mean-value parameterization.
+As in the main text, we set θ+ and θ− as in (39) above with C = 1, which defines ¯P(θ ∣ y). We
+fix an arbitrary compact subinterval Θ○ of Θ. We need to bound, for arbitrary fixed ˆθ ∈ Θ○
+the maximum over θ ∈ Θ○ of
+2 ⋅ D(ˆθ ∣ θ) ¯P(θ ∣ y).
+(48)
+We let β(θ) denote the canonical parameter corresponding to θ.
+Note that, for g(θ∗) =
+D(θ∗∥θ) − D(θ∗∥θ−), we have g′(θ∗) = β(θ−) − β(θ) < 0 constant in θ∗ (here we used that β
+is monotonically increasing in θ). In particular, if ˆθ and θ are both contained in Θ○, then
+g′(ˆθ) ≤ −C− where C− = minθ∈Θ○ β(θ) − β(θ−) > 0, and then, for ˆθ = θ − a, a > 0:
+¯P(θ ∣ y) ≤ 2e−n(D(ˆθ∥θ)−D(ˆθ∥(θ)−)) = 2e−n(D(θ−a∥θ)−D(θ−a∥θ−)
+≤ 2enD(θ∥θ−)−anC∗ = 2eC−anC−.
+28
+
+Analogously, if ˆθ,θ ∈ Θ○, ˆθ = θ + a, a > 0, we can derive ¯P(θ ∣ y) ≤ 2eC−anC+ for C+ =
+minθ∈Θ○ β(θ+) − β(θ) > 0.
+It follows that with C± = min{C−,C+}, there exists another
+constant C′ such that, if θ, ˆθ ∈ Θ○, then the risk assessment at θ satisfies
+2 ⋅ D(ˆθ∥θ) ¯P(θ ∣ y) ≤ bC′a2eC−∣a∣nC±.
+(49)
+Writing ∣a∣ =
+√
+K/n, we see that for large enough K, the right-hand side is smaller than 1/n.
+We thus find that there is a constant K such that, for all θ, ˆθ ∈ Θ○, if ∣ˆθ − θ∣ ≥
+√
+K/n, then
+the risk bound (49), and hence (48), is smaller than 1/n. Without loss of generality we may
+assume K ≥ 1.462 (the need for this condition will become clear later).
+We now fix K′ > 0 and set, for α ∈ [−K′,K′], for ˆθ ∈ Θ○,
+ˆθα ∶= ˆθ + α ⋅
+�
+�
+�
+�
+2
+nI(ˆθ)
+.
+(50)
+where K′ is chosen such that ˆθ−K′ ≤ ˆθ −
+√
+K/n and ˆθK′ ≥ ˆθ +
+√
+K/n for all ˆθ ∈ Θ○. Since I(θ)
+is bounded from below on Θ○ by a positive constant, such a K′ must exist (note we do not
+require that θα ∈ Θ○ for all α ∈ [−K′,K]). We now define Θ′
+ˆθ ∶= Θ○ ∩ {ˆθα ∶ α ∈ [−K′,K]}.
+Our proof strategy for the remainder of the proof is showing that Θ′
+ˆθ is an interval around ˆθ
+that is small enough so that we can re-derive the results for the Gaussian case (up to a factor
+converging to 1) on this set, and then showing that the global maximum of (48) over Θ○ is
+achieved on this set, so that only this set matters.
+We have that ∣I′(θ)/I(θ)∣, where I′ is the derivative if I, is bounded by a constant on Θ○.
+A first-order Taylor approximation of I(θ) then implies that there exist a constant C○ such
+that for all large enough n larger than some n○, for all ˆθ ∈ Θ○, we have
+for all θ,θ○ ∈ Θ′
+ˆθ ∶ 1 −
+√
+C○
+n ≤ I(θ)
+I(θ○) ≤ 1 +
+√
+C○
+n .
+(51)
+We consider, at sample size n > n○ the risk assessment (5), i.e. the maximum over θ ∈ Θ○ of
+(48) that we get for any fixed ˆθ(y) ∈ Θ○. We first consider the maximum over θ ∈ Θ′
+ˆθ.
+Within Θ′
+ˆθ, we can perform essentially the same derivation as in (44), using the Taylor
+approximation (40) in the second step in both numerator and denominator and aggressively
+and repeatedly using (51) above to vary θ in I(θ). This (tediously but straightforwardly)
+gives, for α ∈ [−K′,K′] such that ˆθα ∈ Θ′,
+¯P(ˆθα ∣ y) = 2e1eO(n−1/2) ⋅
+1
+e−2α + e2α = 2e1eO(n−1/2) ⋅
+1
+e−
+√
+(2I(ˆθ)n(θ−ˆθ) + e+
+√
+(2I(ˆθ)n(θ−ˆθ)
+D(ˆθ∥ˆθα) = (1 + O(n−1/2))(ˆθ − ˆθα)2I(ˆθ)
+which holds uniformly for all ˆθ ∈ Θ○ and all α ∈ [−K′,K′] such that ˆθα ∈ Θ′
+ˆθ, i.e. we can pick
+the same constant in O(n−1/2) for all such ˆθ and α. But this means that for all large enough
+n, on the set Θ′, we can repeat the same reasoning as in (46) in the Gaussian case up to a
+factor of order 1 + O(n−1/2): since this factor converges to 0, we must have that, uniformly
+for all ˆθ ∈ Θ, for all n larger than some n0, the maximum will be above (1/n) ⋅ 1.44 and will
+be achieved by one or both elements inside a set Θmax = {ˆθ + γ+
+n/√n, ˆθ − γ−
+n/√n} ∩ Θ′, where
+both γ+
+n and γ−
+n converge to 1.46..., at which point ¯P(θn ∣ y) ≈ 0.68. Note that by requiring
+29
+
+K ≥ 1.462 we made sure K′ is large enough so that both choices for the maximum are in
+{ˆθα ∶ α ∈ [−K′,K]}; at least one of these two choices for θn then also lies in Θ′ for n larger
+than some n0, where n0 can be chosen uniformly, independently of ˆθ.
+Thus, the risk (48) maximized over θ ∈ Θ′
+ˆθ ⊂ Θ○ is bounded from below by 1.44/n, and the
+maximum over Θ○ must be at least as large. But (49) now gives that the maximum of (48)
+over θ ∈ Θ○ ∖Θ′
+ˆθ is at most 1/n which is smaller than the maximum risk within Θ′
+ˆθ. Therefore
+the global maximum is achieved within Θ′
+ˆθ and, just as for the Gaussian location case, is
+given by (1/n)⋅1.45..., which is what we had to prove (the extension to n∗ ≠ n mentioned in
+(41) is now trivial to add; the various ‘for all large enough n’ qualifications we imposed are
+absorbed into C∗)). As a by-product, we also find that, since ¯P(θn ∣ y) ≪ 1, Condition Zero
+holds automatically in this setting.
+B
+Establishing that the MLE is e-posterior minimax
+The proposition below shows, by setting θ′ ∶= ˆθ, that the MLE ˆθ is e-posterior minimax
+both for the Gaussian location family with the two-point prior (Example 8) and for general
+exponential families with a pure Savage-Dickey e-variable with arbitrary prior (Example 11).
+Proposition 3 Consider a regular exponential family given in its mean-value parameter
+space as above. Let θ′ ∈ Θ and let f ∶ R+
+0 → R+
+0 be a continuous function such that limx→∞ xf(x) =
+0. We have
+min
+a max
+θ∈Θ D(a∥θ) ⋅ f(D(θ′∥θ))
+is achieved by a = θ′. In particular,
+1. Suppose that ˆθ = ˆθ(y) ∈ Θ and consider the dampened e-posterior ¯P [γ](θ ∣ y) for any
+0 < γ ≤ 1 and relative to any prior W. It can be written as f(D(ˆθ∥θ)) for a function f
+of the required type.
+2. For the normal location family, the e-posterior based on the two-point discrete prior as
+in (44) can also be written as f(D(ˆθ∥θ)) for a function f of the required type.
+Proof: Let g(a) ∶= maxθ∈Θ D(a∥θ) ⋅ f(D(θ′∥θ)). First consider a = θ′. It follows from (42)
+that the maximum over θ in the definition of g(a) = g(θ′) is achieved by some θ∗
+− < θ′ and
+also by some θ∗
++ > θ′. Now consider a ≠ θ′. We must show that g(a) > g(θ′). If a > θ′, we have
+that D(a∥θ∗
+−) > D(θ′∥θ∗
+−) so
+g(a) ≥ D(a∥θ∗
+−) ⋅ f(D(θ′∥θ∗
+−)) > D(θ′∥θ∗
+−) ⋅ f(D(θ′∥θ∗
+−)) = g(θ′).
+The case a < θ′ goes similarly, with θ∗
++ replacing θ∗
+−. This establishes the first result.
+As to (1), the case with γ = 1 now follows directly from (43). For γ < 1, use the fact that
+1/((1 − γ)x−1 + γ) is increasing in x.
+As to (2): using (44), and once more that D(ˆθ∥θ) = (1/2)(ˆθ − θ)2, and considering sep-
+arately the cases that ˆθ > θ and ˆθ < θ, we find that ¯P(θ ∣ y) = f(D(ˆθ∥θ)), where f is of the
+required form. 2
+30
+
+C
+Pure Savage-Dickey based E-Collections for Gaussian and
+Exponential Families
+Let ¯P [γ](θ ∣ y) be the ‘dampened e-posterior’ as in the main text, based on arbitrary prior
+W on Θ. In Proposition 3 above we showed that for any 0 < γ ≤ 1 the MLE ˆθ is the ¯P [γ]-
+e-posterior minimax estimator irrespective of b. Below we further, via Proposition 4 below,
+show that, for the choice γ = 1/2, the risk assessment bound (5) holds with
+¯R(y, ˆθ(y)) = 2
+nD(P (n)
+ˆθ
+∥P (n)
+W )
+(52)
+for all n such that the expression on the right is larger than 1 — which will be the case for all
+but the smallest n. Here we used the notation D(P (n)
+θ
+∥P (n)
+W ) for the KL divergence between
+distribution Pθ and Bayes marginal PW , both defined on n outcomes. For the normal location
+family with prior with mean 0 and precision λ, we have the exact expression
+D(P (n)
+ˆθ
+∥P (n)
+W ) = 1
+2 log n + λ
+λ
++
+nλ
+n + λ
+ˆθ2
+(53)
+which together with (52) gives (36) in the main text. (53) is found by using the fact that
+D(P (n)
+ˆθ
+∥P (n)
+W ) = ln ¯PW (ˆθ ∣ y), an identity which follows from (43) and holds for general regular
+exponential families. If these have continuous prior w, we get, for ˆθ in any compact subset
+of the parameter space, the expression (Chapter 8 of Gr¨unwald (2007))
+D(P (n)
+ˆθ
+∥P (n)
+W ) = 1
+2 log n
+2π − log w(ˆθ)
+I(ˆθ)1/2 + o(1),
+with I(ˆθ) the Fisher information at ˆθ, allowing us to generalize (36) to general exponential
+families.
+We proceed to derive (52). We start with a proposition that is applicable more generally
+than just for KL loss:
+Proposition 4 Let Θ ⊂ R, b ∈ R+ and let L ∶ Θ×Θ → R+
+0 be a loss function and let ˘θ ∶ Y → Θ
+be an estimator. Fix some y ∈ Y and let ˘θ ∶= ˘θ(y). Consider an e-posterior ¯P(θ ∣ y) with
+¯Psup ∶= supy∈Y,θ∈Θ ¯P(θ ∣ y). and let ¯P ′ be an upper bound on ¯P up to a factor Csup i.e. for all
+θ ∈ Θ, y ∈ Y, ¯P(θ ∣ y) ≤ Csup ¯P ′(θ ∣ y). Fix some θL,θR ∈ Θ (depending on ˘θ) with θL < θR so
+that both:
+1. for all θ ≥ θR, ¯P ′(θ ∣ y) ≤ 1 and ¯P ′(θ ∣ y)L(θ, ˘θ) is decreasing in θ.
+2. for all θ ≤ θL, ¯P ′(θ ∣ y) ≤ 1 and ¯P ′(θ ∣ y)L(θ, ˘θ) is increasing in θ.
+Then our main risk assessment bound (5) holds for ¯P(θ ∣ y) with
+¯R(y, ˘θ(y)) = b(y) ⋅ max{Csup, ¯Psup} ⋅
+max
+θ∈[θL,θR]L(θ, ˘θ).
+(54)
+Proof: For θ ≤ θL, ¯P(θ ∣ y)L(θ, ˘θ) ≤ ¯P ′(θ ∣ y)CsupL(θ, ˘θ) ≤ Csup ¯P ′(θL ∣ y)L(θL, ˘θ) ≤
+CsupL(θL, ˘θ).
+Analogously for θ ≥ θR, we have ¯P(θ ∣ y)L(θ, ˘θ) ≤ CsupL(θR, ˘θ).
+Finally
+for θ ∈ [θL,θR], we have ¯P(θ ∣ y)L(θ, ˘θ) ≤ ¯Psup ⋅ maxθ∈[θL,θR] ⋅L(θ, ˘θ). The result follows. 2
+31
+
+We now use Proposition 4 to show the bound (52). Assume the setting of that bound. We
+set ˘θ to the MLE and L(θ, ˘θ) ∶= D(˘θ∥θ). We first apply Proposition 4 with ¯P ′(θ∣y) ∶= ¯P(θ∣y)
+(and Csup = 1) set to the dampened e-posterior ¯P [1/2] relative to a prior W (independent
+of θ). Because of the dampening with γ = 1/2, we know that ¯Psup ≤ 2. We will apply the
+proposition with θL < θR such that ¯P [1/2](θL ∣ y) = ¯P [1/2](θR ∣ y) = 1. These must exist (use
+(42)), and by (43) they satisfy
+D(ˆθ∥θL) = D(ˆθ∥θR) =
+D(P (n)
+ˆθ
+∥P (n)
+W )
+n
+.
+To verify the conditions of Proposition 4, we will show, using (43), that
+D(ˆθ∥θ)exp(−nD(ˆθ∥θ) + D(P (n)
+ˆθ
+∥P (n)
+W ))
+(55)
+is increasing for θ < θL and decreasing for θ > θR. For this, setting C = D(P (n)
+ˆθ
+∥P (n)
+W ), it is
+sufficient to show that g(u) ∶= uexp(−nu + C) is decreasing if u ≥ D(ˆθL∥θL), i.e. if u ≥ C/n.
+Differentiation gives that g(u) is decreasing if u > 1/n, so Proposition 4 can be applied if
+C ≥ 1 and then (54) gives (52).
+32
+
diff --git a/WtAzT4oBgHgl3EQfYfxK/content/tmp_files/load_file.txt b/WtAzT4oBgHgl3EQfYfxK/content/tmp_files/load_file.txt
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@@ -0,0 +1,1011 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf,len=1010
+page_content='The E-Posterior  Peter D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Gr¨unwald∗  January 5, 2023  Abstract  We develop a representation of a decision maker’s uncertainty based on e-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Like the Bayesian posterior, this e-posterior allows for making predictions against arbi-  trary loss functions that may not be specified ex ante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Unlike the Bayesian posterior,  it provides risk bounds that have frequentist validity irrespective of prior adequacy: if  the e-collection (which plays a role analogous to the Bayesian prior) is chosen badly, the  bounds get loose rather than wrong, making e-posterior minimax decision rules safer than  Bayesian ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The resulting quasi-conditional paradigm is illustrated by re-interpreting  a previous influential partial Bayes-frequentist unification, Kiefer-Berger-Brown-Wolpert  conditional frequentist tests, in terms of e-posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Accepted for publication in a special issue of the  Philosophical Transactions of the Royal Society A, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='1098/rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  1  Introduction and Overview  It is one of the main attractions of Bayesian inference: one may assess the posterior-expected  performance of arbitrary decision rules in arbitrary decision tasks, and, as a consequence,  perform posterior-optimal decisions for any such task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' An important special case arises if we  are given a statistical model P = {Pθ ∶ θ ∈ Θ} equipped with a prior distribution W on Θ,  a space of actions A and a loss function L ∶ Θ × A → R, and we observe data Y = y, with  Y ∼ Pθ for some θ ∈ Θ, taking values in some set Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We call such a combination of P and L a  decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' A decision rule δ is then any function mapping data y ∈ Y to action a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  we write δ(y) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We then assess the expected performance of δ given data Y = y as  Eθ∼W∣y[L(θ,δ(y))],  (1)  with W ∣ y denoting the Bayes posterior distribution based on prior W and data Y = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  In their classic Bayesian theory (1994), Jos´e Bernardo and Adrian Smith adopt, as they  make clear already on page 2, a wholehearted subjectivist position on the interpretation of  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Such a subjective Bayesian stance implies that if the prior distribution truly  describes one’s initial degrees of belief, then the assessment (1) is not just correct for any  decision rule δ, it also implies optimality of the Bayes decision rule, defined to be any rule  δBayes such that, for all y ∈ Y,  ¯Rbayes(W ∣ y) ∶= Eθ∼W∣y[L(θ,δBayes(y))] = min  a∈A Eθ∼W∣y[L(θ,a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (2)  ∗CWI, Amsterdam, and Mathematical Institute, Leiden University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' CWI is the National Resarch Institute  for Mathematics and Computer Science in the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='01335v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='ST]  3 Jan 2023  Unfortunately, in practice, for lack of time, computational power or imagination, one often —  in fact, nearly always — uses pragmatic priors in combination with potentially ‘misspecified’  (wrong) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Then the priors do not truly represent one’s beliefs, and as a consequence (1)  may give a highly misleading impression of δ’s quality in the real world, and using δBayes may  lead to disastrous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Much of my own research in the past has concerned misspecified  models (Gr¨unwald, 1999, Gr¨unwald and van Ommen, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In the present paper, I assume  the model is correct and concentrate on the case in which ”good” priors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' sufficiently  sophisticated to justify using (2) in practice) are hard to come by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The pragmatic priors  that are then used often lead to overconfident posteriors, examples of which abound in the  literature: decisions based on W ∣ y are then substantially worse in practice than predicted  by W ∣ y itself via (1) (Oelrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2020), especially in nonparametrics (Szab´o et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=',  2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' naturally such assessments can lead to (very) bad decisions (Balch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In  this paper, we provide the e-posterior as an alternative for such cases: while remaining close  to the Bayesian ideal, it has a frequentist justification that avoids overconfidence and other  misleading assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' As an additional advantage, in contrast to Bayesian methods, e-  posteriors are readily used in highly nonparametric situations in which a likelihood is not  easy to define — we defer discussion of this bonus until the discussion Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  The E-Posterior  The e-posterior is an analogue to the Bayes posterior for which fre-  quentist analogues ((5) and (6) below) of Bayesian assessments ((1) and (2) above) hold  simultaneously under all θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It is based on e-values, a far-reaching generalization of like-  lihood ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Crucially, just as with the Bayesian posterior, the same e-posterior can be  combined with arbitrary decision problems and loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' That is, if a given prior W  truly represents beliefs, the Bayesian (1) and (2) are appropriate simultaneously for arbi-  trary sets of loss functions {Lb ∶ b ∈ B}, substantiating the common Bayesian claim that the  Bayesian posterior summarizes all one’s uncertainty about the parameter given the data —  since all conceivable decisions can be based on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The same holds for the e-posterior ana-  logues below: they are simultaneously valid for all loss functions, so that, analogously to the  Bayesian case, the e-posterior can also be thought of as a single function summarizing one’s  uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' As discussed in Section 2, this is what sets out the e-posterior from previous  attempts to provide frequentist guarantees for Bayesian posteriors or frequentist analogues  thereof such as confidence distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' There we also explain why we regard validity under  post-hoc choice of loss function as crucial in statistical testing and estimation in the applied  sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Both the Bayesian and the e-representation of uncertainty are relative: whereas the  Bayesian posterior is defined relative to a prior defined upon a statistical model P, the  e-posterior is defined relative to a collection S of e-variables, which is itself defined relative  to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This ‘e-collection’ S in turn may itself be implicitly based on a prior (Example 2) or  a set of priors (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Because it gives rise to stochastic upper bounds on risks rather  than precise risks, we are able to obtain the result that the e-posterior is valid in a frequentist  sense irrespective of the collection S taken;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' but the bounds may get better or worse depend-  ing on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This makes it considerably more robust than the Bayesian risk assessment (1),  which, at least under a subjective Bayes interpretation, one can only trust if one truly fully  believes one’s prior W and one’s model P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It puts the e-posterior approach in the luckiness-  framework (Gr¨unwald, 2007, Shawe-Taylor and Williamson, 1997, Herbrich and Williamson,  2002, Gr¨unwald and Mehta, 2019) in which Bayesian belief (one has to believe one’s prior to  2  trust one’s assessment) is replaced by guarantees and hope that the guarantees are good: if  the prior is well-aligned with the data, one is lucky and gets tight bounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' if not, the bounds  one gets are loose but still valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It goes substantially further though than standard ‘lucki-  ness’ approaches (such as PAC-Bayesian methods (Gr¨unwald and Mehta, 2019)), since these  invariably incorporate an a-priori given loss function into the likelihood (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' via a Gibbs  posterior) so that the same (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Gibbs) posterior can only be used for a single loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Our e-posterior being combinable with arbitrary loss functions, we chose to call the resulting  type of inference quasi-conditional, a name we explain further in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Before we start  with the details,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' we should point out that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' of course,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' not all can be rosy: the specific type  of stochastic upper bound that provides our sense of quasi-conditional frequentist validity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  as given in Proposition 1 and interpreted underneath in terms of a De Finetti-type of game  between a decision-maker and a bookie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' while holding universally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' may in some instances  provide a weaker inference than one would really like to see — we consider it as the main  open problem of this work to further investigate this novel type of validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Formal Definitions  Formally, we associate P = {Pθ ∶ θ ∈ Θ} with some corresponding  collection S of e-variables S = {Sθ ∶ θ ∈ Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We focus on what we call the simple setting, in  which all parameters in parameter vector θ ∈ Θ ⊆ Rk are of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' An e-variable Sθ relative  to any θ ∈ Θ is then simply any nonnegative statistic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' a random variable that can be  written as a function of the data, Sθ = Sθ(Y )) that satisfies the inequality:  EY ∼Pθ[Sθ] ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (3)  We defer the case of nonparametrics and nuisance parameters, requiring a slightly extended  definition, to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Over the last three years, interest in e-variables has exploded  (Gr¨unwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2019, Shafer, 2021, Vovk and Wang, 2021, Henzi and Ziegel, 2021, Ramdas  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2022), as tools to extend traditional Neyman-Pearson tests to situations with optional  stopping and continuation while keeping Type-I error guarantees — see Section 5 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Here we use them in a novel way: we define the e-posterior based on S and Y to be the  reciprocal of Sθ:  ¯P(θ ∣ y) ∶=  1  Sθ(y),  (4)  where by convention we set 1/0 to be ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' and we define the e-posterior risk assessment of  decision rule δ based on S by  e-posterior risk bound: ¯R(δ) ∶= ¯R(Y,δ(Y )) with ¯R(y,a) ∶= sup  θ∈Θ  ¯P(θ ∣ y) ⋅ L(θ,a),  (5)  where we use the convention that ∞ ⋅ a ∶= 0 if a = 0 and ∞ if a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  ¯R(δ) is the random  variable that maps Y to the corresponding risk bound (5), which is to be interpreted as a  specific type of stochastic upper bound on the risk that holds under all θ ∈ Θ simultaneously  — Proposition 1 below makes this precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It holds for all loss functions L ∶ Θ × A → R  satisfying the condition that supθ∈Θ L(θ,a) ≥ 0 for each a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We call this the no-sure-gain  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It is satisfied if L ≥ 0, but in general it also allows negative losses (as will be useful  in Example 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In contrast to the upper bound-nature of (5), the Bayesian (1) does not just  bound but precisely gives the conditional risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Next, in analogy to (2), we suggest, once S  3  has been fixed, the S-based e-posterior-minimax decision rule as any rule δmm satisfying, for  all y ∈ Y, any loss function L,  e-posterior minimax rule: sup  θ∈Θ  ¯P(θ ∣ y) ⋅ L(θ,δmm(y)) = min  a∈A sup  θ∈Θ  ¯P(θ ∣ y) ⋅ L(θ,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (6)  Contents of this Paper  Section 2 provides Proposition 1 which establishes the frequentist  sense in which ¯R(δ) as defined by (5) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We discuss this quasi-conditional validity at  length and provide extended examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The paper continues in Section 3 by listing various  types of e-collections, and investigating the risk assessment and the corresponding e-posterior  minimax rules that ensue from them when instantiated to simple yet important situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In  particular, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='1 considers the Gaussian location family and a weighted squared error  loss, with data-dependent weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In this situation (extended to 1-dimensional exponential  families in the Supplementary Material), Bayesian risk assessment based on standard objec-  tive Bayes posteriors can fail spectacularly if the weights strongly vary with the data, whereas  our type of risk assessment via (5) is always valid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' moreover, if the weights are constant,  the assessment for e-posterior minimax rules can match minimax optimal frequentist and  Bayesian rates up to a small constant factor, thereby alleviating concerns that the sup in (5)  may make assessments overly pessimistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='2, we consider a Bayesian-frequentist  unification for hypothesis testing with a simple null that was introduced in an influential  paper by Berger, Brown and Wolpert (1994) (BBW from now on), building on earlier work  by Kiefer (1976, 1977), Brownie and Kiefer (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' BBW’s approach was later extended with  various collaborators (Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 1994, Wolpert, 1996, Berger and Guglielmi, 2001, Dass  and Berger, 2003), culminating in Berger’s (2003) IMS Lecture paper Could Fisher, Neyman  and Jeffreys have agreed on testing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We show how to re-interpret the BBW conditional error  probabilities, as well as Kiefer’s earlier conditional confidence estimators (1976, 1977) and  Vovk’s (1993) extension of it, in terms of (quite special) e-posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Our re-interpretation  suggests that in practice, we will usually want to use other e-variable collections than the  BBW one — which is not at all to criticize their pioneering and highly original work, that,  together with Vovk’s (1993), has served as the main inspiration for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Then, in Sec-  tion 4, we describe a variation of the e-posterior risk assessment (5) in which the e-posterior  is replaced by its capped version (see Figure 1) and we investigate when this arguably more  intuitive form of the e-posterior gives rise to the same assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This turns out to be the  case under Condition Zero, a fundamental condition on the decision rules employed, which  may or may not be enforceable in any given decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Finally, in Section 5, we give  more background on the traditional use and interpretation of e-variables and generalize to  testing with composite nulls, estimating with nuisance parameters and nonparametrics, and  we provide some concluding discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' But first, we introduce several running examples of  e-posteriors, highlighting via (9) the connection with the Bayes posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='1  Examples of E-Collections  Example 1 [A Simple E-Posterior: Savage-Dickey] A simple way to design e-variables  is to start with a prior distribution W on Θ and setting pW (y) ∶= ∫ pθ(y)dW(θ) to be the  Bayes marginal based on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Here, as in the rest of the paper, we assume that for all θ ∈ Θ,  Pθ has density pθ relative to some common underlying measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' One then defines, for each  4  θ ∈ Θ,  Ssd  θ = pW (y)  pθ(y)  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  ¯P sd(θ ∣ y) = pθ(y)  pW (y) = w(θ ∣ y)  w(θ)  (7)  where the latter equality holds if W has density w and we denote prior/posterior densities  by w(θ) and w(θ ∣ y), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Ssd  θ  is an e-variable, since it satisfies EY ∼Pθ[Ssd  θ ] =  ∫Y pW (y)dy = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Pawel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (2022) study this e-variable, focusing on an ‘evidential’ in-  terpretation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' it also figures prominently in (Gr¨unwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2019) and, under the name  ‘prior-to-posterior e-variable’, in (Neiswanger and Ramdas, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We chose our alternative  name because, in the Bayesian literature, the quantity w(θ ∣ y)/w(θ) is generally known as  the Savage-Dickey density ratio after the prominent Bayesians L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Savage and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Dickey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The  Savage-Dickey e-variables are by no means the only reasonable ones to base an e-posterior  on, but they are educationally insightful because they give a first interpretation of risk assess-  ment (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Namely, using Bayes’ theorem we can trivially but unusually rewrite the standard  Bayesian uncertainty assessment (1) as follows:  Eθ∼W∣y[L(θ,δ(y))] = ∫  pθ(y)  pW (y) ⋅ L(θ,δ(y))dW(θ) = Eθ∼W [ ¯P sd(θ ∣ y) ⋅ L(θ,δ(y))],  (8)  whereas with Y = y, we have  ¯R(δ) =  max  θ  ¯P sd(θ ∣ y) ⋅ L(θ,δ(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (9)  Replacing prior expectation (8) by maximum (9), the Savage-Dickey e-posterior is always  more pessimistic than the Bayesian posterior based on the same prior, and might thus not  look so outlandish to Bayesians any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Example 2 [Savage-Dickey for the Normal Location Family] Let {Pθ ∶ θ ∈ R} repre-  sent the normal location family, where Pθ with density pθ has mean θ and variance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We  take as prior a normal distribution with mean θ0 and variance ρ2 > 0, and define the precision  λ ∶= ρ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Suppose we observe Y = xn = (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=',xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' By standard calculations, the standard  Bayesian posterior is given by a normal distribution with mean ˘θ = (n/(n+λ))ˆθ+(λ/(n+λ))θ0,  with ˆθ = ˆθ(y) the MLE (∑n  i=1 xi)/n, and posterior variance 1/(n + λ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' with density  w(θ ∣ xn) =  √  n + λ  2π  ⋅ e− (n+λ)(θ−˘θ)2  2  (10)  so that, by (7),  ¯P sd(θ ∣ y) =  √  n + λ  λ  ⋅ e− n+λ  2 (θ−˘θ)2+ λ  2 ⋅(θ−θ0)2 =  √  n + λ  λ  ⋅ e− n  2 (θ−ˆθ)2+ 1  2 ⋅ nλ  n+λ⋅(ˆθ−θ0)2  (11)  where the latter equality follows by simple calculus when θ0 = 0 and reducing the general  case to this case by considering translated data x1 − θ0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=',xn − θ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Example 3 [Two-Point Prior-Based E-Posterior for Normal Location] Fix some  C > 0 and set n∗ equal to the sample size n (in Example 8 we generalize to and explain cases  with n∗ ≠ n) and, for each θ, define θ− < θ and θ+ > θ such that  1  2(θ − θ+)2 = 1  2(θ − θ−)2 = C  n∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (12)  5  Then Sθ(y) = (1/2)pθ−(y)+(1/2)pθ+(y)  pθ(y)  is an e-variable (just plug it into the definition to check).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  It is not of Savage-Dickey form, since θ− and θ+ depend on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In fact it coincides with the  two-sided version of Johnson’s (2013) uniformly most powerful Bayes factor at level α if we  were to set C ∶= −log α, which might, according to Johnson, motivate its use for testing  H0 = {Pθ} against H1 = {Pθ′ ∶ θ′ ∈ Θ,θ′ ≠ θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Here we will use it for estimating and not just  testing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' for general θ ∈ Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' then its Bayesian interpretation is not so clear, but it will tend  to give better risk assessments (5) than the Savage-Dickey e-variable above, as we explain in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Both Example 2 and 3 are extended to general regular (Barndorff-Nielsen, 1978)  1-dimensional exponential families in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Example 4 [Simple Hypothesis Testing with LR E-Variables] Consider testing a  simple H0 = {P0} vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' a simple H1 = {P1} (we remark on the composite case in Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We can embed this in our setting by taking P = H0 ∪ H1 and as our loss function a Wald-  Neyman-Pearson-type loss given by A = A = {0,1}, L ∶ {0,1}2 → R+  0 nonnegative, and  L(0,1) > 0, L(1,0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We further say that the loss function satisfies Condition Zero,  henceforth abbreviated to C0, if L(0,0) = L(1,1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The interpretation is that action 0  stands for ‘accept the null’ leading to loss 0 if the true state of nature is P0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' and 1 stands for  ‘reject the null’, leading to loss 0 if the alternative P1 is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The use of such loss functions is  implicit in the Neyman-Pearson testing paradigm, as noted in 1939 by Wald (1939) who stated  that one may ‘of course’ set L(0,0) = L(1,1) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' impose C0, and because it simplifies  treatment we will do so in most testing examples in this paper — but in Example 6 we will  deviate from it, and see that this in a sense makes for the most interesting applications of  our risk assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We associate both P0 and P1 with an e-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' A natural choice is to  take the likelihood ratio, S1 = p0(Y )/p1(Y ) and S0 = p1(Y )/p0(Y ), seen to be e-variables by  noting EY ∼Pθ(pθ′(Y )/pθ(Y )) = ∫ pθ′(y)dy = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In fact, they are the ‘prototypical’ e-variables  for simple-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='-simple testing (Gr¨unwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The risk assessment (5) now becomes  ¯R(δ) = max{p1(Y )  p0(Y )L(1,δ(Y )), p0(Y )  p1(Y )L(0,δ(Y ))} under C0  =  p1(Y )  p0(Y )L(1,δ(Y ))+p0(Y )  p1(Y )L(0,δ(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (13)  Example 5 [Simple Testing with BBW E-Variables] Several other e-variables may  be considered for simple hypothesis testing as well, including the Savage-Dickey e-variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  we treat these in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' There we also show that BBW’s conditional error probabilities  (Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 1994, Wolpert, 1996, Berger, 2003) can be interpreted as e-posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Formally,  we obtain a parameterized set of e-posteriors { ¯P bbw  w  (θ ∣ y)} for 0 < w < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We consider the  special case w = 1/2 here, deferring the general case and definitions to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' As we will  see there, this e-posterior is quite special: for the e-posterior minimax decision rule δmm we  get, with ¯R defined relative to the BBW e-posterior, under C0,  ¯R(δmm)  =  p1(Y )  p0(Y ) + p1(Y )L(1,δmm(Y )) +  p0(Y )  p0(Y ) + p1(Y )L(0,δmm(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (14)  Thus, it can again be rewritten as a sum rather than a supremum and we see that it is formally  identical to the Bayes assessment (1) when P0 and P1 are equipped with the uniform prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We thus have a frequentist risk assessment, not requiring a prior assumption on P, that is  as strong as a Bayesian one, which is optimal under a specific prior assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This seems  almost too good to be true, especially since it is slightly, but uniformly, better than the  6  Figure 1:  (left and middle) The brown solid curve is the pure Savage-Dickey e-posterior  ¯P(θ ∣ y) for the normal location family based on sample y = (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=',x100) with ˆθ(y) = 1 and  prior W with λ = 1 and θ0 = 0 as in (11) (it reaches a maximum of about 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The green curve  represents the 1/2-dampened version (see (35)) of this e-posterior, by design bounded by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  The dashed brown curve is the capped version ¯P cap(θ ∣ y) (Definition 1) of ¯P(θ ∣ y), equal  to it whenever the latter is ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The blue curve represents the generalized Savage-Dickey  two-point prior-based e-posterior optimized for n∗ = n = 100 as in Example 3 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The  red curve is twice the tail area of the objective Bayes posterior w○(θ ∣ y) of Example 7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  given by f(θ) = 2W ○(¯θ ≥ θ ∣ y) = 2∫  ∞  θ  w○(θ ∣ y)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' the remark underneath Definition 1,  the set {θ ∶ ¯P(θ ∣ y) ≥ α} is a conservative 2-sided (1 − α)-confidence interval, whereas, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Example 7, the set {θ ∶ f(θ) ≥ α} is an exact nonconservative 2-sided (1 − α)-confidence  interval, so that plotting f(θ) provides a better comparison to ¯P(θ ∣ y) than directly plotting  w○(θ ∣ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The middle display zooms in on the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='05 right tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (right) The value of (θ − ˆθ)2 ¯P(θ ∣ y) is depicted for the same four e-posteriors, using the  same colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The two-point prior-based e-posterior is optimized to give good assessments  relative to squared error, its maximum is considerably smaller than for the others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' for the  pure Savage-Dickey prior, the maximum is excessively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  alternative (13) — and as we will see in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='2, there is indeed no free lunch: (14) only  holds for δmm, which for the BBW e-variable can sometimes be a rather bad decision rule,  whereas (13) holds for arbitrary rules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' there are some other prices to pay as well when using  BBW’s e-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In fact, a major motivation for the author’s involvement in e-variables was  my realization, some fifteen years ago, that when deriving frequentist Type-I error bounds  for decisions based on Bayes factors/likelihood ratios in simple testing, one ends up with  the expression p1/p0 as in (13) if one follows the reasoning of classic Bayesian texts such as  (Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 1963, Good, 1991) or Royall’s universal bound (Royall, 1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' whereas using  BBW’s ideas, one obtains p1/(p1 +p0) as in (14): the small but essential discrepancy between  the two assessments tripped me up for about 15 years until I finally developed the necessary  tool to understand it — the much more general e-posterior of which both approaches are  simply a special case, each with its own (dis)advantages, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  2  Quasi-Conditional Risk Assessment  We want risk bounds that are valid, in some precise sense, simultaneously for all loss functions  of the form L ∶ Θ × A → R that satisfy the no-sure-gain condition (see underneath (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  7  45  3-  2  1-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='0  12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='4  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='01-  12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='8  60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='0Proposition 1 below shows that, for any choice of e-variable collection S = {Sθ ∶ θ ∈ Θ} and  any given decision rule δ, we have that ¯R(δ) as in (5) satisfies this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Proposition 1 [Validity of E-Risk Assessments] Consider a decision problem as defined  in the introduction relative to any model {Pθ ∶ θ ∈ Θ} for random variable Y ∈ Y and loss  function L as above that satisfies the no-sure-gain condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Let ¯R(δ) be as in (5) relative  to some e-variable collection S = {Sθ ∶ θ ∈ Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For all θ ∈ Θ we have:  EY ∼Pθ [L(θ,δ(Y ))  ¯R(δ)  ] ≤ 1,  (15)  where we adopt the convention 0/0 ∶= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In particular, this implies, for every function b ∶  Y × U × Θ × A → R+, and for any random variable U taking values in some set U and any  distribution P ′  θ for (Y, U) whose marginal on Y coincides with Pθ, that  E(Y,U)∼P ′  θ [b(Y,U,θ, δ(Y )) ⋅ L(θ,δ(Y ))] ≤  sup  (y,u,θ)∈Y×U×Θ  b(y,u,θ,δ(y)) ⋅ ¯R(y,δ(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (16)  Moreover, for any e-collection S, the risk bound (5) with any minimax decision rule δmm  as given by (6) dominates (5) when applied with any other decision rule δ: for all y ∈ Y:  ¯R(y,δmm(y)) ≤ ¯R(y,δ(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Proof: Fix any θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' By definition of ¯R(y,a), we have for all y ∈ Y that ¯R(y,δ(y)) ≥ ¯P(θ ∣  y)L(θ,δ(y)) so that  EY ∼Pθ [L(θ,δ(Y ))  ¯R(δ)  ] ≤ EY ∼Pθ [  L(θ,δ(Y ))  S−1  θ (Y )L(θ,δ(Y ))] = EY ∼Pθ [Sθ] ≤ 1,  (17)  noting that the derivation is valid as long as the denominator ¯R(δ) inside the first expectation  cannot evaluate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In case it can, we split up the expectand into two terms, 1 ¯R(δ)=0 ⋅  L(θ,δ(Y ))/ ¯R(δ) + 1 ¯R(δ)>0 ⋅ L(θ,δ(Y ))/ ¯R(δ), and use that the first fraction is 0/0 which by  our convention evaluates to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' we proceed with the second term as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This proves (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (16) and domination of δmm is then immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' 2  The bounds (15) and (16) (we mostly use the latter) are unusual and for them to be practically  interesting, they should satisfy several desiderata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In particular:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' They should have a clear interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We show that this is the case directly below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' They should exceed standard risk bounds obtainable for standard estimators (actions)  in standard decision problems by only a small factor, at least if we choose a reasonable  e-collection S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This is indeed the case in our running examples (see Example 8, for the  Gaussian location family with standard squared error loss, Example 10 for an extension  to exponential families, and Example 9, underneath (28) for simple-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='-simple testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' There should exist settings in which, just like Bayesian risk assessments, they allow  one to make more risky decisions conditional on more extreme data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This happens in  Example 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' There should exist situations in which they give useful bounds whereas a standard  Bayesian assessment, based on a standard convenience prior, is really highly misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We give such a case in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  8  Bookie Interpretation  The following interpretation of (16) has a De Finettian flavour to  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Suppose a bookie offers decision-maker (DM) to play the following game, really a meta-  decision problem: DM first specifies a decision-rule δ for the original decision-problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' DM  then gets paid some amount ℓ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' then θ and Y are revealed to both DM and bookie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Then  bookie presents a collection of numbers {Bθ,a ∶ θ ∈ Θ,a ∈ A}, all assumed > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Bookie may  choose Bθ,a as functions of the data Y and external data, here encoded as U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' hence we have  Bθ,a = b(Y,U, θ,a) for some function b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' DM then has to pay back Bθ,δ(Y ) ⋅ L(θ,δ(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' If the  rules are such that (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' bookie a priori guarantees that), no matter what Y,U are observed,  bookie will provide a B such that Bθ,δ(Y ) ≤ ℓ/ ¯R(δ) then DM should accept to play this game,  for her expected gain is ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' if this cannot be guaranteed, then DM has no guarantee that the  expected gain is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This directly follows from (16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' note there is no need for DM to  know the details of how B = b(Y,U,θ, δ(Y )) is defined, as long as the promise Bθ,δ(Y ) ≤ ¯R−1(δ)  always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  In all examples to follow, we consider a simple bookie, by which we mean that the B =  b(y,u) presented is a single number that can be written as a function of Y and U only, and  does not vary with θ or a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The effect of this is that, both for a Bayesian DM and for an  e-posterior minimax DM, knowledge of B does not give any incentive to change the action  taken: for any given prior W, a decision rule δ is Bayes relative to loss function L iff it is  Bayes relative to loss function L′(θ,a) ∶= b(y,u) ⋅ L(θ,a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' and the same holds for e-posterior  minimaxity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Bookies who provide a Bθ,a depending on θ and a are certainly worth looking  at as well, but the analysis is substantially more difficult and will be left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Example 6 [Policy Makers vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' the Decision Maker Interpretation] While the bookie  interpretation may seem rather abstract, here we show that it can translate into practically  relevant settings in which risk bound (16) has actual decision-theoretic consequences, allowing  one to reliably make more extreme decisions if the data are more extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  In standard  hypothesis testing problems, this cannot happen, since there are only two actions, ‘reject’  and ‘accept’, and then ‘more extreme actions’ do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' But in practical testing problems  there is often an additional factor, measuring the importance of the decision in light of the  observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The following real-world example (in essence taken from Gr¨unwald (2022))  illustrates: a study about vaccine efficacy (ve) in a pandemic has been set up as a test between  null hypothesis ve ≤ 30% and alternative ve ≥ 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The original plan was to vaccinate all  people above 60 years of age if the null is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  But suppose the null actually gets  rejected with unexpectedly strong evidence for the alternative — there is substantially more  evidence than policy makers had hoped for — and at the same time the virus’ reproduction  rate may be much higher than anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Based on the observed data and the changed  circumstances, the policy makers might now contemplate an alternative action: vaccinate  every adult, instead of everyone above 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This would change the losses of all decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We  can model this (in an admittedly highly stylized way) by assuming that the loss incurred is  really B ⋅ L(θ,a) where B is some positive number depending on how the decision ‘δ(y) = a’  really gets translated into practical measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For example, δ(y) = 1 (vaccine effective) may  translate into ‘vaccinate everyone over the age of V ’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' If L(1,1) and L(0,0) are both negative  (as we allow), then the lower V , the higher the value of B will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' As indicated above, we  simplify things by assuming that the B = b(y,u) ultimately chosen does not depend on θ or  the action taken;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' all losses are multiplied by the same amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In practice, the employed  weight B might arise from a government committee (say the ‘policy makers’) that analyzes  the vaccine study results independently from DM, potentially in a very different way (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  9  fully Bayesian, or just informally by eyeballing graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The more convincing they determine  the evidence to be, the higher the B they may propose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' similarly, B may depend on external  U — if the question is more pressing then one might want to take more drastic measures so  the B’s go up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The role of the DM (who should now perhaps be called the ‘final risk checker’)  is simply to formulate an absolute upper bound ℓ for the risk and tell the policy makers: as  long as you guarantee policies so that B ⋅ ¯R(Y,δ(Y )) ≤ ℓ, I can guarantee you a risk of no  more than ℓ (here ℓ should be determined before the data become available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' in contrast, the  details of how B arises need not be known and can be arrived at by ‘vague’ means, as long  as B is conditionally (given the data) independence of θ and a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  The Quasi-Conditional Stance  By stressing the importance of (16), we adopt a philos-  ophy closely related to Neyman’s (1950) influential inductive behaviour view of statistics —  we cannot say anything about our performance for an individual study, but we can make  claims about our expected performance (or, by the law of large numbers and concentration  inequalities, with high probability, about our average performance in several runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' One can  say less than under strict Bayesian assumptions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e the assumption that one fully believes  one’s prior: in that case, the assessments one makes will be conditionally correct, and not just  in the average sense of (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' But there is still a ‘conditional’ aspect to (16), in the informal  sense that more extreme data (small ¯R(δ)) leads to a decision rule being acceptable at higher  ‘stakes’ (value of B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' More formally, upon observing ¯R(δ) = r, a Bayesian who insists on  risk bounded by ℓ would condition on this event and hence accept importance weight B iff  B⋅r ≤ ℓ — a simple bookie can present any B he likes and the Bayesian can decide based on r  whether to accept or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Our DM can safely accept B if the subtly stronger property holds  that it is known a priori that the bookie must present a B with B ⋅ ¯R(δ) ≤ ℓ, no matter the  value ¯R(δ) takes, or equivalently, DM can tell, after observing ¯R(δ) = r, bookie that he must  present a B ≤ ℓ/r and bookie is guaranteed to obey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Thus, our DM can safely accept higher  stakes if better bounds are observed and is thus more flexible than a DM in the unconditional  frequentist Neyman-Pearson setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' but only a subset of the games that are acceptable to  the conditioning Bayesian is acceptable to the DM — in this sense, she is ‘quasi-conditioning’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Such substantial consequences ensuing from subtle differences in the protocol between bookie  and DM are also found, in a different context, in (Gr¨unwald and Halpern, 2011), and are  a subject matter of the Vovk-Shafer theory of game-theoretic probability (Shafer and Vovk,  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' They motivate the term quasi-conditional, emphasizing that the DM may behave in  some but not all senses as if conditioning on the data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' and that her final performance is  evaluated in expectation over all possible data, and not conditionally on any given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  To our knowledge, this is a new paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  It is certainly different from the main exist-  ing attempts to unify Bayesian ideas (represent posterior uncertainty by a data-dependent  distribution over parameters) and frequentist guarantees — here we think of (a) objective  Bayes methods with matching priors chosen to get frequentist coverage (Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (b) confidence and the related fiducial distributions (Schweder and Hjort, 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' and (c)  conditionalist frequentist methods such as BBW’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The example below illustrates this for (a)  and (b), focusing on the normal location family, for which matching objective Bayes priors,  standard objective Bayes priors, confidence distribution and fiducial distribution all agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  It shows that — despite the matching!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' — if one were to use them as if they were standard  posteriors in decision problems with an incentive to play more extreme actions with more ex-  treme data, they fail to give proper frequentist error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (c) is more subtle: in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='2  10  we show that BBW and other conditionalist frequentist approaches can be re-interpreted in  terms of e-posteriors — so they can be used against arbitrary loss functions even though  there were not designed as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' However, these e-posteriors only give meaningful (< ∞)  bounds for a small subset of all decision rules (in the BBW case, effectively only for a single  one (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' )), making them highly restricted in practice — whereas other e-collections such as the  Savage-Dickey and LR ones give meaningful bounds for essentially any decision rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Example 7 [Weighted Squared Error] Take the Gaussian location family with variance  1 as in Example 2 and 3, where we allow data-dependent stopping times such as ‘τ is the  smallest n at which xn > 5’), defined in the standard manner (Williams, 1991, Hendriksen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Hence, data Y takes the form Y = (τ,Xτ) with τ a stopping time and Xτ a  vector of τ outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Suppose we aim to find the estimator with smallest mean square error,  where the importance of the problem at hand can depend on the data via some function  B = b(Y ) (we will ignore external circumstances U);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' it is known to DM that B is some  deterministic function of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We thus set  L(θ,a) = (θ − a)2,Ly(θ,a) ∶= b(y)L(θ,a)  (18)  where Ly is the loss function we are interested in in the end, a squared error loss with  importance weighted by some function b(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In general, B = b(Y ) may again be determined  by policy makers who may reconsider, using potentially vague ways, the importance of the  problem in light of the observed data as in Example 6 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We can think of any estimator  ˘θ for θ as a decision rule, that, upon observing Y = (n,Xn), outputs ˘θ ∶= ˘θ(n,Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We first consider how one would go about this problem in an ‘objective Bayes’ man-  ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The standard (uniform, improper) ‘objective Bayes’ prior for this family and data xn  corresponds to the limit for λ ↓ 0 in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  The ensuing posterior W ○ ∣ xn has a normal  density w○(θ ∣ xn) with mean and median equal to the maximum likelihood estimator (MLE)  ˆθ(xn) = n−1 ∑n  i=1 xi and variance 1/n, and it is ‘matching’ (Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2022) in the sense that  1 − α credible posterior intervals (taken symmetrically around the posterior mean) coincide  exactly, for each fixed n, with 1−α confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In this case the objective Bayes pos-  terior also coincides with the fiducial and the confidence distribution (Schweder and Hjort,  2016) based on xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The name ‘confidence distribution’ already warns that this posterior  should perhaps only be used for ‘confidence’ (statements about confidence intervals) but in  the fiducial and objective Bayes world, it seems to have been advocated more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' That  this is problematic was already exhibited in different settings by (Gr¨unwald, 2018, Example 7)  and Balch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' As is well-known, Bayesian posteriors do not depend on the definition  of the stopping time (Berger and Wolpert, 1988, Hendriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' also, they condition  on Y = (n,Xn) so further conditioning on B = b(Y ) also does not affect them, since it is com-  pletely determined by Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Thus we have w○(θ ∣ Y = (n,Xn),B = b0) = w○(θ ∣ xn) (Hendriksen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2021) so that the Bayes optimal action based on (18) and W ○(θ ∣ Y = (n,xn),B = b0)  is simply the MLE, ˆθ(xn) irrespective of the definition of τ, the definition of b and the value  of b(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Now, based on this posterior, the loss we think we make upon observing Y = (n,xn)  is given, as in (2), by  ¯Rbayes(W ○ ∣ n,xn) = E¯θ∼W ○∣xn[Ly(¯θ, ˆθ)] = b(n,xn)  n  ,  (19)  where we used that W ○ ∣ Y is a normal with mean ˆθ and variance 1/n, independently of the  definition of the stopping time random variable τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Therefore, if data are in fact sampled from  11  θ∗, then the average of the losses we expect to make, in several studies, is given by  E(τ,Xτ)∼Pθ∗[E¯θ∼W ○∣Xτ [LY (¯θ, ˆθ(Xτ))]] = E(τ,Xτ)∼Pθ∗ [b(τ,Xτ)  τ  ],  (20)  whereas the loss we actually make on average is  ¯R∗ ∶= E(τ,Xτ)∼Pθ∗[LY (θ∗, ˆθ(Xτ))] = E(τ,Xτ)∼Pθ∗[b(τ,Xτ) ⋅ (θ∗ − ˆθ(Xτ))2]  (21)  We now show that the difference between believed risk (20) and actual (21) can be unbounded,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' we can get it to be at least k∗, for arbitrarily large k∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' To this end, simply set  b(τ,Xτ) ∶= τ and set the stopping time τ to be the smallest n such that (ˆθ(xn)−θ∗)2/n ≥ k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  By the law of the iterated logarithm (Williams, 1991) we know that τ is finite, Pθ∗-almost  surely, under any θ∗ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' With this definition of b, the believed risk (20) is 1 yet the actual  risk (21) is at least k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' By Proposition 1 such a discrepancy is impossible if we base our risk  assessment on e-posteriors via (5) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This will be made concrete in Example 8 and 11  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  3  A Small Taxonomy of E-Specifications  Existing papers on e-variables such as Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (2021), P´erez-Ortiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (2022), Wasser-  man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (2020) consider Pθ that define a random process X1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', and then they provide  constructions for an e-variable for sample size n, for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Thus, they really provide  e-specifications: a sequence of e-variables S[1]  θ ,S[2]  θ ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', for each θ ∈ Θ, where S[n]  θ  is an  e-variable for data Y = (X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=',Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Straightforwardly, upon defining q(Xn) = pθ(Xn)S[n]  θ  (Xn), we have ∫ q(xn)dµ(xn) ≤ 1  where µ is the underlying measure, so q can be thought of as a probability density (if ∫ qdµ < 1,  we may think of the missing mass being put on an outcome that, under Pθ, has probability  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' S[n]  θ  can then be written as a likelihood ratio q(Xn)/pθ(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Thus, in the simple (no-  nuisance) parameter case, we can rewrite every e-variable as a likelihood ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Importantly  though, q is allowed to depend on n and θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' the specification S[1]  θ ,S[2]  θ ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' may deliver, for  different n and θ, entirely different q’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' To emphasize this possibility, we will from now on  make the potential dependence on n and θ explicit, and henceforth write  S[n]  θ  = q[n]  �θ(Xn)  pθ(Xn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (22)  For an important subclass of e-variable specifications, the corresponding q[n]  �θ = q�θ does not  depend on n after all (it may still depend on θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In that case q�θ defines a random process  for X1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' and the specification (S[n]  θ  )n is called an e-process (Ramdas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' E-  processes have a major advantage: for every e-process, for every stopping time τ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' ‘stop  at the smallest n at which S[n] ≥ 1/α, and set τ to this n’), we have (Ramdas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2022):  EXτ∼Pθ [S[τ](Xτ)] = EXτ∼Pθ [q�θ(Xτ)  pθ(Xτ) ] ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  As a result, with e-processes we can engage in optional stopping — under any stopping time  τ — even the aggressive stopping rule mentioned above or some externally imposed rule,  12  the details of the definition of which we do not need to know — S[τ]  θ  is still an e-variable,  and decision making based on (5) as above can proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' With e-specifications that do not  provide e-processes, one cannot engage in such optional stopping (see the extended discussion  by (Gr¨unwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2019, Section 5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Indeed, in Example 7, which involved a non-constant  stopping time τ, one only gets valid assessments with e-processes rather than mere e-variables,  and we shall use these in Example 8 and 11 to provide risk assessments for that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We will now reconsider the e-collections of all our running examples — as we will see,  some provide e-processes, others do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='1  Generalized Savage-Dickey E-Processes  We have already seen two instances of e-processes: the (specification, for each n, of) Savage-  Dickey e-variables Ssd and the likelihood ratio e-variables Slr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Both are instances of what  we may call generalized Savage-Dickey e-processes (S[n])θ, with  S[n]  θ  = pW�θ(Xn)  pθ(Xn)  (23)  where pW (Xn) = ∫ pθ(Xn)dW(θ) is a Bayes marginal distribution on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Thus, the e-variable  numerator is a Bayes marginal, but the prior W one uses is allowed to depend on the param-  eter θ one is comparing to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Example 8 [Gaussian Location, Example 3 Continued] Fix some anticipated sample  size n∗, representing our best a priori guess of the actual sample size n that we will observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Our aim is to obtain an e-process that is valid for all n, including n ≠ n∗, but is optimized to  give the best possible risk bound if we happen to observe y = xn with n equal or close to n∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' To  this end, for each θ ∈ R we define θ− < θ and θ+ > θ to satisfy (12) with C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The construction  below gives a valid e-process for every C > 0, but C = 1 gives the best bounds for the squared  error, at the cost of potentially somewhat worse yet of course still valid, bounds for other  loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Now define, for each n, the e-variable S[n]  θ  (y) = (1/2)pθ−(y)+(1/2)pθ+(y)  pθ(y)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Clearly  the S[n]  θ  form a generalized Savage-Dickey e-process, with W � θ the uniform distribution  on {θ−,θ+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In the Supplementary Material (Section B, Proposition 3) we show that for the  corresponding e-posterior ¯P(θ ∣ y), we have that  min  a max  θ∈R (θ − a)2 ¯P(θ ∣ y)  is achieved by setting a = ˆθ (independently of n and n∗), so that δmm(y) = ˆθ, and we also  show that (Section A) (5) evaluates to  max  θ∈R (θ − ˆθ)2 ¯P(θ ∣ y) = 1  n ⋅ g(n,n∗) ⋅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' if n∗=n  =  1  n ⋅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (24)  where g(n,n∗) = (n∗  n ) ⋅ en/n∗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Note that g(⋅,n∗) has a minimum at n = n∗ with g(n,n∗) = 1  and increases very slowly for n/n∗ close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The maximum over θ is, for n = n∗, achieved  approximately at θ = ˆθ ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='46/√n at which ¯P(θ ∣ y) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='68, independently of the value of  n = n∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' S[n]  θ  gives an e-process for any n∗ we choose in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In the Supplementary Material,  Section A, we show that (24) still holds up to lower order factors, for general 1-dimensional  families, as long as they are restricted to a compact subinterval of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  13  Incidentally, this shows that for standard decision problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  n fixed in advance,  uniform weight function), the e-posterior based risk bounds (5) are not much worse than the  Bayesian risk assessment and minimax risk bounds for the same model, as long as sufficiently  ‘clever’ e-collections are used: the factor is a modest 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='45 if n can be assumed known in  advance, and a log n-factor otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Example 9 [Hypothesis Testing with Generalized Savage-Dickey and LR] Consider  a simple-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='-simple test as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' From (23) we see that the generalized Savage-Dickey  e-variable at fixed sample size n, with Y = Xn, and based on priors W � θ with mass function  wθ, for θ ∈ Θ = {0,1}, is given by  Sθ(y) = wθ(0)p0(y) + wθ(1)p1(y)  pθ(y)  with wθ(1) = 1 − wθ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (25)  For degenerate priors with w0(0) = w1(1) = 0 this reduces to the LR e-process of (13), and  for ‘pure’ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' nongeneralized) Savage-Dickey e-posteriors, we have w0 = w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Consider Wald-  Neyman-Pearson loss functions under Condition Zero (C0) as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Generalizing  (13), the e-posterior minimax decision rule δmm upon observing Y = y is then given by  δmm(y) = a, for a achieving  min  a∈{0,1}max{L(0,a) ⋅  p0(y)  w0(0)p0(y) + w0(1)p1(y),L(1,a) ⋅  p1(y)  w1(0)p0(y) + w1(1)p1(y)}  C0  =  min  a∈{0,1}(L(0,a) ⋅  p0(y)  w0(0)p0(y) + w0(1)p1(y) + L(1,a) ⋅  p1(y)  w1(0)p0(y) + w1(1)p1(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (26)  It follows that under C0, δmm selects hypothesis 0 (a = 0) iff f (p0(y)  p1(y)) ≥ L(1,0)  L(0,1) where f(r) =  r ⋅ (1−w1)r+w1  (1−w0)r+w0 with wj = wj(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Straightforward calculus gives that, irrespective of the choice  for w0 and w1, f(r) is strictly increasing in r, so that there is a unique point r∗ depending  on w1 and w0, such that for all r ≥ r∗, δmm(y) = 0 and for all r < r∗, δmm(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Further, for  any pure Savage-Dickey e-collection with w1 = w0, we have f(r) = r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' we select action 1 if  p1(y) ⋅ (L(1,0))γ ≥ p0(y) ⋅ (L(0,1))γ  (27)  for γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This is equivalent to taking the Bayes decision under a uniform prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' On the other  hand, for the LR e-collection, δmm(y) selects action 1 if the above holds for γ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Finally,  if the loss is symmetric, L(0,1) = L(1,0), then in fact we select action 1 iff p1(y) > p0(y)  even for general w1 ≠ w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  How about our basic risk assessment (5) for general decision rules, including these min-  imax ones?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' While (5) does not give rise to easily interpretable formulae that work for all  generalized Savage-Dickey e-variables, under C0 we do get intuitive expressions for pure  Savage-Dickey e-variables with w0 = w1 = 1/2 the uniform prior, and for LR e-collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For  the former,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' we can replace the ‘max’ by a ‘+’ using C0 as in (26) and we get:  L(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='δ(y)) ⋅  p0(y)  1  2p0(y) + 1  2p1(y) + L(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='δ(y)) ⋅  p1(y)  1  2p0(y) + 1  2p1(y) =  2 ⋅ L(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='δ(y)) ⋅ wu(0 ∣ y) + 2 ⋅ L(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='δ(y)) ⋅ wu(1 ∣ y) = 2 ⋅ Eθ∼Wu∣y [L(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='δ(y))]  (28)  14  where wu(j ∣ y) =  pj(y)  p0(y)+p1(y) is the posterior based on uniform prior wu with wu(0) = wu(1) =  1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' showing that our bound is always within a factor 2 of the best Bayes-with-uniform-prior  bound and hence within a factor of less than 2 of the standard frequentist minimax bound  for this setting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' obtainable by a procedure that depends on the specific L of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For the  LR e-variables, we similarly get, refining (13),  L(0,δ(y)) ⋅ p0(y)  p1(y) + L(1,δ(y)) ⋅ p1(y)  p0(y) =  L(0,δ(y)) ⋅ wu(0 ∣ y)  wu(1 ∣ y) + L(1,δ(y)) ⋅ wu(1 ∣ y)  wu(0 ∣ y) = Eθ∼wu∣y [  1  wu(δ(y) ∣ y) ⋅ L(θ,δ(y))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (29)  We note that for δ ∶= δmm and the symmetric case L(0,1) = L(1,0), we choose action 1 iff  p1(y) ≥ p0(y) so that wu(δmm(y) ∣ y) ≥ 1/2: the LR bound (29) dominates the uniform-prior  Savage-Dickey bound (28), and 1/wu(δmm(y) ∣ y) in (29) is a luckiness factor, bounded by  2: if we happen to assign high posterior to the selected hypothesis, we are ‘lucky’ and our  loss assessment will get tighter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' still, it is valid irrespective of the value of this posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' If  asymmetric losses are observed, 1/wu(δmm(y) ∣ y) can in general be both larger and smaller  than 2, and the LR risk assessment bound does not dominate the Savage-Dickey one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='2  Conditionalist-Frequentist E-Specifications — not E-Processes  We first continue the Berger-Brown-Wolpert (BBW) simple-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='-simple hypothesis testing  setting of Example 5, later extending it to more general conditionalist frequentist inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Let V ∶= p1(Y )/p0(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We will assume that (a) V is strictly increasing in Y , which (b)  has a (Lebesgue) density under P0 and P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Both assumptions are merely for simplicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' for  example (a) can be dropped by directly analyzing V instead of Y , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' ‘the likelihood ratio of  the likelihood ratio’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Fix some w with 0 < w < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Since, for any y∗, the sum of both integrals below is 1, there  must be a specific y∗ such that  ∫y<y∗((1 − w)p0(y) + wp1(y))dy = w ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' ∫y≥y∗((1 − w)p0(y) + wp1(y))dy = 1 − w,  (30)  so that, upon setting  Sbbw(w)  0  (y) = 1y≥y∗ (1 − w)p0(y) + wp1(y)  (1 − w)p0(y)  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Sbbw(w)  1  (y) = 1y<y∗ (1 − w)p0(y) + wp1(y)  wp1(y)  (31)  we can verify that EY ∼Pθ[Sbbw(w)  θ  ] = 1 for θ ∈ {0,1} so that {Sbbw(w)  θ  ∶ θ ∈ {0,1}} is an e-  collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The corresponding e-posteriors only give useful risk assessment bounds (5) in quite  restricted situations, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' To be sure, the bound (5) holds (as it must by Proposition 1),  but it must be trivial for all decision rules except one as soon as we assume — as will be the  case for every Wald-Neyman-Pearson loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' with L(0,1) > 0,L(1,0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The reason is that  ¯P bbw(w)(θ = 1 ∣ y) = ∞ if y ≥ y∗ and ¯P bbw(w)(θ = 0 ∣ y) = ∞ if y < y∗: for every loss function  L and decision rule δ there will be y such that ¯R(y,δ(y)) = ∞, unless L(0,0) ≤ 0,L(1,1) ≤ 0  and δ is the bbw(w)-decision rule, which satisfies δbbw(w)(y) = 1 iff y ≥ y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This is, in fact,  the only decision rule that BBW themselves study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Note that it must then also be the case  that the minimax decision rule δmm based on ¯P bbw(w) must coincide with δbbw(w), and thus  15  does not depend on the loss L of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This indicates that, for many loss functions, it  will not be a very good rule — it is simply the only rule for which the risk assessment (5)  is nontrivial (and then evaluates to (14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' To be sure, BBW provide a result implying that  if the loss function L satisfies C0, then we may set w as a function of this loss function so  that the rule δbbw(w) = δmm is to some extent reasonable after all in the following sense: it  coincides with the Bayes decision rule δBayes(y) based on prior W with w(1) = w as long as  y /∈ [y−,y+] for some specific values y− ≤ y+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' They call [y−,y+] the no-decision region and  notice that in practice it is often quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  This support for δbbw(w) notwithstanding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' we feel that the pure Savage-Dickey (with  w = 1/2) and LR e-posteriors should be preferred: at the small price of the slightly larger risk  assessments (28) and (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' they (a) allow us to use these assessments for general decision rules  rather than the potentially defective (since independent of b) δbbw(w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' and (b) they provide  e-processes rather than just e-specifications (it is easy to see that the likelihood ratios (22)  corresponding to BBW e-specifications have q[n]  θ  depending on n — BBW e-variables do not  remain e-variables under optional stopping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' generalized Savage-Dickey e-variables do).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  BBW’s Original, Conditionalist-Frequentist Interpretation  For general p0 and p1  as above, we may partition the sample space Y into regions {Yα ∶ α ∈ [0,1]} where Yα =  {y+  α,y−  α} are chosen such that P0(Y ≥ y+  α) = α and P1(Y ≤ y−  α) = β(α) for some function  β ∶ [0,1] → [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We may think of observing Y = y as first observing α so that y ∈ Yα, and  only then observing Y = y itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We may then, by construction, after observing α, perform  a conditional frequentist test with Type-I and Type-II ‘conditional’ error probabilities α and  β(α), by rejecting θ = 0 if Y = y+  α and accepting if Y = y−  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The BBW approach was originally  conceived as a specific way to construct such a partition, with the special property that for  any α we have that y+  α ≥ y∗,y−  α ≤ y∗ (so that δmm invariably rejects if y+  α is observed for some  α) and  α =  (1 − w)p0(y+  α)  (1 − w)p0(y+α) + wp1(y+α),  β(α) =  wp1(y−  α)  (1 − w)p0(y−α) + wp1(y−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  These are exactly our BBW e-posteriors ¯P bbw(w)(0 ∣ y+  α) and ¯P bbw(w)(1 ∣ y−  α) — BBW in  fact call them conditional error probabilities, because that is how they derived them, and  note that they are formally identical to Bayesian posteriors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' the interpretation in terms of  e-posteriors is ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Kiefer and Vovk Confidence Estimators  Essentially the same analysis as for BBW  conditional error probabilities also applies to general conditionalist frequentist e-variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  based on Kiefer’s and Vovk’s conditional confidence estimators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' to which we now turn: every  such confidence estimator gives a valid e-specification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' hence e-posteriors for each n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' but in  general these will not be e-processes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' and they can evaluate to infinity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' hence only giving  useful,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' nontrivial risk assessments for a restricted set of loss functions and decision rules —  although this set may not be as restricted as in the BBW case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' where only a single rule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  δbbw(w) gave nontrivial risk assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Consider a partition {Yα ∶ α ∈ [0,1]} of Y together with a function CI(y) that maps  each y ∈ Y into a subset of Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In case Θ is a continuum, we usually take CI(y) to be an  interval, so that CI(y) = [θL,θR] with endpoints θL and θR determined by y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We say that  ({Yα ∶ α ∈ [0,1]},CI) define a Kiefer-type conditional frequentist confidence set estimator if  16  for all θ ∈ Θ, all α ∈ [0,1],  Pθ(θ /∈ CI(Y ) ∣ Y ∈ Yα) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Such confidence estimators were first considered by Kiefer (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' If β(α) = α, the BBW  approach can be reinterpreted as a Kiefer-type confidence estimator: we set Yα as above and  CI(y) = {δbbw(w)(y)} so that CI(y+  α) = {1},CI(y−  α) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (32)  If β(α) ≠ α then the BBW approach does not give a Kiefer-type confidence estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' How-  ever, Vovk (1993) provided a notion of confidence estimator that generalizes both BBW’s  conditional error probabilities and Kiefer-type confidence estimators: for any function CI(y)  that maps each y ∈ Y into a subset of Θ combined with any function ˆα(Y ) taking values in  [0,1], we say that CI and ˆα define a Vovk-type confidence set estimator if for all θ,  EY ∼Pθ [1θ/∈CI(Y ) ⋅  1  ˆα(Y )] ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (33)  Vovk (1993) defines confidence set estimators in this partition-free manner, motivated by a  game-theoretic interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' He gives a (straightforward) proof showing that any confidence  set estimator in the sense of Kiefer is also a confidence set estimator in the sense of Vovk,  with ˆα(y) set to the α such that y ∈ Yα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Similarly, if we set CI as in (32) and, for each  α ∈ [0,1], ˆα(y+  α) ∶= α, ˆα(y−  α) = β(α), then it is easily seen that (33) hold with equality for  both θ ∈ {0,1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' we obtain a Vovk-type confidence estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Now, if (CI(Y ), ˆα(Y )) is a confidence set estimator in sense of Vovk, it directly follows  from (33) that for all θ, Sθ = 1θ/∈CI(Y )1/ˆα(Y ) is an e-variable, so ¯P(θ ∣ y) is an e-posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Just as in the specific BBW scenario, such a posterior may become ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  4  Condition Zero and Capped Posteriors  Consider Example 2 again, the normal location family with the squared error loss (18) as  in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' If we base our risk assessment directly on the Savage-Dickey e-posterior as  defined there, we get bounds that, while correct, are not always sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This is due to the  fact that, as seen from Figure 1, ¯P(θ ∣ Y ) = S−1  θ (Y ) can become very high for θ close to ˆθ —  a milder form of the issues with the confidence-estimator based e-variables of the previous  section, for which it can become even infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' There, the issues were avoided if the loss  function satisfied Condition Zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It turns out that a weakened version of this condition is  sufficient to avoid this problem, for decision problems more general than testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We now  introduce a new concept, and weaken and generalize Condition Zero with this in mind:  Definition 1 [Capped E-Posterior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Condition Zero] For any given e-posterior ¯P(θ ∣ y),  its capped version ¯P cap(θ ∣ y) is defined by setting, for all θ ∈ Θ, y ∈ Y:  ¯P cap(θ ∣ y) ∶=  min{1, ¯P(θ ∣ y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Let δ be a decision rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We say that δ satisfies C0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Condition Zero,  if  sup  θ∈Θ  ¯P(θ ∣ y)L(θ,δ(y)) = sup  θ∈Θ  ¯P cap(θ ∣ y)L(θ,δ(y))  (34)  for every y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We say that the loss function L itself satisfies Condition Zero if there exists  a decision rule δ such that (34) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  17  Our original Condition Zero in simple-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='-simple testing was defined as a condition on a given  loss function L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' the present condition is strictly weaker as a condition on loss functions and  generalizes the idea to decision rules δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Why would we want to impose such a condition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  First note that setting Sθ = 1 for all θ ∈ Θ gives a trivial, yet valid, e-collection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' it provides  no indication at all about θ based on data y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This suggests that, when using a nontrivial  collection, based on data y, we cannot make any inferences about θ for which Sθ(y) < 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  ¯P(θ ∣ y) > 1): our e-variable provides less evidence against θ than what can be obtained  trivially: in a sense, y gives us no information at all for making inferences about θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It is  therefore of interest to restrict to decision rules satisfying the new Condition Zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For such  problems, our risk assessment will never be based on evaluating ¯P(θ ∣ y) at a value of θ for  which ¯P(θ ∣ y) > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' for which it really does not give any information about this θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Also,  it may be argued that the capped e-posterior is a more intuitive tool than the e-posterior:  for any α ≥ 0, the set {θ ∈ Θ ∶ ¯P(θ ∣ y) ≥ α} is a 1 − α-confidence interval (Gr¨unwald,  2022) (see Figure 1, middle panel), whereas for sets containing θ with ¯P(θ ∣ y) ≥ 1, no clear  interpretation seems to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Thus, by insisting on C0 one really insists on inferences that  can be based on the more intuitive ¯P cap while still satisfying (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Example 10 [Two-Point prior for exponential families:  Condition Zero holds]  The generalized Savage-Dickey two-point prior-based e-posterior that we employed for the  Gaussian location family in Example 8 gives rise to e-collections such that the corresponding  minimax δmm(y) = ˆθ(y), also satisfy Condition Zero whenever the b-weighted squared error  (18) is used: independently of n, ¯P(θ ∣ y) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='68 for the maximizing θ, as was established  below (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For exponential families, we show in the Supplementary Material (Section A,  inside the proof of Theorem 1) that the same holds if the model {Pθ ∶ θ ∈ Θ○} is restricted  to an (arbitrary) compact subset of the parameter space and n is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We do not  know if the result can be extended beyond exponential families, since its proof makes heavy  use of their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  In practice, the DM may be presented with a loss function for which Condition Zero does not  hold, for any reasonable decision rule, and the high-risk problem looms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' What to do in such a  case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' There are two alternative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' As to the first (we describe the second underneath  Example 11), one can (sometimes vastly) improve risk assessments by slightly modifying any  given e-variable Sθ to another e-variable S′  θ that provides almost as much evidence against θ  as Sθ whenever Sθ provides more evidence against y than the trivial 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' when Sθ(y) > 1),  and almost as much as the trivial S = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The corresponding ¯P ′(θ ∣ y) = S′−1  θ  (y) will  then be similar to ¯P(θ ∣ y) if ¯P(θ ∣ y) ≤ 1 and bounded by a small constant otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Three  straightforward ways of doing this are provided by setting S′  θ equal to any of the following  options:  S∗  θ (y) ∶=  max{Sθ(y),1}  EY ∼Pθ[max{Sθ(Y ),1}]  ,  S○  θ(y) ∶= 1  2 ⋅ max{1,Sθ(y)}  S[γ]  θ (y) ∶= (1 − γ) + γSθ(y) for γ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (35)  One immediately verifies that all three options produce e-variables: S[γ] produces one for  all γ, because mixtures of e-variables are e-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Using that the expectation in S∗  θ is  bounded by 2 (to see this, upper bound max by a sum), we see that S○ is dominated by both  S∗  θ and S[1/2]  θ  hence must also be an e-variable and we also see that for all three choices, we  18  have supθ ¯P ′(θ ∣ y) ≤ 2sup(S○  θ(y))−1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' S∗  θ may be the most direct solution, but it can turn  an e-process into an e-specification that is not an e-process anymore, whereas S[1/2]  θ  preserves  the e-process property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Since S[1/2]  θ  dominates S○  θ, we prefer it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' it is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Example 11 [Improving Pure Savage-Dickey for Exponential Families] We illus-  trate the idea above for the case that ¯P(θ ∣ y) = 1/Sθ(y) is the Savage-Dickey e-posterior  based on an arbitrary prior W on the Gaussian location family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Let ¯P [γ](θ ∣ y) ∶= 1/S[γ]  θ (y) be  the corresponding ‘dampened’ e-posterior as defined in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In the Supplementary Material,  Section C we describe this setting in more detail and extend it to 1-dimensional exponential  families, also showing (in Proposition 3) that for any 0 < γ ≤ 1 the MLE ˆθ is the ¯P [γ]-e-  posterior minimax estimator irrespective of the weight b in the loss (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Now for simplicity  let W be a Gaussian prior with some fixed precision λ as in Example 2 and with prior mean  θ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We also show in the Supplementary Material (Proposition 4), that, for the choice  γ = 1/2 that we advocated above, the bound (15) and (16) hold with  ¯R(δmm) = ¯R(Y, ˆθ(Y )) = 2  n (1  2 log n + λ  λ  +  nλ  n + λ  ˆθ(Y )2),  (36)  losing just a log n factor compared to the e-posterior that achieves the standard frequentist  minimax bound up to a constant factor, and that we considered in Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Again we  extend this result to general 1-dimensional families with smooth and strictly positive prior  densities (and again we do not know if it extends to more general 1-dimensional families).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In  contrast, for the original Savage-Dickey e-posterior ¯P(θ ∣ y) we get a much larger ¯R(δmm) ⪰  n−1/2, as can be seen by evaluating ⋅ ¯P(θ ∣ y)(θ − ˆθ)2 at θ = ˆθ + n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' By ‘dampening’ the  posterior we improved the factor of order n−1/2 to a factor of order (log n)n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  As a second solution to the high-risk problem, we can alternatively modify our risk assessment  (5) for decision problems and rules that do not satisfy C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We then find that if we replace the  e-posterior in the risk assessment ¯R(δ) as in (5) by its capped version, the risk assessment itself  may become invalid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' ¯R(δ) does not satisfy (15)/(16)), but, for arbitrary data-dependent  decision problems, 2 ¯R(δ) does remain valid:  Proposition 2 Set, for arbitrary multi-loss decision problem and rule δ, not necessarily  satisfying C0,  ¯Rcap(Y, A) ∶= sup  θ∈Θ  ¯P cap(θ ∣ y) ⋅ L(θ,a) ,  ¯Rcap(δ) ∶= ¯Rcap(Y,δ(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (37)  Then (15) and (16) hold with 2 ¯Rcap(δ) in the role of ¯R(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Proof: Calculating ¯Rcap(y,a) based on the capped posterior is equivalent to calculating  ¯R(y,a) based on the modified posterior ¯P ′(θ ∣ y) = (S○  θ(y))−1 as defined in (35) and multi-  plying the answer by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Since ¯P ′(θ ∣ y) is a standard e-posterior and no requirements on δ  are made, we can use Proposition 1 based on ¯P ′(θ ∣ y) to get the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' 2  This solution to the high-risk problem of some e-collections shows that, for some collections,  the original e-posterior in its capped version may not be so bad after all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This solution may in  fact be preferable, since it does not interfere with the use of e-values in optional continuation:  we can keep multiplying e-values as explained in the next section, and get a factor of 2 in  the bound for the final product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In contrast, if we use any of the modifications in (35), we  must modify the e-variable at each study, making the final (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' when we stop adding studies)  product of e-values potentially much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  19  5  Additional Background and Discussion  The General Definition of E-Values  The definition of e-collection, e-posterior, risk as-  sessment (5) and e-posterior minimax decision rule readily generalize to models with nuisance  parameters, composite hypothesis testing and nonparametric settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' All are covered by the  following extended definition: let P be a set of distributions for random variable Y and  consider a parameter (vector) of interest defined by a function δ ∶ P → Θ with Θ ⊂ Rk for  some k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We now call {Sθ ∶ θ ∈ Θ} an e-collection for P if for all θ ∈ Θ, Sθ = Sθ(Y ) is  a nonnegative function of Y and for all P ∈ P with δ(P) = θ, we have the corresponding  analogue of (3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' for all such P:  EP [Sθ] ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (38)  The e-posterior ¯P(θ ∣ y) relative to such a collection is still defined as S−1  θ (y), and the  definition of risk assessment (5) and e-posterior minimax rule (6) remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It is  easily checked that the results that validate the e-posterior risk assessment and minimax rule,  Proposition 1 and Proposition 2, still hold in this more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  The Original Interpretation of E-Variables  E-variables, while being implicit in earlier  work going back at least to Darling and Robbins (1967), were largely unknown until 2019,  when the concept was given a name and was popularized and extensively developed by papers  such as (Gr¨unwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2019, Shafer, 2021, Vovk and Wang, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In the mean time they  have been studied in a wide variety of contexts (Henzi and Ziegel, 2021, Ren and Barber, 2022,  Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2022) and even had an international workshop devoted to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Originally, they  were used as tools to extend traditional Neyman-Pearson tests to situations with optional  continuation while keeping Type-I error guarantees: suppose one observes a sequence of  studies concerning the same null hypothesis {P ∈ P ∶ δ(P) = θ0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' One can then multiply the  e-values derived from the individual studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Broadly speaking, the product will be a new  e-variable, irrespective of whether the decision to perform a subsequent study depends on  previous study outcomes (the ‘continuation rule’), and irrespective of the rule used to decide  when to stop performing new studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Since, for any fixed α, the probability, under the null,  that an e-variable gets larger than 1/α, is bounded by α, one can use e-values, combined by  multiplication, as an alternative to p-values in a setting in which optional continuation of  experiments is allowed (note that this is always possible, whereas optional stopping within a  study is only possible if we deal with e-processes rather than e-variables, as in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Composite Testing  In light of this original motivation, much initial work on e-values  focused on hypothesis testing in Neyman-Pearson (NP) style, but under optional continua-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Just as in the original NP paradigm, Type-I error of decision rules was required to be  bounded by a pre-given significance level α, and other desiderata such as small Type-II error  only came second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This is different from the treatment of simple-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='-simple testing in the  present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Here, as in Bayesian and minimax approaches, we do allow asymmetric losses  (L(0,1) ≠ L(1,0)), but these potentially different losses appear in the risk assessment (5) in  a symmetric fashion after all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Existing papers on e-values within the original paradigm in fact emphasize composite  hypotheses: the theoretical papers Gr¨unwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (2019), P´erez-Ortiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (2022) and  the practical (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', 2021, Turner and Gr¨unwald, 2022) consider situations in which  the null has nuisance parameters (such as, for example, the variance in the t-test or the  20  proportion in 2 × 2 tables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  For such cases (the null P = {Pλ ∶ λ ∈ Λ} is characterized  by a (potentially) high-dimensional parameter vector λ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' the parameter of interest δ is of  (potentially) much smaller dimension, often a scalar), Gr¨unwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (2019) offer a generic  method to construct e-variables as in (38) for such situations via the reverse information  projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This method is employed by all papers mentioned above to construct e-variables,  with the explicit goal of providing Type-I error bounds under optional continuation in the  Neyman-Pearson style;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' but we can equally use them to provide e-posterior risk assessments  and minimax rules with general loss functions as in this paper — in our examples above,  we only used ‘simple’ e-collections for ease of illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Of particular interest is the paper  (Gr¨unwald, 2022), which may be viewed as a companion to the present one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It also extends  the use of e-variables to deal with post-hoc specified loss functions but it sticks to a variation  of the Neyman-Pearson tenet: there is a distinction between Type-I risks (expected losses),  on which a hard constraint is imposed (similar to the significance level α as a constraint  on Type-I error probability in traditional testing), and Type-II risks, which are minimized  subject to this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Nonparametrics  The same generalization (38) covers nonparametric settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  For ex-  ample, in Cox’s celebrated proportional hazards model, the P ∈ P specify an underlying  continuous time process with dynamics conditional on some covariate vector ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' While the  full likelihood of this process is intractable, Ter Schure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (2021) show that, for any vector  θ of covariate coefficients, the ratio between any two partial likelihoods defined relative to the  same set of covariates constitute an e-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This implies that all the results on e-posterior  risk assessment of the present paper also pertain to ratios of Cox’s partial likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In fact,  for any P and any θ ∈ Θ, any ratio between conditional, marginal or partial likelihoods that  has the same distribution under all distributions in P with δ(P) = θ is trivially an e-variable,  and can be used to define an e-collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We regard all this as an additional advantage of  e-posteriors over Bayesian ones: strictly speaking, a Bayesian needs to specify the full likeli-  hood for each distribution in P in order to get a valid a posterior — as has been argued by  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Robins and Wasserman (2012), this may be viewed as ‘overkill’ and poses practical (and  unnecessary) difficulties for Bayesian nonparametric inferences especially when the samples  are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In contrast, Waudby-Smith and Ramdas (2022) provides an e-collection which  can be used to learn the mean of a sequence of bounded random variables based on a small  sample without any assumptions on the underlying random variables except boundedness  and existence of a common mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Conclusion and Future Work  We have presented the idea of representing uncertainty  by e-posteriors based on e-collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' While e-collections already existed, the idea to use  them in risk assessments such as (5) and to define e-posterior minimax rules as in (6) is,  to our knowledge, completely new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' There are of course many open questions regarding this  quasi-conditional methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The main one is perhaps: what e-collection should be used in  practice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' If one takes as one’s goal “collecting as much evidence as possible in subsequent  studies against a single θ ∈ Θ”, and one is able to formulate a prior on Θ representing  a ‘best guess’ (note that Θ does not necessarily index a set of distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' rather a set of  properties) then there is a good case to be made for the GRO (growth-rate optimal) e-variable  as defined by Gr¨unwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (2019), which is always well-defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' we refer to that paper for  extensive discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For simple nulls, the GRO e-variable is always of pure Savage-Dickey  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' However, if we are interested not in testing but in estimation and want an e-posterior  21  over the full range of θ ∈ Θ, then the e-posterior based on the GRO e-variables may not be  intuitively optimal, as we saw in this paper: the pure Savage-Dickey e-variable based on a  normal prior W in the Gaussian location family (which is GRO relative to W) gives worse  risk assessments than the one based on the two-point prior as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We do not have  a general recommendation (in the style of ‘reference priors’ for Bayesians) for e-collections,  and obviously more research is needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This paper is really just a first, exploratory one,  comparing various possibilities such as two-point, capped- and dampened e-collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' A  second related important question is: in what situations is the bound (16) sufficiently tight  to be practically relevant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Essentially, if we are in the setting that (bookie, or policy makers)  provide B such that B ⋅ ¯R(δ) is constant or varies only very little over y, then the bound  (15) looks like a standard expected risk bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' If ¯R(δ) can vary highly depending on the  data, then the bounds (15) and (16) are still valid, but the supremum in (16) may make them  rather weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Can we get better bounds in such cases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' And what if we let B also vary with θ  and a?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Then, upon learning {Bθ,a ∶ θ ∈ Θ,a ∈ A}, DM might want to re-consider the originally  planned decision rule δ, significantly complicating the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Related to this is the question  what loss functions L correspond most tightly to real-world problems at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In our hypothesis  testing examples, we considered asymmetric loss functions which seems reasonable, but in our  estimation example, we used the squared error loss — which is mathematically convenient  and incentives one to output the posterior mean;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' but in practical situations one may be more  interested in very different, hard-to-formalize, problem-dependent losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' All in all, there is  plenty of room for future work here!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  References  Michael Scott Balch, Ryan Martin, and Scott Ferson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
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+page_content=' Soloff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Principal-agent hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='06812, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
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+page_content=' Polasek, editors, Data  Analysis and Information Systems: Statistical and Conceptual Approaches, pages 289–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Springer, Berlin, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  25  Supplementary Material  In this Supplementary Material, in Section A we provide details and proofs concerning the  two-point prior-based generalized Savage-Dickey e-variables for the Gaussian location family  and its extension to one-dimensional exponential families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In the final Section C we provide  details and proofs concerning the ’pure’ Savage-Dickey e-variables for Gaussian location and  one-dimensional exponential families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In between in Section B we show how the MLE is e-  posterior minimax both for Gaussian location with two-point prior and for general exponential  families with pure Savage-Dickey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  A  Two-point prior-based E-Collections  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='1  A Theorem for One Dimensional Exponential Families  Here we present Theorem 1, which extends the result (24) for two-point prior-based e-  collections of Example 8 to 1-dimensional exponential families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' More precisely, we let {Pθ ∶  θ ∈ Θ} be any regular (Barndorff-Nielsen, 1978) 1-dimensional exponential family given in  any diffeomorphic parameterization (for concreteness, take the mean-value one) and extended  to n outcomes by independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We write the KL divergence between two members of the  family defined on a single outcome as D(θ′∥θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We denote by ˆθ(y) the MLE based on data  y = xn, which is known to be unique and equal to the empirical average n−1 ∑n  i=1 φ(Xi), with  φ the sufficient statistic, whenever this average lies in Θ, which for regular families is an open  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Suppose we observe Y = xn with ˆθ(Y ) ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We now use as our loss function  L(θ, ˘θ) ∶= 2 ⋅ D(˘θ∥θ),  which is parameterization independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' but with the mean-value parameterization, in the  Gaussian location family, 2D(˘θ∥θ) = (θ − ˘θ)2 becomes the squared error loss as in Example 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  of course the factor 2 is merely for mathematical convenience in the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  As we show below (42), for every C > 0, every θ ∈ Θ, there exist θ− < θ < θ+ such that,  consistent with the Gaussian case,  D(θ∥θ+) = C  n∗ , D(θ∥θ−) = C  n∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (39)  As in the Gaussian example, we take C = 1, and we construct, for each n, the e-variables  S[n]  θ  (y) = (1/2)pθ−(y)+(1/2)pθ+(y)  pθ(y)  , obtaining again a generalized Savage-Dickey e-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Based  on the standard Taylor expansion  D(θ∗∥θ) = 1  2I(θ○)(θ∗ − θ)2  (40)  where I(⋅) is the Fisher information, which must hold exactly for some θ○ in between θ∗  and θ, we may expect risk assessments based on this e-variable to behave similarly as in the  Gaussian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Indeed, below we first prove the result (24) of the main text for the Gaussian  case, after which we proceed to prove, heavily using (40), the following extension to regular  1-dimensional exponential families:  Theorem 1 Let Θ○ be an arbitrary compact subinterval of Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Consider restricted model  {Pθ ∶ θ ∈ Θ○} and the collection of e-processes {Sθ ∶ θ ∈ Θ○}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Then, for every 0 < c ≤ 1,  26  there is a constant C∗ such that whenever ˆθ ∈ Θ, and c ≤ n∗/n ≤ 1/c, (24) still holds up to a  vanishing 1 ± O(n−1/2) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' That is, there is C∗ such that, for all n, uniformly for all xn  with ˆθ(xn) ∈ Θ○,  max  θ∈Θ○ L(θ, ˆθ(xn)) ⋅ ¯P(θ ∣ xn) = 1  n ⋅ (n∗  n ) ⋅ expn/n∗ ⋅fn ⋅ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' if n∗=n  =  1  n ⋅ fn ⋅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (41)  where 1 −  √  C∗/n ≤ fn ≤ 1 +  √  C∗/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Thus, even though now ˆθ may not be precisely equal to the e-posterior minimax estimator  δmm any more, we may use it as our decision rule anyway, and then still get, for each compact  sub-model Θ○, risk bounds that, at least for sufficiently large n, are of the same order as  standard minimax frequentist risk bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='2  Proofs for Example 8, 10 in main text, and Theorem 1 above  Below we prove all results that were stated in the main text for the Gaussian location family  and 1-dimensional exponential families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We will freely, without further reference, use standard  terminology and results concerning exponential families (all to be found in (Barndorff-Nielsen,  1978)), but we first do highlight two slightly nonstandard results that we will use repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  The first (easily proved using steepness of regular exponential families) is that for each fixed  θ′ ∈ Θ, D(θ′∥θ) is a continuous function of θ ∈ Θ satisfying  sup  θ<θ′ D(θ′∥θ) = sup  θ>θ′ D(θ′∥θ) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (42)  The second result we need is the KL robustness property (Gr¨unwald, 2007) that holds for all  regular exponential families: for any fixed y = xn such that ˆθ(y) is well-defined, any θ ∈ Θ  and any prior W on Θ (W does not need to have a density), we have:  pθ(y)  pW (y) = exp(−nD(ˆθ∥θ) + D(P (n)  ˆθ  ∥P (n)  W )),  (43)  where PW is the Bayes mixture distribution as defined in Example 1, and, for any distribution  P underlying random process X1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', we denote the marginal distribution of the first n  outcomes as P (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Proof of (24) in Example 8 and Claim in Example 10  Fix C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' By (42) above,  for general regular 1-dimensional exponential families, there exist θ− < θ < θ+ such that (39)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Now define, for i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', xi ∈ Ri, the e-processes Sl  θ(xi) = pθ−(xi)  pθ(xi) and Sr  θ (xi) = pθ+(xi)  pθ(xi)  and Sθ ∶= (1/2)Sl  θ +(1/2)Sr  θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' While these e-processes provide valid e-variables at each sample  size i we have that when evaluated at sample size n (on which their definition depends), Sl  θ  coincides with the uniformly most powerful Bayes factor for a 1-sided test at sample size n and  level α∗/2, for any n and α∗ such that C− = −log(α∗/2), for H0 = {Pθ} vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' H1 = {Pθ′ ∶ θ > θ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  similarly for Slr  θ  with C+ and H1 = {Pθ′ ∶ θ′ < θ} respectively (Johnson, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We first focus on the Gaussian location family for which D(θ′∥θ) = (1/2)(θ′−θ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' ¯P(θ ∣ y)  can be written by (43) with W a prior putting mass 1/2 on θ+ and 1/2 on θ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For the Gaussian  27  location family, (43) then gives:  ¯P(θ ∣ y) =  pθ(y)  1  2pθ−(y) + 1  2pθ+(y)  =  2e−nD(ˆθ∥θ)  e−nD(ˆθ∥θ−) + e−nD(ˆθ∥θ+)  =  2e−(n/2)(ˆθ−θ)2  e−(n/2)⋅(ˆθ−θ+  √  2C/n)2 + e−(n/2)(ˆθ−θ−  √  2C/n)2 =  =  2eC⋅(n/n∗)  e−√n⋅(ˆθ−θ)  √  2C(n/n∗) + e  √n⋅(ˆθ−θ)  √  2C(n/n∗)  (44)  We will assume that, for any observed y, the MLE ˆθ = ˆθ(y) is chosen as action (we will derive  that this is the e-posterior minimax action later on, in Proposition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Thus, to calculate the  risk assessment bound (5) we need to find the maximum over θ of  (ˆθ − θ)2 ¯P(θ ∣ y)  (45)  We find, using (44), and the fact that the maximum (45) must be achieved at the same θ as  the maximum of 2Cn(n/n∗)(ˆθ − θ)2 ¯P(θ ∣ y), that the maximizing θn satisfies  √  2Cn ⋅ n  n∗ (θn − ˆθ)2 = x∗ where x∗ maximizes  x2  exp(x) + exp(−x),  (46)  a function with a unique maximum, found numerically to be achieved at x∗ = ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='065338138969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='.  (45) then becomes,  1  n ⋅  (x∗)2  2C(n/n∗) ⋅ 2eC(n/n∗)  e−x∗ + ex∗  (47)  We want to find the C giving the best possible bounds for the choice n = n∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' when the  anticipated sample size turns out correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This can be found by minimizing (47) for n = n∗:  by differentiation we find C = 1, corresponding to the choice in Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We then find  (still at n = n∗) that θn = ˆθ ± x∗/  √  2n ≈ ˆθ ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='46/√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Similarly, ¯P(θ ∣ y) for the maximizing  θn at n = n∗ is given by 2e1/(ex∗ + e−x∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='6783206434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='. The desired maximum (45),  at general n, is then given, using (47) with C = 1, by (1/n)(n∗/n)en/n∗x∗2/(ex∗ + e−x∗) =  (1/n)(n∗/n)en/n∗ ⋅ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='53222078, which, if n = n∗, evaluates to (1/n) ⋅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='446729604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' and (24)  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Proof of Theorem 1 above and generalized Claim in Example 10  For simplicity we  only consider the case with n = n∗ here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' the extension to c ≤ n/n∗ ≤ 1/c can then be derived  analogously to the Gaussian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We now let Θ represent the mean-value parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  As in the main text, we set θ+ and θ− as in (39) above with C = 1, which defines ¯P(θ ∣ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We  fix an arbitrary compact subinterval Θ○ of Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We need to bound, for arbitrary fixed ˆθ ∈ Θ○  the maximum over θ ∈ Θ○ of  2 ⋅ D(ˆθ ∣ θ) ¯P(θ ∣ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (48)  We let β(θ) denote the canonical parameter corresponding to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Note that, for g(θ∗) =  D(θ∗∥θ) − D(θ∗∥θ−), we have g′(θ∗) = β(θ−) − β(θ) < 0 constant in θ∗ (here we used that β  is monotonically increasing in θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In particular, if ˆθ and θ are both contained in Θ○, then  g′(ˆθ) ≤ −C− where C− = minθ∈Θ○ β(θ) − β(θ−) > 0, and then, for ˆθ = θ − a, a > 0:  ¯P(θ ∣ y) ≤ 2e−n(D(ˆθ∥θ)−D(ˆθ∥(θ)−)) = 2e−n(D(θ−a∥θ)−D(θ−a∥θ−)  ≤ 2enD(θ∥θ−)−anC∗ = 2eC−anC−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  28  Analogously, if ˆθ,θ ∈ Θ○, ˆθ = θ + a, a > 0, we can derive ¯P(θ ∣ y) ≤ 2eC−anC+ for C+ =  minθ∈Θ○ β(θ+) − β(θ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  It follows that with C± = min{C−,C+}, there exists another  constant C′ such that, if θ, ˆθ ∈ Θ○, then the risk assessment at θ satisfies  2 ⋅ D(ˆθ∥θ) ¯P(θ ∣ y) ≤ bC′a2eC−∣a∣nC±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (49)  Writing ∣a∣ =  √  K/n, we see that for large enough K, the right-hand side is smaller than 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We thus find that there is a constant K such that, for all θ, ˆθ ∈ Θ○, if ∣ˆθ − θ∣ ≥  √  K/n, then  the risk bound (49), and hence (48), is smaller than 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Without loss of generality we may  assume K ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='462 (the need for this condition will become clear later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We now fix K′ > 0 and set, for α ∈ [−K′,K′], for ˆθ ∈ Θ○,  ˆθα ∶= ˆθ + α ⋅  �  �  �  �  2  nI(ˆθ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (50)  where K′ is chosen such that ˆθ−K′ ≤ ˆθ −  √  K/n and ˆθK′ ≥ ˆθ +  √  K/n for all ˆθ ∈ Θ○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Since I(θ)  is bounded from below on Θ○ by a positive constant, such a K′ must exist (note we do not  require that θα ∈ Θ○ for all α ∈ [−K′,K]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We now define Θ′  ˆθ ∶= Θ○ ∩ {ˆθα ∶ α ∈ [−K′,K]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Our proof strategy for the remainder of the proof is showing that Θ′  ˆθ is an interval around ˆθ  that is small enough so that we can re-derive the results for the Gaussian case (up to a factor  converging to 1) on this set, and then showing that the global maximum of (48) over Θ○ is  achieved on this set, so that only this set matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We have that ∣I′(θ)/I(θ)∣, where I′ is the derivative if I, is bounded by a constant on Θ○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  A first-order Taylor approximation of I(θ) then implies that there exist a constant C○ such  that for all large enough n larger than some n○, for all ˆθ ∈ Θ○, we have  for all θ,θ○ ∈ Θ′  ˆθ ∶ 1 −  √  C○  n ≤ I(θ)  I(θ○) ≤ 1 +  √  C○  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (51)  We consider, at sample size n > n○ the risk assessment (5), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' the maximum over θ ∈ Θ○ of  (48) that we get for any fixed ˆθ(y) ∈ Θ○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We first consider the maximum over θ ∈ Θ′  ˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Within Θ′  ˆθ, we can perform essentially the same derivation as in (44), using the Taylor  approximation (40) in the second step in both numerator and denominator and aggressively  and repeatedly using (51) above to vary θ in I(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This (tediously but straightforwardly)  gives, for α ∈ [−K′,K′] such that ˆθα ∈ Θ′,  ¯P(ˆθα ∣ y) = 2e1eO(n−1/2) ⋅  1  e−2α + e2α = 2e1eO(n−1/2) ⋅  1  e−  √  (2I(ˆθ)n(θ−ˆθ) + e+  √  (2I(ˆθ)n(θ−ˆθ)  D(ˆθ∥ˆθα) = (1 + O(n−1/2))(ˆθ − ˆθα)2I(ˆθ)  which holds uniformly for all ˆθ ∈ Θ○ and all α ∈ [−K′,K′] such that ˆθα ∈ Θ′  ˆθ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' we can pick  the same constant in O(n−1/2) for all such ˆθ and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' But this means that for all large enough  n, on the set Θ′, we can repeat the same reasoning as in (46) in the Gaussian case up to a  factor of order 1 + O(n−1/2): since this factor converges to 0, we must have that, uniformly  for all ˆθ ∈ Θ, for all n larger than some n0, the maximum will be above (1/n) ⋅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='44 and will  be achieved by one or both elements inside a set Θmax = {ˆθ + γ+  n/√n, ˆθ − γ−  n/√n} ∩ Θ′, where  both γ+  n and γ−  n converge to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', at which point ¯P(θn ∣ y) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Note that by requiring  29  K ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='462 we made sure K′ is large enough so that both choices for the maximum are in  {ˆθα ∶ α ∈ [−K′,K]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' at least one of these two choices for θn then also lies in Θ′ for n larger  than some n0, where n0 can be chosen uniformly, independently of ˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Thus, the risk (48) maximized over θ ∈ Θ′  ˆθ ⊂ Θ○ is bounded from below by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='44/n, and the  maximum over Θ○ must be at least as large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' But (49) now gives that the maximum of (48)  over θ ∈ Θ○ ∖Θ′  ˆθ is at most 1/n which is smaller than the maximum risk within Θ′  ˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Therefore  the global maximum is achieved within Θ′  ˆθ and, just as for the Gaussian location case, is  given by (1/n)⋅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=', which is what we had to prove (the extension to n∗ ≠ n mentioned in  (41) is now trivial to add;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' the various ‘for all large enough n’ qualifications we imposed are  absorbed into C∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' As a by-product, we also find that, since ¯P(θn ∣ y) ≪ 1, Condition Zero  holds automatically in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  B  Establishing that the MLE is e-posterior minimax  The proposition below shows, by setting θ′ ∶= ˆθ, that the MLE ˆθ is e-posterior minimax  both for the Gaussian location family with the two-point prior (Example 8) and for general  exponential families with a pure Savage-Dickey e-variable with arbitrary prior (Example 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Proposition 3 Consider a regular exponential family given in its mean-value parameter  space as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Let θ′ ∈ Θ and let f ∶ R+  0 → R+  0 be a continuous function such that limx→∞ xf(x) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We have  min  a max  θ∈Θ D(a∥θ) ⋅ f(D(θ′∥θ))  is achieved by a = θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In particular,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Suppose that ˆθ = ˆθ(y) ∈ Θ and consider the dampened e-posterior ¯P [γ](θ ∣ y) for any  0 < γ ≤ 1 and relative to any prior W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It can be written as f(D(ˆθ∥θ)) for a function f  of the required type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For the normal location family, the e-posterior based on the two-point discrete prior as  in (44) can also be written as f(D(ˆθ∥θ)) for a function f of the required type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Proof: Let g(a) ∶= maxθ∈Θ D(a∥θ) ⋅ f(D(θ′∥θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' First consider a = θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' It follows from (42)  that the maximum over θ in the definition of g(a) = g(θ′) is achieved by some θ∗  − < θ′ and  also by some θ∗  + > θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Now consider a ≠ θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We must show that g(a) > g(θ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' If a > θ′, we have  that D(a∥θ∗  −) > D(θ′∥θ∗  −) so  g(a) ≥ D(a∥θ∗  −) ⋅ f(D(θ′∥θ∗  −)) > D(θ′∥θ∗  −) ⋅ f(D(θ′∥θ∗  −)) = g(θ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  The case a < θ′ goes similarly, with θ∗  + replacing θ∗  −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' This establishes the first result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  As to (1), the case with γ = 1 now follows directly from (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For γ < 1, use the fact that  1/((1 − γ)x−1 + γ) is increasing in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  As to (2): using (44), and once more that D(ˆθ∥θ) = (1/2)(ˆθ − θ)2, and considering sep-  arately the cases that ˆθ > θ and ˆθ < θ, we find that ¯P(θ ∣ y) = f(D(ˆθ∥θ)), where f is of the  required form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' 2  30  C  Pure Savage-Dickey based E-Collections for Gaussian and  Exponential Families  Let ¯P [γ](θ ∣ y) be the ‘dampened e-posterior’ as in the main text, based on arbitrary prior  W on Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' In Proposition 3 above we showed that for any 0 < γ ≤ 1 the MLE ˆθ is the ¯P [γ]-  e-posterior minimax estimator irrespective of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Below we further, via Proposition 4 below,  show that, for the choice γ = 1/2, the risk assessment bound (5) holds with  ¯R(y, ˆθ(y)) = 2  nD(P (n)  ˆθ  ∥P (n)  W )  (52)  for all n such that the expression on the right is larger than 1 — which will be the case for all  but the smallest n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Here we used the notation D(P (n)  θ  ∥P (n)  W ) for the KL divergence between  distribution Pθ and Bayes marginal PW , both defined on n outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For the normal location  family with prior with mean 0 and precision λ, we have the exact expression  D(P (n)  ˆθ  ∥P (n)  W ) = 1  2 log n + λ  λ  +  nλ  n + λ  ˆθ2  (53)  which together with (52) gives (36) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' (53) is found by using the fact that  D(P (n)  ˆθ  ∥P (n)  W ) = ln ¯PW (ˆθ ∣ y), an identity which follows from (43) and holds for general regular  exponential families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' If these have continuous prior w, we get, for ˆθ in any compact subset  of the parameter space, the expression (Chapter 8 of Gr¨unwald (2007))  D(P (n)  ˆθ  ∥P (n)  W ) = 1  2 log n  2π − log w(ˆθ)  I(ˆθ)1/2 + o(1),  with I(ˆθ) the Fisher information at ˆθ, allowing us to generalize (36) to general exponential  families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  We proceed to derive (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We start with a proposition that is applicable more generally  than just for KL loss:  Proposition 4 Let Θ ⊂ R, b ∈ R+ and let L ∶ Θ×Θ → R+  0 be a loss function and let ˘θ ∶ Y → Θ  be an estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Fix some y ∈ Y and let ˘θ ∶= ˘θ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Consider an e-posterior ¯P(θ ∣ y) with  ¯Psup ∶= supy∈Y,θ∈Θ ¯P(θ ∣ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' and let ¯P ′ be an upper bound on ¯P up to a factor Csup i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' for all  θ ∈ Θ, y ∈ Y, ¯P(θ ∣ y) ≤ Csup ¯P ′(θ ∣ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Fix some θL,θR ∈ Θ (depending on ˘θ) with θL < θR so  that both:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' for all θ ≥ θR, ¯P ′(θ ∣ y) ≤ 1 and ¯P ′(θ ∣ y)L(θ, ˘θ) is decreasing in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' for all θ ≤ θL, ¯P ′(θ ∣ y) ≤ 1 and ¯P ′(θ ∣ y)L(θ, ˘θ) is increasing in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Then our main risk assessment bound (5) holds for ¯P(θ ∣ y) with  ¯R(y, ˘θ(y)) = b(y) ⋅ max{Csup, ¯Psup} ⋅  max  θ∈[θL,θR]L(θ, ˘θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  (54)  Proof: For θ ≤ θL, ¯P(θ ∣ y)L(θ, ˘θ) ≤ ¯P ′(θ ∣ y)CsupL(θ, ˘θ) ≤ Csup ¯P ′(θL ∣ y)L(θL, ˘θ) ≤  CsupL(θL, ˘θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Analogously for θ ≥ θR, we have ¯P(θ ∣ y)L(θ, ˘θ) ≤ CsupL(θR, ˘θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Finally  for θ ∈ [θL,θR], we have ¯P(θ ∣ y)L(θ, ˘θ) ≤ ¯Psup ⋅ maxθ∈[θL,θR] ⋅L(θ, ˘θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' 2  31  We now use Proposition 4 to show the bound (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Assume the setting of that bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We  set ˘θ to the MLE and L(θ, ˘θ) ∶= D(˘θ∥θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We first apply Proposition 4 with ¯P ′(θ∣y) ∶= ¯P(θ∣y)  (and Csup = 1) set to the dampened e-posterior ¯P [1/2] relative to a prior W (independent  of θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' Because of the dampening with γ = 1/2, we know that ¯Psup ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' We will apply the  proposition with θL < θR such that ¯P [1/2](θL ∣ y) = ¯P [1/2](θR ∣ y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' These must exist (use  (42)), and by (43) they satisfy  D(ˆθ∥θL) = D(ˆθ∥θR) =  D(P (n)  ˆθ  ∥P (n)  W )  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  To verify the conditions of Proposition 4, we will show, using (43), that  D(ˆθ∥θ)exp(−nD(ˆθ∥θ) + D(P (n)  ˆθ  ∥P (n)  W ))  (55)  is increasing for θ < θL and decreasing for θ > θR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' For this, setting C = D(P (n)  ˆθ  ∥P (n)  W ), it is  sufficient to show that g(u) ∶= uexp(−nu + C) is decreasing if u ≥ D(ˆθL∥θL), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content=' if u ≥ C/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  Differentiation gives that g(u) is decreasing if u > 1/n, so Proposition 4 can be applied if  C ≥ 1 and then (54) gives (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
+page_content='  32' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtAzT4oBgHgl3EQfYfxK/content/2301.01335v1.pdf'}
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+Tight Data Access Bounds for Private Top-k Selection
+Hao Wu
+whw4@student.unimelb.edu.au
+The University of Melbourne
+Australia
+Olga Ohrimenko
+oohrimenko@unimelb.edu.au
+The University of Melbourne
+Australia
+Anthony Wirth
+awirth@unimelb.edu.au
+The University of Melbourne
+Australia
+Abstract
+We study the top-k selection problem under the differ-
+ential privacy model: m items are rated according to
+votes of a set of clients. We consider a setting in which
+algorithms can retrieve data via a sequence of accesses,
+each either a random access or a sorted access; the goal
+is to minimize the total number of data accesses. Our
+algorithm requires only O(
+√
+mk) expected accesses:
+to our knowledge, this is the first sublinear data-access
+upper bound for this problem. Accompanying this, we
+develop the first lower bounds for the problem, in three
+settings: only random accesses; only sorted acceses;
+a sequence of accesses of either kind. We show that,
+to avoid Ω(m) access cost, supporting either kind of
+access, i.e. the freedom to mix, is necessary, and that in
+this case our algorithm’s access cost is almost optimal.
+1. Introduction
+We consider the differentially private top-k selection prob-
+lem; there are m items to be rated according to n clients’
+votes. Each client can either vote or not vote for each item,
+and can vote for an unlimited number of items. Since this
+data can be sensitive (e.g., visited websites, purchased items,
+or watched movies), the goal is to identify a set of k items
+with approximately the highest number of votes, while con-
+cealing the votes of individual clients.
+Private top-k selection is a fundamental primitive and under-
+lies a wide range of differentially private machine learning
+and data analytics tasks such as discovering frequent pat-
+terns from data (Bhaskar et al., 2010), training wide neural
+networks (Zhang et al., 2021), tracking data streams (Car-
+doso & Rogers, 2022), false discovery rate control in hy-
+pothesis testing (Qiao et al., 2021), etc.
+In recent years, a significant progress has been made towards
+understanding how accurate the algorithms for this problem
+can be. For example, (Bafna & Ullman, 2017; Steinke &
+Ullman, 2017) provide lower bounds for the problem in
+terms of sample complexity, which can be achieved by a
+number of existing algorithms (Durfee & Rogers, 2019;
+Qiao et al., 2021).
+Another line of research is devoted to improving the effi-
+ciency of the algorithms. Early works such as the peeling
+solution (Bhaskar et al., 2010) need to iterate k times over
+all items. The improved mechanisms (Durfee & Rogers,
+2019; Qiao et al., 2021) iterate over each item only once.
+Since k can be much smaller than m, the research commu-
+nity remains interested in the following question:
+Research Question: Can we develop private top-k se-
+lection algorithms that access only a sublinear number
+of items?
+Although it seems to be an unachievable target, it is possi-
+ble to address this question by considering how items are
+accessed. For example, Durfee & Rogers (2019) consider
+the setting where the data has been pre-processed and re-
+sides in an existing data analytics system, that can return
+the items in sorted order (which we refer to as sorted access
+in our paper). Their top-k algorithm can make a sublinear
+number of accesses at a cost of potentially returning fewer
+than k items, while to guarantee that k items are returned,
+the number of retrieved items can be m. Since retrieving
+information from an existing system incurs corresponding
+query processing and communication cost, it is crucial to
+minimize the number of data accesses.
+In this paper, we systematically investigate the minimum
+number of items an algorithm needs to evaluate (a.k.a. ac-
+cess cost) , in order to answer the private top-k selection
+problem. In addition to sorted access, we also consider
+another common way of accessing items’ data, i.e., the ran-
+dom access, in which an algorithm can actively request the
+data of an arbitrary item.1 Both types of accesses have been
+considered by previous literature for the non-private ver-
+sion of top-k selection problem (see Ilyas et al. (2008) for a
+comprehensive survey).
+Example 1.1. Consider the example of a movie ranking
+1Here random carries the sense of RAM, rather than the out-
+come of a random process.
+arXiv:2301.13347v1  [cs.CR]  31 Jan 2023
+
+Tight Data Access Bounds for Private Top-k Selection
+2
+database. It can present the movies in sorted order, accord-
+ing to their rating by the clients, or it can return directly the
+rating of a specific movie.
+Our Contributions. Our results are threefold. On the upper
+bound side,
+• If the system supports both sorted access and random
+access, we design an algorithm with expected access
+cost O(
+√
+mk).
+To our knowledge, this is the first asymptotically sublin-
+ear bound of the access cost for the private top-k selection
+problem. Our algorithm builds on existing works (Dur-
+fee & Rogers, 2019; Qiao et al., 2021) and inherits their
+error bounds, which are known to be asymptotically opti-
+mal (Bafna & Ullman, 2017; Steinke & Ullman, 2017).
+On the lower bound side,
+• If the system supports either only sorted accesses or only
+random accesses, but not both, we show a lower bound
+of Ω(m).
+• If the system supports both sorted accesses and random
+accesses, we show a lower bound of Ω(
+√
+mk).
+These statements are informal versions of Theorems 5.1, 5.3,
+and 5.4, which impose modest assumptions on the privacy
+guarantee, and relatively weak assumptions on the accuracy
+guarantee of the algorithms. They show that supporting
+sorted and random access to the items’ data simultaneously
+is necessary to break the linear barrier, and the access cost
+of our algorithm is essentially optimal.
+Organization. Our paper is organized as follows. Section 2
+introduces the problem formally. Section 3 discusses the
+preliminaries for our algorithm. Section 4 introduces our
+algorithm. Section 5 presents the lower bounds for the
+problem. Section 6 discusses the related works.
+2. Model Description
+Let C
+.= {1, . . . , m} be a set of m items, and U
+.=
+{1, . . . , n} be a set of n clients. Each client v ∈ U can
+cast at most one vote for each item, and can vote for an
+unlimited number of items. Hence, client v’s votes, denoted
+by ⃗xv, can be viewed as a vector in D .= {0, 1}m, such that
+for each i ∈ C, ⃗xv[i] = 1 if v votes for item i, where ⃗xv[i] is
+the i(th) entry of ⃗xv. We regard the collection of voting vec-
+tors from all n clients as a dataset X = {⃗x1, . . . , ⃗xn} ∈ Dn.
+For each item i ∈ C, let its score ⃗h[i] .= �
+v∈U ⃗xv[i] be
+the number of clients that vote for i. The dataset X can
+be described by its histogram ⃗h .= (⃗h[1], . . . ,⃗h[m]) ∈ Nm.
+We also define π : C → C to be a permutation that puts the
+entries of ⃗h in nonincreasing order2, s.t., ⃗h[π(1)] ≥ · · · ≥
+⃗h[π(m)].
+Our goal is to design a differentially private algorithm that
+returns a set S of k items with (approximately) largest
+scores, while minimizing its data access cost.
+In what follows, we discuss the privacy guarantee, the utility
+guarantee, the data access model of an algorithm formally.
+Privacy Guarantee. We call two datasets X, X ′ neigh-
+boring, denoted by X ∼ X ′, if they differ in addition
+or deletion of one client vector, e.g., X ′ = X ∪ {⃗xn+1}
+or X ′ = X \ {⃗xv} for some v ∈ [n].
+Let ⃗h and ⃗h′ be the histograms corresponding to X and X ′,
+respectively. It is easy to see that if X ∼ X ′, then the score
+of each item can differ by at most 1, i.e., ∥⃗h − ⃗h′∥∞ ≤ 1.
+Hence, for every ⃗h,⃗h′ ∈ Nm, we also call them neighboring
+histograms, written as ⃗h ∼ ⃗h′, if ∥⃗h − ⃗h′∥∞ ≤ 1.
+Let A be an algorithm for the top-k selection problem.
+To protect the voting information of individual clients, we
+would like its output distributions to be similar for neighbor-
+ing inputs, as defined thus.
+Definition 2.1 ((ε, δ)-Private Algorithm (Dwork & Roth,
+2014)). Given ε, δ > 0, a randomized algorithm A : Nm →
+Z is called (ε, δ)-differentially private (DP), if for for ev-
+ery ⃗h,⃗h′ ∈ Nm such that ⃗h ∼ ⃗h′, and all (measurable)
+Z ⊆ Z,
+Pr
+�
+A(⃗h) ∈ Z
+�
+≤ eε · Pr[A(⃗h′) ∈ Z] + δ .
+(1)
+We call ε and δ the privacy parameters. Typically, it is
+required that δ is cryptographically negligible, i.e., δ ≤
+1/mω(1) (Vadhan, 2017; Dwork & Roth, 2014). An algo-
+rithm A is also called ε-DP for short, if it is (ε, 0)-DP.
+Utility Guarantee. In line with previous research (Bafna
+& Ullman, 2017; Durfee & Rogers, 2019), we measure
+the error of an output S by the maximum amount by
+which ⃗h[π(k)] exceeds the score of an item in S, defined
+formally as follows.
+Definition 2.2 ((α, k)-Accuracy). Given a vector ⃗h, param-
+eters k ∈ N+, and α ∈ R+, an output S ⊆ [m] of size k is
+called (α, k)-accurate, if for each i ∈ S, ⃗h[i] ≥ ⃗h[π(k)]−α.
+Data Access. We assume that the histogram ⃗h has been
+preprocessing by an existing data management system, and
+an algorithm A can access ⃗h only through the system. We
+consider two access models that abstract common function-
+alities supported by a system: sorted access and random
+access. Such access models have been widely accepted by
+the community for non-private top-k selection problems
+(see Ilyas et al. (2008) for a survey).
+2We break ties arbitrarily.
+
+Tight Data Access Bounds for Private Top-k Selection
+3
+Sorted Access. Let Cs be the set of items already returned
+by sorted access (initially, Cs = ∅). When a new sorted-
+access request is submitted, the system returns an item-
+score pair (i,⃗h[i]), where i ∈ C \ Cs has the largest score,
+i.e., i = arg maxj∈C\Cs ⃗h[j]. An alternative view is that
+the system returns
+�
+π(1),⃗h[π(1)]
+�
+,
+�
+π(2),⃗h[π(2)]
+�
+, . . . in
+order, one tuple at a time.
+Random Access. A request of random access consists of a
+reference i ∈ C to an item. In response, the system returns
+the corresponding item-score pair (i,⃗h[i]). We emphasize
+that a random access does not imply that i must be a ran-
+domly chosen item.
+Access Cost. Given an algorithm A and a histogram ⃗h,
+the access cost of an algorithm on ⃗h, cost
+�
+A,⃗h
+�
+, is the
+total number of accesses – either sorted or random – to the
+system. Note that this is an upper bound of distinct number
+of entries A learns from ⃗h, as a random access may retrieve
+a previously encountered item-score pair.
+3. Preliminaries
+In this section, we review two building blocks for construct-
+ing our algorithm: a technique for aggregating ranked data
+from multiple sources, and a framework for the existing
+one-shot algorithms for private top-k selection.
+Threshold Algorithm
+The threshold algorithm (Fagin et al., 2003) is a top-k selec-
+tion algorithm, when the information of an item needs to be
+aggregated from multiple resources. In this scenario, there
+are m items, each associated with t attributes. W.L.O.G.,
+assume that each attribute is a real number. Therefore,
+each item i can be represented by a vector ⃗yi ∈ Rt. The
+score of item i is computed by a function f : Rt → R,
+which is assumed to be monotone, s.t., for each ⃗y, ⃗y′ ∈ Rt,
+if ⃗y[j] ≤ ⃗y′[j], ∀j ∈ [t], then f(⃗y) ≤ f(⃗y′).
+The vectors ⃗y1, . . . , ⃗ym do not reside in a single data man-
+agement system, but distributed in t systems L1, . . . , Lt, s.t.,
+for each item i, its jth attribute ⃗yi[j] resides on Lj. Each Lj
+allows for both sorted access and random access. We can
+view it as an array of m tuples Lj[1], . . . , Lj[m], each of
+the form (i, val) ∈ [m] × R, where val equals ⃗yi[j]. The tu-
+ples in Lj are sorted in descending order by their val’s.
+Further, Lj is augmented with an inverted index σj :
+[m] → [m] to support random access, such that, for each an
+item i ∈ [m], Lj[σj(i)] contains the tuple (i, ⃗yi[j]).
+The aim is to identify the top-k items with highest scores
+according to f, while minimizing the access cost, i.e.,
+the total number of data accesses performed by the algo-
+rithm.to L1, . . . , Lt. The algorithm is described in Algo-
+rithm 1.
+Algorithm 1 Threshold Algorithm ATA (Fagin et al., 2003)
+1: Input: Sorted array Lj and inverted index σj, ∀j ∈ [t].
+2: S ← ∅.
+3: repeat
+4:
+Do sorted access in parallel to each of the t array Lj.
+As a tuple (i, val) is seen under sorted access:
+5:
+Retrieve from L1, . . . , Lt by random access (with
+the help of σ1, . . . , σt) all attributes of item i, to
+compute f(⃗yi).
+6:
+If f(⃗yi) is among the-k highest scores seen so far,
+add (i, f(⃗yi)) to S; if |S| > k, remove the tuple
+with lowest score from S .
+7:
+For each Lj, let yj
+.= ⃗yi[j], where i is the last item
+seen in Lj under sorted access.
+8:
+Define the threshold τ .= f(y1, . . . , ym).
+9: until there are k tuples in S with score at least τ.
+10: Return the set of items contained in the tuples in S.
+The correctness of the algorithm is obvious: when the algo-
+rithm stops, since f is monotone, the scores of all unseen
+items are at most τ, which are lower than the score of each
+tuples in S.
+For each tuple (i, val) encountered during sorted access, ATA
+retrieves all entries ⃗yi[j] of ⃗yi by random accesses. This
+seems to be redundant, as i might have been encountered
+previously, and therefore f(⃗yi) has already been evaluated.
+The algorithm ATA performs so, to keep the buffer size
+(memory usage) strictly bounded by O(k). If this is not
+required, ATA can be augmented with an additional data
+structure to keep track of previously evaluated i’s, to avoid
+repeated computation.
+Access Cost. Fagin et al. (2003) did not provide asymptotic
+bound for the access cost. Instead, they proved that ATA
+is instance optimal. Informally, instance optimally implies
+that for every algorithm A which solves the top-k selection
+problem correctly and whose first access to an item must
+be sorted access as opposed to random access, the access
+cost of ATA is at most the access cost of A (up to some
+multiplicative constant). In Section 4, we apply a different
+technique to asymptotically bound the access cost of our
+algorithm.
+One-shot Private Top-k Algorithm
+We review an existing framework for the differentially pri-
+vate top-k selection algorithms (Durfee & Rogers, 2019;
+Qiao et al., 2021). The framework is described in Algo-
+rithm 2. The framework does not consider a specific data
+access model, and instead needs to learn all entries of ⃗h.
+Definition 3.1 (Noise Distributions). Given parameter b ∈
+R, the Laplace distribution, Lap (b), and the Gumbel
+distribution, Gumbel (b), have probability density func-
+
+Tight Data Access Bounds for Private Top-k Selection
+4
+Algorithm 2 Private Top-k Algorithm M
+1: Input: vector ⃗h
+2: for each item i ∈ [m] do
+3:
+⃗v[i] ← ⃗h[i] + Zi, where Zi is an independent noise
+random variable;
+4: end for
+5: Return a set S of k items that the maximizes the ⃗v[i]’s.
+tions p(z) =
+1
+2b · exp
+�
+− |z|
+b
+�
+, ∀z ∈ R, and p(z) =
+1
+b · exp
+�
+−
+� z
+b + exp
+�
+− z
+b
+���
+, ∀z ∈ R, respectively.
+Candidates noise distributions for Zi in Algorithm 2 in-
+clude Lap (1/ε) (Qiao et al., 2021) and Gumbel (1/ε) (Dur-
+fee & Rogers, 2019). The corresponding privacy guarantees,
+are stated as follows.
+Fact 3.2 ((Qiao et al., 2021)). Assume that Zi ∼ Lap (1/ε),
+then M is 2kε-DP. Given δ
+∈
+[0, 0.05], if it holds
+that m ≥ 2 and 8ε
+�
+k log (m/δ) ≤ 0.2, then M also
+satisfies
+�
+8ε
+�
+k log (m/δ), δ
+�
+-DP.
+Fact 3.3 ((Durfee & Rogers, 2019)). Assume that Zi ∼
+Gumbel (1/ε).
+For each δ ∈ [0, 1], M is (ε′′, δ)-DP,
+where ε′′ .= min
+�
+kε, kε
+�
+eε−1
+eε+1
+�
++ ε
+�
+k ln 1
+δ
+�
+.
+Next, we discuss their utility guarantees.
+Fact 3.4. Given β ∈ (0, 1), if the Zi ∼ Lap (1/ε),
+or Gumbel (1/ε), then with probability at least 1 − β, the
+returned solution by Algorithm 2 is (α, k)-accurate, for
+α ∈ O
+�
+ln (m/β)
+ε
+�
+.
+Remark 3.5. Compared to Laplace noise, the Gumbel noise
+allows the algorithm to return a ranked list of indices, in-
+stead of a set which contains on order information. For
+consistency of presentation, we assume that Algorithm 2
+returns a set for both choices.
+4. Sublinear Access and Time Algorithm
+In this section, we present an algorithm for top-k selec-
+tion problem, which achieves optimal privacy utility trade-
+off, and with high probability, has an expected access
+cost O(
+√
+mk) and computation time O(
+√
+mk log log m).
+Our presentation follows two steps: we first present a straw-
+man algorithm with sublinear access cost but only linear
+computation; next we show how to improve its time com-
+plexity to O(
+√
+mk log log m).
+4.1. A Strawman Approach
+A natural idea is to combine the threshold algorithm ATA
+with the oneshot private top-k algorithm. Each item i ∈ [m]
+now has two attributes, namely, ⃗h[i] and Zi, where Zi is
+sampled independently from Lap (1/ε) or Gumbel (1/ε).
+Since the histogram ⃗h is stored in a database management
+system, which allows for two types of access: sorted access
+and random access, we can think of this as a sorted array L1
+of m tuples, each of the form (i, val) ∈ [m] × R, where val
+equals ⃗h[i].The tuples in L1 are sorted in descending or-
+der by their val’s. Further, L1 has an inverted index σ1 to
+support random access.
+Additionally, we can construct another sorted array L2 of m
+tuples, each of the form (i, val) ∈ [m] × R, where val
+equals Zi. The tuples in L2 are also sorted in descending
+order by their val’s. L2 also has an inverted index σ2 to
+support random access.
+Then we can run the algorithm ATA, with input L1, L2,
+σ1, σ2, and an aggregating function f(⃗h[i], Zi) .= ⃗h[i]+Zi.
+It is easy to see that f is monotone. The pseudo-code is in
+Algorithm 3.
+Algorithm 3 Private Threshold Algorithm APrivTA
+1: Let I1 = 1, I2 = 2, . . . , Im = m.
+Generate m
+tuples (I1, Z1), . . . , (Im, Zm), where the Zi’s are
+i.i.d. random variables; sort the tuples in descend-
+ing order by the values of the Zi’s, denote the sorted
+sequence by
+�
+I(1), Z(1)
+�
+, . . . ,
+�
+I(m), Z(m)
+�
+, and store
+this sequence in an array L2; construct σ2 : [m] → [m],
+s.t., L2[σ2(i)] = (i, Zi) for each item i ∈ [m].
+2: Run Algorithm 1 on input L1, L2, σ1, σ2, with an ag-
+gregation function f
+�
+(⃗h[i], Zi)
+� .= ⃗h[i] + Zi;
+Privacy and Utility Guarantee. The privacy and utility guar-
+antee of the algorithm inherits directly from Algorithm 2.
+Access and Time Complexity. It is easier to first discuss
+the time complexity and then the access cost. Generating
+the random variables takes O(m) time, and sorting them
+takes O(m log m) time. Hence the total running time is
+bounded by O(m log m).
+It remains to discuss the number of accesses the algorithm
+performs on L1. Our analysis relies on the following impor-
+tant observation.
+Lemma 4.1. The I(1), . . . , I(m) are distributed uniformly
+over all possible permutations over [m], and are indepen-
+dent of the random variables Z(1), . . . , Z(m).
+Intuitively, the claim holds since each Zi in Algorithm 3
+follows the same distribution independently. The proof of
+the lemma is included in Appendix B.1.
+Theorem 4.2. The expected access cost of Algorithm 3
+on L1, E [cost (APrivTA, L1)], is bounded by O
+�√
+mk
+�
+.
+The proof of the Theorem is in Appendix B.1.
+
+Tight Data Access Bounds for Private Top-k Selection
+5
+4.2. An Online Sampling Approach
+Pre-generating all m noise values may be excessive. For
+problems with small values of k, e.g., k = 10, the Algo-
+rithm 3 may need to know only a small subset of tuples in L2.
+It is of interest whether we can also reduce the number of
+noisy random variables generated to O(
+√
+mk log log m), by
+constructing the L2 (and σ2) on the fly. The main result of
+this section is stated as follows.
+Theorem 4.3. There is an algorithm Aoracle, that,
+• does not require to pre-generate L2;
+• answers sorted access and random access query to L2
+in O(log log m) time in expectation.
+Further, the tuples returned by Aoracle have the same
+marginal distribution as those generated by Algorithm 3.
+Algorithm 4 Algorithm Aoracle
+Initialization
+1: J ← ∅, idx ← 0;
+2: L2[i] ← nil, σ2(i) ← nil, ∀i ∈ [m]
+Sorted Access
+1: idx ← idx + 1;
+2: if idx /∈ J then
+3:
+Sample I(idx) uniformly from [m] \ I(J );
+4:
+Invoke Aord-stat to sample Z(idx);
+5:
+σ2(I(idx)) ← idx;
+6:
+L2[idx] ← (I(idx), Z(idx)); J ← J ∪ {idx}.
+7: end if
+8: return L2[idx]
+Random Access (Input: item i ∈ [m])
+1: if σ2(i) = nil then
+2:
+Sample j uniformly from [m] \ J ;
+3:
+Invoke Aord-stat to sample Z(j);
+4:
+σ2(i) ← j;
+5:
+L2[j] ← (i, Z(j)); J ← J ∪ {j}.
+6: end if
+7: return L2[σ2(i)]
+There are two key ingredients for constructing Aoracle.
+Sampling the I(j)’s. The first ingredient is Lemma 4.1,
+which allows Aoracle to sample the I(j) and the Z(j) inde-
+pendently according to their marginal distributions, without
+changing the joint distribution of the I(j) and the Z(j). The
+lemma states that the I(j)’s are distributed uniformly over
+all possible permutations over [m]. It is not hard to sample
+an I(j) on the fly: let J be the set of indexes such that the
+values of I(j′), j ∈ J have been determined; if j /∈ J ,
+then I(j) just distributes uniformly over the subset of unseen
+items, i.e., [m] \ I(J ), where I(J ) .=
+�
+I(j′) : j′ ∈ J
+�
+. Cor-
+respondingly, we can also construct the inverted index σ2 on
+the fly: given an item i ∈ [m], if it has not been encountered,
+then σ2(i) should equal one of the undetermined indexes,
+namely [m] \ J , uniformly at random.
+Sampling the Z(j)’s.
+The second ingredient is an al-
+gorithm Aord-stat which generates the Z(j)’s on the
+fly.Formally, for each J
+⊆
+[m], define Z(J )
+.=
+(Z(j), j ∈ J ), and let z(J ) refer to a vector (z(j), j ∈ J ) ∈
+R|J |. Denote by pZ(J )(·) the marginal density of Z(J ), in-
+duced by the generating procedure of Algorithm 3. Call z(J )
+a feasible realization of Z(J ), if pZ(J )(z(J )) > 0. Given
+such a feasible realization, let pZ(j)|Z(J )(· | z(J )) be the
+density function of Z(j), conditioned on Z(J ) = z(J ). The
+property of Aord-stat is stated as follows.
+Lemma 4.4. For each J ⊆ [m] s.t., J ̸= [m], each j ∈
+[m] \ J , and each feasible realization z(J ) of Z(J ),
+Aord-stat samples a random variable with the conditional
+density pZ(j)|Z(J )(· | z(J )) in O(log log m) expected time.
+The proof of the Lemma is discussed in Section 4.2.1 . Now,
+we return to the construction of Aoracle. The algorithm is
+described in Algorithm 4.
+Initialization. The algorithm creates an empty array L2 and
+an empty inverted index σ2. Further, it creates a set J , to
+record the positions of L2 which are already sampled, and
+a variable idx, to record the last visited position by sorted
+access. In practice, L2 and σ2 need not to be physically ini-
+tialized, and can be implemented by hash sets with constant
+initialization time.
+Sorted Access. Indeed, it is trivial to handle the sorted ac-
+cess. We just maintain an index, idx ∈ N, of last tuple
+returned by sorted access. When a new request of sorted
+access arrives, we increase idx by 1. Let J be the set of
+indexes of tuples in L2 that have already been generated.
+If idx ∈ J , then Aoracle returns L2[idx] directly; other-
+wise, it generates L2[idx] before returning it.
+Random Access. A random access request comes with a
+reference to an item i ∈ [m]. We need to identify the
+tuple L2[j] = (I(j), Z(j)) s.t., I(j) = i. There are two cases:
+if item i has been encountered previously (σ2(i) ̸= nil),
+then Aoracle returns L2[σ2(i)] directly; otherwise, Aoracle
+randomly pick an index j from [m] \ J , and set I(j) ←
+i, σ2(i) ← j, and calls Aord-stat to generate Z(j). Finally,
+it returns L2[j].
+4.2.1. SAMPLING ORDERED NOISES
+In this section, we prove Lemma 4.4. Deciding the condi-
+tional distribution of the Z(j)’s and sampling them directly
+from such distribution can be non-trivial. As a result, we
+follow the three-step approach outlined below:
+• First,
+we
+show
+that
+Z1, . . . , Zm
+can
+be
+con-
+verted
+from
+a
+sequence
+of
+i.i.d.
+uniform
+ran-
+dom variables U1, . . . , Um, and that the sorted se-
+
+Tight Data Access Bounds for Private Top-k Selection
+6
+Figure 1. A pictorial illustration for Lemma 4.6.
+quence Z(1), . . . , Z(m) of Z1, . . . , Zm can therefore be
+transferred from the sorted sequence U(1), . . . , U(m)
+of U1, . . . , Um.
+• Next, to avoid generating the entire sequence of random
+variables, we study the distribution of U(j), conditioned
+on a set of U(j′) which have already been sampled.
+• Finally, we show how to sample an U(j) from such dis-
+tribution in O(log log m) expected time.
+Transform U(j) to Z(j). Since all potential noise distribu-
+tions (Lap (1/ε) or Gumbel (1/ε)) of the Zi’s (the noise
+random variables, before sorting) have continuous cumula-
+tive distribution function, we can sample them indirectly via
+uniform random variables, based on the inversion method.
+Fact 4.5 (Inversion Method (Devroye, 1986)). Let F be
+a continuous cumulative distribution function on R with
+inverse F −1 defined by
+F −1(u) .= inf {x : F(x) = u, 0 < u < 1} .
+If U is a uniform [0, 1] random variable, then F −1(U) has
+distribution function F.
+Let F be the cumulative distribution function of the Zi,
+and U1, . . . , Um be independent uniform random variables
+on [0, 1]. We compare the two post-processing procedures:
+• By Fact 4.5, we can obtain Z1, . . . , Zm by com-
+puting
+F −1(U1), . . . , F −1(Um);
+then
+we
+ob-
+tain Z(1), . . . , Z(m)
+by sorting this sequence in
+descending order.
+• Alternatively, we first sort U1, . . . , Um in descending
+order, to obtain a sequence U(1), . . . , U(m); then we
+compute F −1(U(1)), . . . , F −1(U(m)).
+Lemma 4.6. The two procedures output the same sequence.
+The proof of the lemma is included in Appendix B.2.
+Based on the lemma, we can sample U(j) first then com-
+pute F −1(U(j)) to generate Z(j). It reduces to study the
+distribution of U(j) (conditioned on the subset of U(j′)’s
+which have been sampled), and algorithm for sampling a
+random variable efficiently from such distribution.
+Distribution of U(j).
+Recall that J is the set of in-
+dexes which have been queried already . Denote U(J ) .=
+�
+U(j′) : j′ ∈ J
+�
+a shorthand for the order statistics
+that have been sampled.
+Further,
+write u(J )
+.=
+�
+u(j′) : j′ ∈ J
+�
+∈
+[0, 1]|J | as a set of numbers
+within [0, 1], indexed by J . Call u(J ) a feasible realiza-
+tion of U(J ), if for each j, j′ ∈ J s.t. j ≤ j′, it holds
+that u(j) ≥ u(j′).
+Given a new query index j ∈ [m] \ J , we are interested in
+the conditional probability density, pU(j)|U(J )(u(j) | u(J )),
+of U(j), given the occurrence of a feasible realization u(J )
+of U(J ). For ease of reading, we omit the subscripts of the
+conditional probability densities, whenever their meaning
+can be unambiguously determined from their parameters.
+Depending on the relative position of j w.r.t. the indexes
+in J , we consider the following three cases:
+• J is empty. It reduces to study the un-conditional proba-
+bility density p(u(j)) of U(j).
+• J is not empty, and j is greater than the largest index
+in J ; in this case, j has a predecessor (the largest index
+that is smaller than j), denoted by ℓ, in J .
+• J is not empty, and j is smaller than the largest index
+in J ; in this case, j has both a predecessor, denoted by ℓ,
+and a successor (the smallest index that is larger than j),
+denoted by r, in J .
+Hereafter, if J ̸= ∅, we consider only feasible realiza-
+tion u(J ) of U(J ).
+Theorem 4.7. (1) If J is empty, then the density p(u(j))
+of U(j) is given by: ∀ u(j) ∈ [0, 1],
+p(u(j)) =
+m!
+(j−1)!(m−j)! · (1 − u(j))j−1(u(j))m−j.
+(2)
+(2) If J is not empty, and j is greater than the largest
+index in J , then given U(ℓ) = u(ℓ), U(j) is indepen-
+dent of all other random variables U(j′) for all j′ ∈
+J \ {ℓ}, i.e., p
+�
+u(j) | u(J )
+�
+= p
+�
+u(j) | u(ℓ)
+�
+; further, for
+each u(j) ∈ [0, u(ℓ)],
+p
+�
+u(j) | u(ℓ)
+�
+=
+(m−ℓ)!
+(j−ℓ−1)!(m−j)!·
+� u(ℓ)−u(j)
+u(ℓ)
+�j−ℓ−1� u(j)
+u(ℓ)
+�m−j
+1
+u(ℓ) .
+(3)
+(3) If J is not empty, and j is smaller than the largest in-
+dex in J , then given U(ℓ) = u(ℓ) and U(r) = u(r), U(j) is
+independent of all other random variables U(j′) for all j′ ∈
+J \ {ℓ, r}, i.e., p
+�
+u(j) | u(J )
+�
+= p
+�
+u(j) | u(ℓ), u(r)
+�
+; fur-
+ther, for each u(j) ∈ [u(r), u(ℓ)],
+p
+�
+u(j) | u(ℓ), u(r)
+�
+=
+(r−ℓ−1)!
+(j−ℓ−1)!(r−j−1)!·
+� u(ℓ)−u(j)
+u(ℓ)−u(r)
+�j−ℓ−1� u(j)−u(r)
+u(ℓ)−u(r)
+�r−j−1
+1
+u(ℓ)−u(r) .
+(4)
+The theorem removes the dependency of U(j) from all but
+at most two variables in U(J ). The detailed proof is in
+the Appendix B.3. We provide an informal but intuitive
+
+sort
+sort
+F-1()Tight Data Access Bounds for Private Top-k Selection
+7
+explanation of the conditional probability densities. Take
+the Equation 4 for example. Conditioned on U(r) = u(r)
+and U(ℓ) = u(ℓ), r − ℓ − 1 uniform random variables fall
+into the interval [u(r), u(ℓ)]. Of these r − ℓ − 1 random
+variables, j − ℓ − 1 of them are ≥ ui, and r − j − 1 of them
+are < ui. The number of possible combinations is given
+by
+(r−ℓ−1)!
+(j−ℓ−1)!(r−j−1)!. For a fixed combination, the former
+happens with probability
+� u(ℓ)−u(j)
+u(ℓ)−u(r)
+�j−ℓ−1
+, the latter hap-
+pens with probability
+� u(j)−u(r)
+u(ℓ)−u(r)
+�r−j−1
+, and the probability
+density of U(j) = u(j) is
+1
+u(ℓ)−u(r) .
+Sampling U(j). We now discuss how to sample the U(j)
+efficiently from their conditional distributions. First, note
+that determining the conditional distributions may need to
+find the index j’s predecessor or successor in J . This can
+be done by Van Emde Boas tree (van Emde Boas, 1975)
+in O(log log m) time. Next, we show that sampling from
+such conditional distributions takes O(1) expected time.
+Specifically, we will sample random variables with Beta
+distributions, then convert them into ones which follow
+desired conditional distributions.
+Definition 4.8 (Beta Distribution (Ross, 2018)). The beta
+distribution Beta (α, β) is a distribution defined on [0, 1]
+whose density is given by
+f(x) = xα−1(1−x)β−1
+B(α,β)
+,
+∀x ∈ [0, 1],
+(5)
+where α, β
+>
+0 are shape parameters, B(α, β)
+.=
+� 1
+0 xα−1(1 − x)β−1 dx is a normalisation constant.
+It is known that, when α ≥ 1, β ≥ 1, a random vari-
+able X ∼ Beta (α, β) can be generated in O(1) expected
+time (Devroye, 1986; Gentle, 2009).
+Theorem 4.9. Assume that ℓ < j < r ≤ m, and 1 ≥
+u(ℓ) > u(r) ≥ 0. Then
+1. If X ∼ Beta (m − j + 1, j), then the density function
+of X is the same as Equation (2).
+2. If X ∼ Beta (m − j + 1, j − ℓ), then the density func-
+tion of Y .= u(ℓ) · X is the same as Equation (3).
+3. If X ∼ Beta (r − j, j − ℓ), then the density function
+of Y .= (u(ℓ) − u(r)) · X is the same as Equation (4).
+The proof of the Theorem is included in Appendix B.2.
+5. Lower Bounds
+In this section, we generate the lower bounds for the prob-
+lem. Following the setting in Section 2, since⃗h is the sum of
+voting vectors of n clients, we have ||⃗h||∞ ≤ n. It follows
+that an arbitrary subset S ⊆ [m] of size k is (n, k)-accurate.
+All lower bounds in this section hold for algorithms that
+are (n − O(1), k)-accurate, which is just slightly better than
+the trivial error guarantee.
+5.1. Random Access
+Theorem 5.1. Assume that 0 ≤ β < 0.1. Let A be an
+algorithm that has only random access to ⃗h, does not return
+items which it has not seen , and for each input, with prob-
+ability at least 1 − β, returns a solution that is (n − 1, k)-
+accurate. Then there exists a family of histograms H, and
+a distribution µ on H, if ⃗h is sampled from H according to
+distribution µ, it holds that
+E
+⃗h
+�
+cost
+�
+A,⃗h
+��
+∈ Ω(m).
+(6)
+Note that the theorem does not even require A to be a (ε, δ)-
+DP algorithm. The proof of the theorem is in Appendix C.1.
+At a high level, our construction focuses on a family of
+histograms the values of whose entries are either n or 0.
+Further, if an ⃗h is sampled from H, there are roughly 2k
+entries of ⃗h that have value n, and those entries appear at
+random positions of ⃗h, so that A is unlikely to identify more
+than k of them, before learning the values of Ω(m) entries.
+5.2. Sorted Access
+The lower bound for the sorted access case relies on the
+following lemma.
+Lemma 5.2. Let A be an (ε, δ)-DP algorithm, which for
+each input histogram, with probability at least 1−β, returns
+a solution that is (n − 2, k)-accurate. Let S ⊆ [m], and ⃗hS
+be a histogram, s.t.
+⃗hS[i] .=
+�
+n − 1,
+∀ i ∈ S,
+0,
+∀ i ∈ ¯S,
+(7)
+where ¯S .= [m] \ S. Let SL be a subset of S of size |S|/k
+sampled uniformly at random (without replacement), SH .=
+S \ SL, and ⃗hSH,SL be a histogram neighboring to ⃗h s.t.
+⃗hSH,SL[i] .=
+�
+�
+�
+�
+�
+n,
+∀ i ∈ SH,
+n − 1,
+∀ i ∈ SL,
+0,
+∀ i /∈ S.
+(8)
+Then,
+PrSL,A
+�
+A(⃗hSH,SL) ∩ SL ̸= ∅
+�
+≥ 1−β−δ−e−1
+eε
+,
+(9)
+where the randomness is first over the choice of SL then
+over the output of A.
+The proof of the Lemma is included in Appendix C.2. Note
+that, for each i ∈ SL,⃗hSH,SL[i] is among the |S|/k smallest
+entries of the |S| largest entries in ⃗hSH,SL. Informally, the
+lemma states that the probability that output of A contains
+some item i ∈ SL is not too “small”.
+Theorem 5.3. Let ε, δ, β be non-negative parameters,
+s.t., ε ∈ O(1), δ+β ≤ 0.6. Let A be an (ε, δ)-DP algorithm
+that has only sorted access, does not return items which it
+has not seen, and for each input histogram, with probability
+
+Tight Data Access Bounds for Private Top-k Selection
+8
+at least 1 − β, returns a solution that is (n − 2, k)-accurate.
+Then there exists a family of histograms H so that, if ⃗h is
+sampled uniformly at random from H, it holds that
+E
+⃗h
+�
+cost
+�
+A,⃗h
+��
+∈ Ω(m).
+(10)
+Proof. Let S = [m/2], SL be a uniform random subset
+of S of size |S|/k, and ⃗hSH,SL be a histogram built as
+outlined in Equation (8). Let H be the collection of all
+possible outcomes of ⃗hSH,SL. Then by Lemma 5.2,
+PrSL,A
+�
+A(⃗hSH,SL) ∩ SL ̸= ∅
+�
+≥ 1−β−δ−e−1
+eε
+.
+But for each i
+∈
+SL, ⃗hSH,SL[i] is among the
+top (|S| − |S|/k + 1)(th)
+to |S|(th)
+largest numbers
+in ⃗hSH,SL. Since A has only sorted access to ⃗hSH,SL and
+does not return an item which it has not seen, if it returns an
+item in SL, it needs to invoke at least |S| − |S|/k ∈ Ω(m)
+sorted accesses. It follows that the expected access cost
+of A is at least 1−β−δ−e−1
+eε
+· Ω(m). Inequality (10) follows
+from the assumption that ε ∈ O(1), and β + δ ≤ 0.6.
+5.3. Random and Sorted Access
+Theorem 5.4. Let ε, δ, β be non-negative parameters,
+s.t., ε ≤ 1, δ + β ≤ 0.05. Let A be an algorithm that has
+both sorted access and random access to ⃗h, does not return
+items which it has not seen, and for each input, with prob-
+ability at least 1 − β, returns a solution that is (n − 2, k)-
+accurate. Then there exists a family of histograms H, if ⃗h is
+sampled uniformly at random from H, it holds that
+E
+⃗h
+�
+cost
+�
+A,⃗h
+��
+∈ Ω
+�√
+mk
+�
+.
+(11)
+Proof. Let H be the collection of all possible ⃗hSH,SL gen-
+erated as follows:
+• First, sample a subset S ⊆ [m] of size τ .=
+√
+mk without
+replacement uniformly at random.
+• Then we randomly sample a uniform subset SL of S
+of size |S|/k, and construct a histogram ⃗hSH,SL as de-
+scribed by Equation (8).
+Since Lemma 5.2 holds for each S ⊆ [m], we have
+PrS,SL,A
+�
+A
+�
+⃗hSH,SL
+�
+∩ SL ̸= ∅
+�
+≥
+1
+eε
+�
+1 − β − δ − 1
+e
+�
+(a)
+≥ e−1�
+1 − 0.05 − e−1�
+≥ 0.21,
+where the randomness is first over the choice of S, then
+over the choice of SL, and finally over the output of A,
+and inequality (a) follows from the assumption that ε ≤ 1
+and β + δ ≤ 0.05. In what follows, we omit the sub-
+scripts S, SL, A from the probability notations, when the
+source of randomness is clear from the context.
+Consider the event E : A accesses some item i ∈ SL. As A
+does not return an item which it has not seen, the event E is
+a necessary condition for A
+�
+⃗hSH,SL
+�
+∩ SL ̸= ∅. Hence,
+Pr [E] ≥ Pr
+�
+A
+�
+⃗hSH,SL
+�
+∩ SL ̸= ∅
+�
+.
+Let η .= τ/20. We decompose E into two mutually ex-
+clusive events: E1 : A accesses some item i ∈ SL for the
+first time within η access operations; E2 : A accesses some
+item i ∈ SL for the first time after η access operations.
+Then Pr [E] = Pr [E1] + Pr [E2].
+Lemma 5.5. The probability that A accesses some item i ∈
+SL for the first time within η access operations, denoted
+by Pr [E1], is upper bounded by Pr [E1] ≤ 0.19.
+The proof of Lemma 5.5 is omitted here, and is included
+in Appendix C.3. Intuitively, the lemma holds since: 1) A
+can not access some i ∈ SL within η sorted accesses; 2)
+because of the way it is generated, SL is a random subset
+from [m] of size
+√
+mk/k, hence it is also unlikely for A
+to come across some i ∈ SL with at most η =
+√
+mk/20
+random access. To conclude the justification of (11), and
+hence prove the Theorem, we apply Lemma 5.5.
+Pr [E2]
+= Pr [E] − Pr [E1]
+≥ Pr
+�
+A
+�
+⃗hSH,SL
+�
+∩ SL ̸= ∅
+�
+− Pr [E1]
+≥ 0.21 − 0.19 ∈ Ω(1).
+But when E2 happens, the access cost is Ω(η). Therefore,
+the expected access cost of A is lower bounded by Ω(η) =
+Ω(
+√
+mk).
+6. Related Work
+Private Selection. The private top-1 selection problem is a
+special case of the private top-k problem. The latter has been
+studied extensively, e.g., the exponential mechanism (Mc-
+Sherry & Talwar, 2007), report noisy max (Dwork & Roth,
+2014), permute-and-flip (McKenna & Sheldon, 2020; Ding
+et al., 2021). Of interest is the permute-and-flip mecha-
+nism: when the largest score of the items is known a prior ,
+the mechanism can potentially terminate without iterating
+over all m items. However, in this scenario, an asymptotic
+upper bound for the number of items evaluated remains
+unresolved.
+Private Top-k Mechanisms. Bhaskar et al. (2010) were the
+first to apply the “peeling exponential mechanism”, which
+iteratively invoked the exponential mechanism to select the
+item with highest score, then remove it. They also proposed
+an oneshot Laplace mechanism for private top-k selection.
+Bhaskar et al. (2010) analyzed the pure differential privacy
+guarantees of both algorithms. Subsequently, Durfee &
+Rogers (2019) showed that the peeling exponential mech-
+anism has an equivalent oneshot implementation (i.e., Al-
+gorithm 2 with Gumbel noise), and studied its approximate
+
+Tight Data Access Bounds for Private Top-k Selection
+9
+privacy guarantee. Qiao et al. (2021) provided the approxi-
+mate privacy guarantee for the oneshot Laplace mechanism,
+without the help of the composition theorem.
+Both Bhaskar et al. (2010) and Durfee & Rogers (2019) have
+proposed private algorithms which estimate top-k based on
+the true top ¯k items for some ¯k ≥ k. Given an integer k,
+both algorithm may need to set ¯k = m, in order to return k
+items.
+Accuracy Lower Bound.
+Bafna & Ullman (2017)
+and Steinke & Ullman (2017) show that, for approximate
+private algorithms, the error guarantees of existing algo-
+rithms (McSherry & Talwar, 2007; Dwork & Roth, 2014;
+Durfee & Rogers, 2019; Qiao et al., 2021) are essentially
+optimal.
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+978-3-319-57048-8\ 7. URL https://doi.org/
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+Zhang, H., Mironov, I., and Hejazinia, M.
+Wide
+network learning with differential privacy.
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+abs/2103.01294, 2021. URL https://arxiv.org/
+abs/2103.01294.
+
+Tight Data Access Bounds for Private Top-k Selection
+11
+A. Probability Inequalities
+Fact A.1 (Chernoff Bound, (Mitzenmacher & Upfal 2017, Theorem 4.4 and 4.5)). Let X1, X2, . . . , Xn be independent
+Poisson trials such that, for i ∈ [m], Pr [Xi = 1] = pi, where 0 < pi < 1. Then, for X = �n
+i=1 Xi, µ = E [X],
+Pr [X ≥ (1 + λ)µ] ≤
+�
+eλ
+(1 + λ)1+λ
+�µ
+,
+∀λ > 0,
+(12)
+Pr [X ≥ (1 + λ)µ] ≤
+�
+eλ
+(1 + λ)1+λ
+�µ
+≤ e−λ2µ/3,
+∀λ ∈ (0, 1],
+(13)
+Pr [X ≤ (1 − λ)µ] ≤
+�
+e−λ
+(1 − λ)1−λ
+�µ
+≤ e−λ2µ/2,
+∀λ ∈ (0, 1).
+(14)
+The known concentration inequalities for sampling with replacement can be transferred to the case of sampling without
+replacement, based on a notable reduction technique.
+Fact A.2 ((Hoeffding, 1994) ). Let X = (x1, . . . , xN) be a finite population of N real points, Y1, . . . , Yn denote a random
+sample without replacement from X and X1, . . . , Xn denote a random sample with replacement from X. If f : R → R is
+continuous and convex, then
+E
+�
+f
+� n
+�
+i=1
+Yi
+��
+≤ E
+�
+f
+� n
+�
+i=1
+Xi
+��
+.
+In particular, the lower bound presented in Fact A.1 can be converted into the following one, by combining its proof
+in (Mitzenmacher & Upfal, 2017) and Fact A.2.
+Corollary A.3 (Chernoff bound). Let X = (x1, . . . , xN) ∈ {0, 1}N be a finite population of N binary points and
+Y1, . . . , Yn be a random sample drawn without replacement from the population. Then, for Y = �n
+i=1 Yi,
+Pr
+� n
+�
+i=1
+Yi ≤ (1 − λ)µ
+�
+≤
+�
+e−λ
+(1 − λ)1−λ
+�µ
+≤ e−λ2µ/2,
+∀λ ∈ [0, 1).
+(15)
+where µ = np is the expectation of �n
+i=1 Yi, and p .= 1
+N
+�N
+i=1 xi is the mean of X.
+Note that, in Corollary A.3, we also extend the range of λ from {0, 1}, to [0, 1). When λ = 0,
+�
+e−λ
+(1−λ)1−λ
+�µ
+= e−λ2µ/2 = 1,
+and Inequality (15) holds trivially.
+Fact A.4 ((Blitzstein & Hwang, 2014)). Let Γ(a)
+.=
+� ∞
+0
+xa−1e−x dx, ∀a > 0 be the Gamma function.
+It holds
+that Γ(a + 1) = aΓ(a), and Γ( 1
+2) = √π. Then for each k ∈ N+, we have Γ(k) = (k − 1)! and
+Γ
+�1
+2 + k
+�
+=
+�
+k − 1
+2
+��
+k − 3
+2
+�
+· · · 1
+2Γ
+�1
+2
+�
+= 2k − 1
+2
+2k − 3
+2
+· · · 1
+2
+√π = (2k)!
+4kk!
+√π.
+(16)
+Fact A.5 ((Ross, 2018)). Given shape parameters α, β > 0, define beta function B(α, β) .=
+� 1
+0 xα−1(1 − x)β−1 dx. Then
+B(α, β) = Γ(α)Γ(β)
+Γ(α+β) , In particular,
+B(α, β) = (α − 1)!(β − 1)!
+(α + β − 1)!
+,
+∀α, β ∈ N+.
+(17)
+The factorials can be estimated as follows.
+Fact A.6 (Stirling’s Approximation (Robbins, 1955; Matousek & Nesetril, 2009)). For k = 1, 2, ...
+√
+2πk
+�k
+e
+�k
+exp
+�
+1
+12k + 1
+�
+≤ k! ≤
+√
+2πk
+�k
+e
+�k
+exp
+� 1
+12k
+�
+.
+(18)
+
+Tight Data Access Bounds for Private Top-k Selection
+12
+B. Proofs for Section 4
+B.1. Proofs for Section 4.1
+Proof for Lemma 4.1. For each i ∈ [m], let pZi(·) be the density function of random variable Zi. Since the Zi’s are i.i.d.
+random variables, they share the same density function, i.e., pZ1(·) = · · · = pZm(·).
+Let Sm be the collection of all possible permutations over [m], and s refer to a permutation in Sm . Further, let z(1) ≥
+. . . ≥ z(m) denote a possible realization of Z(1), . . . , Z(m), and z(1:m) .= (z(1), . . . , z(m)). We write I(1:m) = s, if I(j) =
+s(j), ∀j ∈ [m], and Z(1:m) = z(1:m), when Z(j) = z(j), ∀j ∈ [m].
+Let pI(1:m),Z(1:m)(s, z(1:m)) be the probability density when I(1:m) = s and Z(1:m) = z(1:m). The probability den-
+sity, pZ(1:m)(z(1:m)) of Z(1:m) = z(1:m) is given by
+pZ(1:m)(z(1:m)) =
+�
+s∈Sm
+pI(1:m),Z(1:m)(s, z(1:m))
+=
+�
+s∈Sm
+�
+j∈[m]
+pZs(j)(z(j))
+=
+�
+s∈Sm
+�
+j∈[m]
+pZ1(z(j))
+= m!
+�
+j∈[m]
+pZ1(z(j)).
+Hence, for a given s ∈ Sm, the probability density of I(1:m) = s, conditioned on Z(1:m) = z(1:m), is given by
+pI(1:m)|Z(1:m)(s | z(1:m)) =
+pI(1:m),Z(1:m)(s, z(1:m))
+pZ(1:m)(z(1:m))
+=
+�
+j∈[m] pZ1(z(j))
+m! �
+j∈[m] pZ1(z(j)) = 1
+m! ,
+which is independent of the values of the Z(1:m). Finally,
+pI(1:m)(s) =
+E
+Z(1:m)
+�
+pI(1:m)|Z(1:m)(s | Z(1:m))
+�
+= 1
+m!.
+Lemma B.1. Let cost (APrivTA, L1) be the access cost of Algorithm 3.
+Pr [cost (APrivTA, L1) ≥ 2 · r] ≤ ek
+kk · (r2/m)
+k
+er2/m ,
+∀r ∈ N, s.t., r ≥
+√
+mk.
+(19)
+Proof for Lemma B.1. Let i(1), . . . , i(r) ∈ [m] be the top-r items with highest scores in ⃗h. Also consider the items in the
+first r tuples in the array L2, denoted by I(1:r) =
+�
+I(1), . . . , I(r)
+�
+.
+Consider the following event:
+E .=
+���
+i(1), . . . , i(r)
+�
+∩ I(1:r)
+�� > k .
+We claim that
+1. When event E happens, cost (APrivTA, L1) < 2r.
+2. The complement of E, denoted by ¯E, happens with probability Pr
+� ¯E
+�
+≤ ek
+kk · (r2/m)k
+er2/m .
+Combing both claims, we get
+Pr [cost (APrivTA, L1) ≥ 2r] ≤ Pr
+� ¯E
+�
+≤ ek
+kk · (r2/m)
+k
+er2/m .
+
+Tight Data Access Bounds for Private Top-k Selection
+13
+Claim One. It suffices to show that, when event E happens, Algorithm 3 stops before r rounds of parallel sorted access. In
+such a case, the number of random accesses incurred is less than r. Therefore, the total number of access cost to ⃗h is less
+than 2r.
+Assume that Algorithm 1 invokes r rounds of parallel sorted accesses. Since i(r) and I(r) are the last encountered items, the
+corresponding threshold (Algorithm 1, Line 1) is given by τr .= f(⃗h[i(r)], ZI(r)).
+But for each i ∈
+�
+i(1), . . . , i(r)
+�
+∩ I(1:r), it holds that ⃗h[i] ≥ ⃗h[i(r)], and Zi ≥ ZI(r). Since f is monotone, the score of
+item i, f(⃗h[i], Zi), is at least τr. Since event E happens, at least k items encountered by the algorithm have score at least τ.
+Therefore, the stopping condition of Algorithm 1 should be satisfied.
+Claim Two. Via Lemma 4.1, the I(1), . . . , I(m) distribute uniformly over all permutations of [m]. Therefore, the set I(1:r)
+can be viewed as a uniform sample (without replacement) of r elements from [m]. Hence, for each j ∈ [r], the probability
+that i(j) belongs to I(1:r) is given by
+Pr
+�
+i(j) ∈ I(1:r)
+�
+= r/m .
+Let 1[i(j)∈I(1:r)] be the indicator for the event that i(j) ∈ I(1:r), and S .= �
+j∈[r] 1[i(j)∈I(1:r)]. The event ¯E is equivalent
+to S ≤ k. Observe that
+Pr
+�
+1[i(j)∈I(1:r)] = 1
+�
+= Pr
+�
+i ∈ I(1:r)
+�
+= r/m,
+µ .= E [S] =
+�
+j∈[r]
+E
+�
+1[i(j)∈I(1:r)]
+�
+= r2/m ≥ k.
+(20)
+Since k/µ ∈ (0, 1], λ .= 1 − k/µ ∈ [0, 1). By the Chernoff bound for sampling without replacement (Fact A.3), we have
+Pr [S ≤ k] = Pr [S ≤ (k/µ) · µ] ≤
+�
+e−λ
+(1 − λ)1−λ
+�µ
+=
+�
+ek/µ−1
+(k/µ)k/µ
+�µ
+=
+ek−µ
+(k/µ)k = ekµk
+kkeµ = ek
+kk · (r2/m)
+k
+er2/m .
+Proof of Theorem 4.2. The expectation of cost (APrivTA, L1) is given by
+E [cost (APrivTA, L1)] =
+� ∞
+0
+Pr [cost (APrivTA, L1) ≥ s] ds
+= 2 ·
+� ∞
+0
+Pr [cost (APrivTA, L1) ≥ 2r] dr
+r .= s/2
+= 2 ·
+� √
+mk+1
+0
+Pr [cost (APrivTA, L1) ≥ 2r] dr + 2 ·
+� ∞
+√
+mk+1
+Pr [cost (APrivTA, L1) ≥ 2r] dr
+(a)
+≤ 2 ·
+� √
+mk+1
+0
+Pr [cost (APrivTA, L1) ≥ r] dr + 2 ·
+� ∞
+√
+mk+1
+Pr [cost (APrivTA, L1) ≥ 2⌊r⌋] dr
+(b)
+≤ 2 ·
+√
+mk + 2 + 2 ·
+� ∞
+√
+mk+1
+ek
+kk · (⌊r⌋2/m)
+k
+e⌊r⌋2/m
+dr
+(c)
+≤ 2 ·
+√
+mk + 2 + 2 ·
+� ∞
+√
+mk+1
+ek
+kk · ((r − 1)2/m)
+k
+e(r−1)2/m
+dr
+= 2 ·
+√
+mk + 2 + 2 · ek
+kk ·
+� ∞
+4k
+tke−k √m dt
+2
+√
+t
+t .= (r − 1)2/m
+≤ 2 ·
+√
+mk + 2 + √m ek
+kk ·
+� ∞
+0
+tk−1/2e−k dt
+(d)
+≤ 2 ·
+√
+mk + 2 + √m ek
+kk · Γ(k + 1/2)
+(e)
+≤ 2 ·
+√
+mk + 2 + √m ek
+kk · (2k)!
+4kk!
+(f)
+≤
+�
+2 +
+√
+2
+�
+·
+√
+mk + 2,
+
+Tight Data Access Bounds for Private Top-k Selection
+14
+where Inequality (a) follows since Pr [cost (APrivTA, L1) ≥ 2r] ≤ Pr [cost (APrivTA, L1) ≥ 2⌊r⌋] ; Inequality (b) follows
+from Lemma B.1; Inequality (c) follows since the function y .= xk/ex (with y′ = kxk−1ex−xkex
+e2x
+) is decreasing when x ≥ k,
+and that (r − 1)2/m ≥ k, when r ≥
+√
+mk + 1; Inequalities (d) and (e) follows from Fact A.4; Inequalities (f) follows
+Fact A.6 and that
+ek
+kk · (2k)!
+4kk! ≤ ek
+kk ·
+√
+2π · 2k
+� 2k
+e
+�2k exp
+�
+1
+12·2k
+�
+4k√
+2π · k
+� k
+e
+�k exp
+�
+1
+12·k+1
+� ≤
+√
+2.
+(21)
+B.2. Proofs for Section 4.2
+Proof of Theorem 4.3. This is a directly consequence of the facts that
+• Based on a technique by Brassard & Kannan (1988) for sampling a random perturbation on the fly, and on the
+discussions in Section 4.2, it holds that each I(j), j ∈ [m], and each value of inverted index σ2(i), i ∈ [m] can be
+sampled on the fly in O(1) times.
+• Based on Lemma 4.4, each Z(j), j ∈ [m] can be sampled on the fly with O(log log m) expected time.
+Proof of Lemma 4.6. First, observe that the following two multisets are the same
+�
+F −1(U1), . . . , F −1(Um)
+�
+and
+�
+F −1(U(1)), . . . , F −1(U(m))
+�
+. By construction, Z(1), . . . , Z(m) is a sorted sequence of the multiset. To prove
+the claim, it suffices to show that F −1(U(1)), . . . , F −1(U(m)) is also a sorted sequence. This is true: since F is a cumulative
+distribution function on R, it is non-decreasing; it follows that for every two 1 ≤ j < j′ ≤ m, U(j) ≥ U(j′) implies that
+F −1(U(j)) ≥ F −1(U(j′)).
+Proof of Theorem 4.9. If X ∼ Beta (m − j + 1, j), then its density function f(x) satisfies f(x) = (x)m−j(1−x)j−1
+B(m−j+1,j)
+, ∀x ∈
+[0, 1]. Based on Fact A.5, B(m − j + 1, j) = (m−j)!(j−1)!
+m!
+, which proves the first claim.
+If X ∼ Beta (m − j + 1, j − ℓ), then the density of Y , denoted by ˜f(y), is given by ∀y ∈ [0, u(ℓ)],
+˜f(y) = f
+� y
+u(ℓ)
+�
+· dX
+dY =
+1
+B(m − j + 1, j − ℓ) ·
+� y
+u(ℓ)
+�m−j�
+1 −
+� y
+u(ℓ)
+��j−ℓ−1
+·
+1
+u(ℓ)
+,
+Noting that B(m − j + 1, j − ℓ) = (m−j)!(j−ℓ−1)!
+(m−ℓ)!
+proves the second claim.
+The third claim follows from similar argument as the second one.
+B.3. Proof for Theorem 4.7
+We need the following fact.
+Fact B.2 ((Guntuboyina, 2019)). The joint density of U(1), . . . , U(m) is given by
+p(u(1), u(2), . . . , u(m)) = m! ,
+(22)
+where 1 ≥ u(1) ≥ u(2) · · · ≥ u(m) ≥ 0.
+As a sanity check, note that the simplex ∆∗
+n
+.=
+�
+u(1), u(2) . . . , u(m) ∈ Rm : 1 ≥ u(1) ≥ u(2) . . . ≥ u(m) ≥ 0
+�
+has vol-
+ume 1/m!. Since U(1), . . . , U(m) distribute uniformly over ∆∗
+n, each point of the simplex has density m!. Based on this fact,
+we can derive the following lemma.
+
+Tight Data Access Bounds for Private Top-k Selection
+15
+Lemma B.3. Let j1, . . . , jt ∈ [m] be an increasing sequence of indexes. Then the joint distribution of U(j1), . . . , U(jt) is
+given by
+p
+�
+u(j1), . . . , u(jt)
+�
+= m! ·
+�
+u(jt)
+�m−jt
+(m − jt)!
+·
+�
+1 − u(j1)
+�j1−1
+(j1 − 1)!
+·
+t�
+s=2
+�
+u(js−1) − u(js)
+�js−js−1−1
+(js − js−1 − 1)!
+.
+(23)
+Proof of Lemma. p
+�
+u(j1), . . . , u(jt)
+�
+is given by the following integration:
+� 1
+0
+· · ·
+� 1
+0
+m! · 1[0≤u(m)≤...≤u(1)≤1] du(m) . . . du(1+jt) du(−1+jt) . . . du(1+jt−1) . . . du(−1+j1) . . . du(1).
+Integrating with respect to u(m) over the range [0, u(m−1)], we have
+� 1
+0
+· · ·
+� 1
+0
+m! · u(m−1) · 1[0≤u(m−1)≤...≤u(1)≤1] du(m−1) . . . du(1+jt) du(−1+jt) . . . du(1+jt−1) . . . du(−1+j1) . . . du(1).
+Then integrate u(m−1) over the range [0, u(m−2)], all the way down to the integral with respect to u(1+jt) over the
+range [0, u(jt)]. We obtain
+� 1
+0
+· · ·
+� 1
+0
+m! ·
+�
+u(jt)
+�m−jt
+(m − jt)!
+· 1[0≤u(jt)≤u(−1+jt)...≤u(1)≤1] du(−1+jt) . . . du(1+jt−1) . . . du(−1+j1) . . . du(1).
+Integrating with respect to u(−1+jt) over the range [u(jt), u(−2+jt)], we have
+� 1
+0
+· · ·
+� 1
+0
+m!
+�
+u(jt)
+�m−jt
+(m − jt)!
+�
+u(−2+jt) − u(jt)
+�
+· 1[0≤u(jt)≤u(−2+jt)...≤u(1)≤1] du(−2+jt) . . . du(1+jt−1) . . . du(−1+j1) . . . du(1).
+Then integrate u(−2+jt) over the range [u(jt), u(−3+jt)], all the way down to u(1+jt−1) over the range [u(jt), u(jt−1)]:
+� 1
+0
+· · ·
+� 1
+0
+�
+�
+�
+�
+�
+�
+�
+m!·
+�
+u(jt)
+�m−jt
+(m − jt)!
+·
+�
+u(jt−1) − u(jt)
+�jt−jt−1−1
+(jt − jt−1 − 1)!
+· 1[0≤u(jt−1)≤u(−1+jt−1)...≤u(1)≤1]
+�
+�
+�
+�
+�
+�
+�
+du(−1+jt−1) . . . du(1+jt−2) . . . du(−1+j1) . . . du(1).
+Repeating the above efforts proves Equation (23).
+Proof of Theorem 4.7.
+Claim One: Applying Lemma B.3 directly, we have
+p
+�
+u(j)
+�
+= m! ·
+�
+u(j)
+�m−j
+(m − j)! ·
+�
+1 − u(j)
+�j−1
+(j − 1)!
+,
+∀u(j) ∈ [0, 1],
+which proves the first claim.
+Claim Two: let J = {ζ1, . . . , ζc}, s.t., ζ1 < · · · < ζc < j. Following the notation of Theorem 4.7, we have ℓ = ζc. By
+Lemma B.3,
+p
+�
+u(J )
+�
+= m! ·
+�
+u(ζc)
+�m−ζc
+(m − ζc)!
+·
+�
+1 − u(ζ1)
+�ζ1−1
+(ζ1 − 1)!
+·
+c
+�
+s=2
+�
+u(ζs−1) − u(ζj)
+�ζs−ζs−1−1
+(ζs − ζs−1 − 1)!
+,
+p
+�
+u(J ∪{j})
+�
+= m! ·
+�
+u(j)
+�m−j
+(m − j)! ·
+�
+u(ζc) − u(j)
+�j−ζc−1
+(j − ζc − 1)!
+·
+�
+1 − u(ζ1)
+�ζ1−1
+(ζ1 − 1)!
+·
+c
+�
+s=2
+�
+u(ζs−1) − u(ζs)
+�ζs−ζs−1−1
+(ζs − ζs−1 − 1)!
+.
+
+Tight Data Access Bounds for Private Top-k Selection
+16
+Hence,
+p
+�
+u(j) | u(J )
+�
+= p
+�
+u(J ∪{j})
+�
+p
+�
+u(J )
+�
+=
+(m − ζc)!
+(j − ζc − 1)!(m − j)!
+�u(ζc) − u(j)
+u(ζc)
+�j−ζc−1� u(j)
+u(ζc)
+�m−j
+1
+u(ζc)
+.
+Similarly, the densities of p
+�
+u(ζc)
+�
+, p
+�
+u(ζc), u(j)
+�
+are given by
+p
+�
+u(ζc)
+�
+= m! ·
+�
+u(ζc)
+�m−j
+(m − ζc)! ·
+�
+1 − u(ζc)
+�ζc−1
+(ζc − 1)!
+,
+p
+�
+u(ζc), u(j)
+�
+= m! ·
+�
+u(j)
+�m−j
+(m − j)! ·
+�
+u(ζc) − u(j)
+�j−ζc−1
+(j − ζc − 1)!
+·
+�
+1 − u(ζc)
+�ζc−1
+(ζc − 1)!
+.
+It is easy to see that
+p
+�
+u(j) | u(ζc)
+�
+= p
+�
+u(ζc), u(j)
+�
+p
+�
+u(ζc)
+�
+= p
+�
+u(j) | u(J )
+�
+.
+Claim Three: let J = {ζ1, . . . , ζc}, s.t., ζ1 < . . . < ζc, and there exists c′ ∈ [c − 1] for which ζc′ < j < ζc′+1. Following
+the notation of Theorem 4.7, we have ℓ = ζc′, r = ζc′+1. By Lemma B.3,
+p
+�
+u(J )
+�
+= m! ·
+�
+u(ζc)
+�m−ζc
+(m − ζc)!
+·
+�
+1 − u(ζ1)
+�ζ1−1
+(ζ1 − 1)!
+·
+c
+�
+s=2
+�
+u(ζs−1) − u(ζs)
+�ζs−ζs−1−1
+(ζs − ζs−1 − 1)!
+,
+p
+�
+u(J ∪{j})
+�
+= m! ·
+�
+u(ζc)
+�m−ζc
+(m − ζc)!
+·
+�
+1 − u(ζ1)
+�ζ1−1
+(ζ1 − 1)!
+·
+�
+u(ζc′) − u(j)
+�j−ζc′−1
+(j − ζc′ − 1)!
+·
+�
+u(j) − u(ζc′+1)
+�ζc′+1−j−1
+(ζc′+1 − j − 1)!
+·
+c′
+�
+j=2
+�
+u(ζs−1) − u(ζs)
+�ζs−ζs−1−1
+(ζs − ζs−1 − 1)!
+·
+c
+�
+j=c′+2
+�
+u(ζs−1) − u(ζs)
+�ζs−ζs−1−1
+(ζs − ζs−1 − 1)!
+.
+Hence,
+p
+�
+u(j) | u(J )
+�
+= p
+�
+u(J ∪{j})
+�
+p
+�
+u(J )
+�
+=
+(ζc′+1 − ζc′ − 1)!
+(j − ζc′ − 1)!(ζc′+1 − j − 1)!
+·
+�
+u(ζc′) − u(j)
+u(ζc′) − u(ζc′+1)
+�j−ζc′−1�
+u(j) − u(ζc′+1)
+u(ζc′) − u(ζc′+1)
+�ζc′+1−j−1
+1
+u(ζc′) − u(ζc′+1)
+.
+Similarly, the densities of p
+�
+u(ζc′), u(ζc′+1)
+�
+, p
+�
+u(ζc′), u(j), u(ζc′+1)
+�
+is given by
+p
+�
+u(ζc′), u(ζc′+1)
+�
+= m! ·
+�
+u(ζc′+1)
+�m−ζc′+1
+(m − ζc′+1)!
+·
+�
+u(ζc′) − u(ζc′+1)
+�ζc′+1−ζc′−1
+(ζc′+1 − ζc′ − 1)!
+·
+�
+1 − u(ζc′)
+�ζc′−1
+(ζc′ − 1)!
+,
+p
+�
+u(ζc′), u(j), u(ζc′+1)
+�
+= m! ·
+�
+u(ζc′+1)
+�m−ζc′+1
+(m − ζc′+1)!
+·
+�
+u(ζc′) − u(j)
+�j−ζc′−1
+(j − ζc′ − 1)!
+·
+�
+u(j) − u(ζc′+1)
+�ζc′+1−j−1
+(ζc′+1 − j − 1)!
+·
+�
+1 − u(ζc′)
+�ζc′−1
+(ζc′ − 1)!
+.
+It is easy to see that
+p
+�
+u(j) | u(ζc′), u(ζc′+1)
+�
+=
+p
+�
+u(ζc′), u(j), u(ζc′+1)
+�
+p
+�
+u(ζc′), u(ζc′+1)
+�
+= p
+�
+u(j) | u(J )
+�
+.
+
+Tight Data Access Bounds for Private Top-k Selection
+17
+C. Proofs for Section 5
+C.1. Proofs for Section 5.1
+Proof of Theorem 5.1. We do not specify the family H and the distribution µ on H directly. Instead, we show how we can
+sample a ⃗h from H according to µ: for each i ∈ [m], independently set
+⃗h[i] .=
+�
+n,
+w.p. 2k
+m ,
+0,
+w.p. 1 − 2k
+m .
+(24)
+This can also be understood as, for each item i ∈ [m], with probability 2k/m all of the n clients votes for i; and with
+probability 1 − 2k/m, none of the clients votes for i.
+Assume that: 1) A retrieves entries from ⃗h (via random access) without repetition. This only decreases its access cost, since
+accessing a previously encountered entry does not provide additional information. 2) If A terminates before retrieving all
+entries in ⃗h, it continues to read the remaining entries without being charged for the additional accesses. This enables A
+to obtain more entries for free. Now, let J1, . . . , Jm ∈ [m] be the order in which A accesses the entries. The sequence
+constitutes a permutation of [m], and for each t ∈ [m], the choice of Jt can depend on previous choices J1, . . . , Jt−1 and
+outcomes ⃗h[J1], . . . ,⃗h[Jt−1]. However, whatever the choice of Jt is, the distribution of ⃗h[Jt] (conditioned on previous
+choices and outcomes) is till given by Equation (24). Hence, ⃗h[J1], . . . ,⃗h[Jm] can be viewed as independent random
+variables.
+Consider the following events. Event E1 : ⃗h has at least k non-zero entries. Since each entry of ⃗h is generated independently,
+via Chernoff bound (Fact A.1),
+Pr
+� ¯
+E1
+�
+≤ exp
+�
+−(0.5)2 · 2k
+2
+�
+≤ exp (−(0.5)2) ≤ 0.78 .
+Event E2 : the number of non-zero entries among ⃗h[J1] , . . . ,⃗h[Jm/50] is less than k. Since ⃗h[J1], . . . ,⃗h[Jm/50] are
+independent, via Chernoff bound (Fact A.1), and noting that k ≥ 1,
+Pr
+� ¯
+E2
+�
+≤
+� e24
+2525
+�k/25
+≤
+� e24
+2525
+�1/25
+≤ 0.11 .
+Event E3 : A returns an (n − 1, k)-accurate solution. By assumption, we have Pr
+� ¯
+E3
+�
+≤ β < 0.1.
+Hence, Pr [E1 ∩ E2 ∩ E3] ≥ 1 − Pr
+� ¯
+E1
+�
+− Pr
+� ¯
+E2
+�
+− Pr
+� ¯
+E3
+�
+∈ Ω(1). Observe that, when E1 and E3 happens, each
+item i returned by A must have frequency ⃗h[i] = n. However, when E2 happens, A cannot see k items with non-zero
+frequency, from its first m/50 retrievals.
+It follows that , with probability at least Pr [E1 ∩ E2 ∩ E3], A has access cost at least m/50, which proves the theorem.
+C.2. Proofs for Section 5.2
+Proof of Lemma 5.2. Let O .= {o ⊆ [m] : |o| = k} is the collection of all possible outputs of A. Observe that, given an
+input histogram ⃗hS or ⃗hSH,SL, if A’s output is (n − 2, k)-accurate, then it must be a subset of S. Therefore, define G .=
+{o ⊆ S : |o| = k} to be the collection of all (n − 2, k)-accurate outputs. Via the assumption that A outputs an (n − 2, k)-
+accurate solution with probability at least 1 − β, it holds that
+Pr
+�
+A(⃗h) ∈ G
+�
+≥ 1 − β .
+Next, let F .= {sℓ ⊆ S : |sℓ| = |S|/k} be the collection all possible outcomes of SL. For each sℓ ∈ F, define Osℓ
+.=
+{o ∈ O : o ∩ sℓ ̸= ∅}, the collection of sets in O that has nonempty intersection with sℓ, and Gsℓ
+.= {o ∈ G : o ∩ sℓ ̸= ∅},
+the collection of sets in G that has nonempty intersection with sℓ. Conditioned on SL = sℓ,
+Pr
+�
+A(⃗hSH,SL) ∩ SL ̸= ∅ | SL = sℓ
+�
+= Pr
+�
+A(⃗hSH,SL) ∈ Osℓ | SL = sℓ
+�
+≥ Pr
+�
+A(⃗hSH,SL) ∈ Gsℓ | SL = sℓ
+�
+,
+
+Tight Data Access Bounds for Private Top-k Selection
+18
+where the inequality holds as Osℓ ⊇ Gsℓ. Moreover, since A is (ε, δ)-DP, it holds that
+Pr
+�
+A(⃗hSH,SL) ∈ Gsℓ | SL = sℓ
+�
+≥ 1
+eε (Pr [A(h) ∈ Gsℓ] − δ) .
+(25)
+Further, since the events that A(h) = o are mutually exclusive for different values of o ∈ G, we have
+Pr [A(h) ∈ G] =
+�
+o∈G
+Pr [A(h) = o] ≥ 1 − β ,
+(26)
+Pr [A(h) ∈ Gsℓ] =
+�
+o∈G,o∩sℓ̸=∅
+Pr [A(h) = o] =
+�
+o∈G
+1[o∩sℓ̸=∅] · Pr [A(h) = o] .
+(27)
+Finally,
+Pr
+SL,A
+�
+A(⃗hSH,SL) ∩ SL ̸= ∅
+�
+=
+�
+sℓ∈F
+Pr [SL = sℓ] · Pr
+�
+A(⃗hSH,SL) ∈ Osℓ | SL = sℓ
+�
+≥
+�
+sℓ∈F
+Pr [SL = sℓ] · Pr
+�
+A(⃗hSH,SL) ∈ Gsℓ | SL = sℓ
+�
+≥
+�
+sℓ∈F
+Pr [SL = sℓ] · e−ε
+��
+o∈G
+1[o∩sℓ̸=∅] · Pr [A(h) = o] − δ
+�
+= −e−εδ + e−ε �
+o∈G
+Pr [A(h) = o]
+� �
+sℓ∈F
+1[o∩sℓ̸=∅] · Pr [SL = sℓ]
+�
+(a)
+= −e−ε · δ + e−ε �
+o∈G
+Pr [A(h) = o]
+�
+1 −
+�|S|−k
+|S|/k
+�
+� |S|
+|S|/k
+�
+�
+(b)
+≥ −e−ε · δ + e−ε �
+o∈G
+Pr [A(h) = o] (1 − exp (−1))
+≥ −e−ε · δ + e−ε(1 − β)
+�
+1 − e−1�
+≥ e−ε�
+1 − β − δ − e−1�
+,
+where equation (a) follows, since �
+sℓ∈F 1[o∩sℓ̸=∅]·Pr [SL = sℓ] can interpreted as the probability that given a subset o ⊆ S
+of size |k|, the sampled subset SL has nonempty intersection with o; and inequality (b) follows, since
+�|S|−k
+|S|/k
+�
+� |S|
+|S|/k
+� = |S| − k
+|S|
+· · · |S| − k − |S|/k + 1
+|S| − |S|/k + 1
+≤
+�|S| − k
+|S|
+�|S|/k
+≤ exp
+�
+− k
+|S| · |S|
+k
+�
+= exp (−1) .
+C.3. Proofs for Section 5.3
+Proof of Lemma 5.5. Assume that: 1) if A terminates before performing η accesses operations, it continues to perform more
+until it reaches η, and will not be charged for any additional access. Now, let J1, . . . , Jη represents the first η operations
+of A: each Jt is either a character ‘s’, implying that the t(th) operation is a sorted access, or an integer in [m], implying
+that the t(th) operation is a random access to the entry ⃗hSH,SL[Jt]. Further, let J′
+1, . . . , J′
+t be the subsequence of all random
+access in J1, . . . , Jη.
+
+Tight Data Access Bounds for Private Top-k Selection
+19
+Consider an alternative algorithm A′ that operates as follows: it first performs 9η sorted accesses to obtain all frequencies of
+all items from SH, followed by random accesses J′
+1, . . . , J′
+t. It is clear that, A′ retrieves a greater number of entries than the
+first η operations of A.
+We bound the probability that A′ does not retrieve an entry from SL. First consider random access J′
+1. If J′
+1 had already
+been retrieved by sorted access, then clearly J′
+1 /∈ SL. Otherwise, conditioned on the fact that the entries SH have been
+determined, according to the manner S and SL are generated, each item in [m] \ (SH) belongs to SL with equal probability.
+Hence,
+Pr [J′
+1 /∈ SL] ≥ 1 −
+τ/k
+m − (k − 1)τ/k .
+In general, for each 1 < ℓ ≤ t, suppose that J′
+1, . . . , J′
+ℓ−1 does not belong to SL, and has revealed mℓ ≤ ℓ − 1 distinct items
+from [m] \ (SH). Whatever the choice of J′
+ℓ is, and the items J′
+1, . . . , J′
+ℓ−1 are, each of the remaining items that have not
+been queried belong to SL with equal probability. Hence
+Pr [J′
+ℓ /∈ SL] ≥ 1 −
+τ/k
+m − (k − 1)τ/k − mℓ
+.
+Noting that mℓ < t ≤ η, we have
+Pr [J′
+1, . . . , J′
+t /∈ SL] ≥
+t�
+ℓ=1
+�
+1 −
+τ/k
+m − (k − 1)τ/k − mℓ
+�
+≥
+t�
+ℓ=1
+�
+1 −
+τ/k
+m − (k − 1)τ/k − η
+�
+(a)
+≥
+t�
+ℓ=1
+exp
+�
+−
+2τ/k
+m − (k − 1)τ/k − η
+�
+≥ exp
+�
+−
+τ 2/(10k)
+m − (k − 1)τ/k − η
+�
+≥ exp
+�
+−
+m/10
+m − 21
+√
+mk/20
+�
+≥ exp
+�
+−
+m/10
+m − 21m/40
+�
+≥ 0.81.
+where inequality (a) holds, since 1 − x ≥ e−2x, ∀x ∈ [0, 3/4], and
+τ/k ≤ 3
+4(m − (k − 1)τ/k − η)
+(28)
+⇐⇒ τ
+�3
+4 + 1
+4k + 3
+40
+�
+≤ 3
+4m
+(29)
+The last inequality holds, since 3
+4 +
+1
+4k + 3
+40 ≤ 3/2 for k ∈ N+, and 2τ = 2
+√
+mk ≤ m, i.e., k ≤ m/4.
+
diff --git a/b9FQT4oBgHgl3EQfhzZX/content/tmp_files/load_file.txt b/b9FQT4oBgHgl3EQfhzZX/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..91f4a42cbd31cc669bdf02b0c066ee285f81954b
--- /dev/null
+++ b/b9FQT4oBgHgl3EQfhzZX/content/tmp_files/load_file.txt
@@ -0,0 +1,1520 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf,len=1519
+page_content='Tight Data Access Bounds for Private Top-k Selection  Hao Wu  whw4@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='unimelb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='au  The University of Melbourne  Australia  Olga Ohrimenko  oohrimenko@unimelb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='au  The University of Melbourne  Australia  Anthony Wirth  awirth@unimelb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='au  The University of Melbourne  Australia  Abstract  We study the top-k selection problem under the differ-  ential privacy model: m items are rated according to  votes of a set of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We consider a setting in which  algorithms can retrieve data via a sequence of accesses,  each either a random access or a sorted access;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' the goal  is to minimize the total number of data accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Our  algorithm requires only O(  √  mk) expected accesses:  to our knowledge, this is the first sublinear data-access  upper bound for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Accompanying this, we  develop the first lower bounds for the problem, in three  settings: only random accesses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' only sorted acceses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  a sequence of accesses of either kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We show that,  to avoid Ω(m) access cost, supporting either kind of  access, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' the freedom to mix, is necessary, and that in  this case our algorithm’s access cost is almost optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Introduction  We consider the differentially private top-k selection prob-  lem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' there are m items to be rated according to n clients’  votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Each client can either vote or not vote for each item,  and can vote for an unlimited number of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Since this  data can be sensitive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', visited websites, purchased items,  or watched movies), the goal is to identify a set of k items  with approximately the highest number of votes, while con-  cealing the votes of individual clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Private top-k selection is a fundamental primitive and under-  lies a wide range of differentially private machine learning  and data analytics tasks such as discovering frequent pat-  terns from data (Bhaskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2010), training wide neural  networks (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2021), tracking data streams (Car-  doso & Rogers, 2022), false discovery rate control in hy-  pothesis testing (Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2021), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  In recent years, a significant progress has been made towards  understanding how accurate the algorithms for this problem  can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' For example, (Bafna & Ullman, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Steinke &  Ullman, 2017) provide lower bounds for the problem in  terms of sample complexity, which can be achieved by a  number of existing algorithms (Durfee & Rogers, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Another line of research is devoted to improving the effi-  ciency of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Early works such as the peeling  solution (Bhaskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2010) need to iterate k times over  all items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The improved mechanisms (Durfee & Rogers,  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2021) iterate over each item only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Since k can be much smaller than m, the research commu-  nity remains interested in the following question:  Research Question: Can we develop private top-k se-  lection algorithms that access only a sublinear number  of items?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Although it seems to be an unachievable target, it is possi-  ble to address this question by considering how items are  accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' For example, Durfee & Rogers (2019) consider  the setting where the data has been pre-processed and re-  sides in an existing data analytics system, that can return  the items in sorted order (which we refer to as sorted access  in our paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Their top-k algorithm can make a sublinear  number of accesses at a cost of potentially returning fewer  than k items, while to guarantee that k items are returned,  the number of retrieved items can be m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Since retrieving  information from an existing system incurs corresponding  query processing and communication cost, it is crucial to  minimize the number of data accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  In this paper, we systematically investigate the minimum  number of items an algorithm needs to evaluate (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ac-  cess cost) , in order to answer the private top-k selection  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' In addition to sorted access, we also consider  another common way of accessing items’ data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', the ran-  dom access, in which an algorithm can actively request the  data of an arbitrary item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1 Both types of accesses have been  considered by previous literature for the non-private ver-  sion of top-k selection problem (see Ilyas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (2008) for a  comprehensive survey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Consider the example of a movie ranking  1Here random carries the sense of RAM, rather than the out-  come of a random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='13347v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='CR]  31 Jan 2023  Tight Data Access Bounds for Private Top-k Selection  2  database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' It can present the movies in sorted order, accord-  ing to their rating by the clients, or it can return directly the  rating of a specific movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Our Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Our results are threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' On the upper  bound side,  If the system supports both sorted access and random  access, we design an algorithm with expected access  cost O(  √  mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  To our knowledge, this is the first asymptotically sublin-  ear bound of the access cost for the private top-k selection  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Our algorithm builds on existing works (Dur-  fee & Rogers, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2021) and inherits their  error bounds, which are known to be asymptotically opti-  mal (Bafna & Ullman, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Steinke & Ullman, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  On the lower bound side,  If the system supports either only sorted accesses or only  random accesses, but not both, we show a lower bound  of Ω(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  If the system supports both sorted accesses and random  accesses, we show a lower bound of Ω(  √  mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  These statements are informal versions of Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3,  and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='4, which impose modest assumptions on the privacy  guarantee, and relatively weak assumptions on the accuracy  guarantee of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' They show that supporting  sorted and random access to the items’ data simultaneously  is necessary to break the linear barrier, and the access cost  of our algorithm is essentially optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Section 2  introduces the problem formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Section 3 discusses the  preliminaries for our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Section 4 introduces our  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Section 5 presents the lower bounds for the  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Section 6 discusses the related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Model Description  Let C  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , m} be a set of m items, and U  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , n} be a set of n clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Each client v ∈ U can  cast at most one vote for each item, and can vote for an  unlimited number of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Hence, client v’s votes, denoted  by ⃗xv, can be viewed as a vector in D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= {0, 1}m, such that  for each i ∈ C, ⃗xv[i] = 1 if v votes for item i, where ⃗xv[i] is  the i(th) entry of ⃗xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We regard the collection of voting vec-  tors from all n clients as a dataset X = {⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , ⃗xn} ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  For each item i ∈ C, let its score ⃗h[i] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= �  v∈U ⃗xv[i] be  the number of clients that vote for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The dataset X can  be described by its histogram ⃗h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= (⃗h[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ,⃗h[m]) ∈ Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  We also define π : C → C to be a permutation that puts the  entries of ⃗h in nonincreasing order2, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', ⃗h[π(1)] ≥ · · · ≥  ⃗h[π(m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Our goal is to design a differentially private algorithm that  returns a set S of k items with (approximately) largest  scores, while minimizing its data access cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  In what follows, we discuss the privacy guarantee, the utility  guarantee, the data access model of an algorithm formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Privacy Guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We call two datasets X, X ′ neigh-  boring, denoted by X ∼ X ′, if they differ in addition  or deletion of one client vector, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', X ′ = X ∪ {⃗xn+1}  or X ′ = X \\ {⃗xv} for some v ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Let ⃗h and ⃗h′ be the histograms corresponding to X and X ′,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' It is easy to see that if X ∼ X ′, then the score  of each item can differ by at most 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', ∥⃗h − ⃗h′∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Hence, for every ⃗h,⃗h′ ∈ Nm, we also call them neighboring  histograms, written as ⃗h ∼ ⃗h′, if ∥⃗h − ⃗h′∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Let A be an algorithm for the top-k selection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  To protect the voting information of individual clients, we  would like its output distributions to be similar for neighbor-  ing inputs, as defined thus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1 ((ε, δ)-Private Algorithm (Dwork & Roth,  2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Given ε, δ > 0, a randomized algorithm A : Nm →  Z is called (ε, δ)-differentially private (DP), if for for ev-  ery ⃗h,⃗h′ ∈ Nm such that ⃗h ∼ ⃗h′, and all (measurable)  Z ⊆ Z,  Pr  �  A(⃗h) ∈ Z  �  ≤ eε · Pr[A(⃗h′) ∈ Z] + δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (1)  We call ε and δ the privacy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Typically, it is  required that δ is cryptographically negligible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', δ ≤  1/mω(1) (Vadhan, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Dwork & Roth, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' An algo-  rithm A is also called ε-DP for short, if it is (ε, 0)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Utility Guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' In line with previous research (Bafna  & Ullman, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Durfee & Rogers, 2019), we measure  the error of an output S by the maximum amount by  which ⃗h[π(k)] exceeds the score of an item in S, defined  formally as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2 ((α, k)-Accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Given a vector ⃗h, param-  eters k ∈ N+, and α ∈ R+, an output S ⊆ [m] of size k is  called (α, k)-accurate, if for each i ∈ S, ⃗h[i] ≥ ⃗h[π(k)]−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Data Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We assume that the histogram ⃗h has been  preprocessing by an existing data management system, and  an algorithm A can access ⃗h only through the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We  consider two access models that abstract common function-  alities supported by a system: sorted access and random  access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Such access models have been widely accepted by  the community for non-private top-k selection problems  (see Ilyas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (2008) for a survey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  2We break ties arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Tight Data Access Bounds for Private Top-k Selection  3  Sorted Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let Cs be the set of items already returned  by sorted access (initially, Cs = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' When a new sorted-  access request is submitted, the system returns an item-  score pair (i,⃗h[i]), where i ∈ C \\ Cs has the largest score,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', i = arg maxj∈C\\Cs ⃗h[j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' An alternative view is that  the system returns  �  π(1),⃗h[π(1)]  �  ,  �  π(2),⃗h[π(2)]  �  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' in  order, one tuple at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Random Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' A request of random access consists of a  reference i ∈ C to an item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' In response, the system returns  the corresponding item-score pair (i,⃗h[i]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We emphasize  that a random access does not imply that i must be a ran-  domly chosen item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Access Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Given an algorithm A and a histogram ⃗h,  the access cost of an algorithm on ⃗h, cost  �  A,⃗h  �  , is the  total number of accesses – either sorted or random – to the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Note that this is an upper bound of distinct number  of entries A learns from ⃗h, as a random access may retrieve  a previously encountered item-score pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Preliminaries  In this section, we review two building blocks for construct-  ing our algorithm: a technique for aggregating ranked data  from multiple sources, and a framework for the existing  one-shot algorithms for private top-k selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Threshold Algorithm  The threshold algorithm (Fagin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2003) is a top-k selec-  tion algorithm, when the information of an item needs to be  aggregated from multiple resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' In this scenario, there  are m items, each associated with t attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=',  assume that each attribute is a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Therefore,  each item i can be represented by a vector ⃗yi ∈ Rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The  score of item i is computed by a function f : Rt → R,  which is assumed to be monotone, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', for each ⃗y, ⃗y′ ∈ Rt,  if ⃗y[j] ≤ ⃗y′[j], ∀j ∈ [t], then f(⃗y) ≤ f(⃗y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The vectors ⃗y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , ⃗ym do not reside in a single data man-  agement system, but distributed in t systems L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Lt, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=',  for each item i, its jth attribute ⃗yi[j] resides on Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Each Lj  allows for both sorted access and random access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We can  view it as an array of m tuples Lj[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Lj[m], each of  the form (i, val) ∈ [m] × R, where val equals ⃗yi[j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The tu-  ples in Lj are sorted in descending order by their val’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Further, Lj is augmented with an inverted index σj :  [m] → [m] to support random access, such that, for each an  item i ∈ [m], Lj[σj(i)] contains the tuple (i, ⃗yi[j]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The aim is to identify the top-k items with highest scores  according to f, while minimizing the access cost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=',  the total number of data accesses performed by the algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='to L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The algorithm is described in Algo-  rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Algorithm 1 Threshold Algorithm ATA (Fagin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2003)  1: Input: Sorted array Lj and inverted index σj, ∀j ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  2: S ← ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  3: repeat  4:  Do sorted access in parallel to each of the t array Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  As a tuple (i, val) is seen under sorted access:  5:  Retrieve from L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Lt by random access (with  the help of σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , σt) all attributes of item i, to  compute f(⃗yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  6:  If f(⃗yi) is among the-k highest scores seen so far,  add (i, f(⃗yi)) to S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' if |S| > k, remove the tuple  with lowest score from S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  7:  For each Lj, let yj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= ⃗yi[j], where i is the last item  seen in Lj under sorted access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  8:  Define the threshold τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= f(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , ym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  9: until there are k tuples in S with score at least τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  10: Return the set of items contained in the tuples in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The correctness of the algorithm is obvious: when the algo-  rithm stops, since f is monotone, the scores of all unseen  items are at most τ, which are lower than the score of each  tuples in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  For each tuple (i, val) encountered during sorted access, ATA  retrieves all entries ⃗yi[j] of ⃗yi by random accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' This  seems to be redundant, as i might have been encountered  previously, and therefore f(⃗yi) has already been evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The algorithm ATA performs so, to keep the buffer size  (memory usage) strictly bounded by O(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' If this is not  required, ATA can be augmented with an additional data  structure to keep track of previously evaluated i’s, to avoid  repeated computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Access Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Fagin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (2003) did not provide asymptotic  bound for the access cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Instead, they proved that ATA  is instance optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Informally, instance optimally implies  that for every algorithm A which solves the top-k selection  problem correctly and whose first access to an item must  be sorted access as opposed to random access, the access  cost of ATA is at most the access cost of A (up to some  multiplicative constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' In Section 4, we apply a different  technique to asymptotically bound the access cost of our  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  One-shot Private Top-k Algorithm  We review an existing framework for the differentially pri-  vate top-k selection algorithms (Durfee & Rogers, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The framework is described in Algo-  rithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The framework does not consider a specific data  access model, and instead needs to learn all entries of ⃗h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1 (Noise Distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Given parameter b ∈  R, the Laplace distribution, Lap (b), and the Gumbel  distribution, Gumbel (b), have probability density func-  Tight Data Access Bounds for Private Top-k Selection  4  Algorithm 2 Private Top-k Algorithm M  1: Input: vector ⃗h  2: for each item i ∈ [m] do  3:  ⃗v[i] ← ⃗h[i] + Zi, where Zi is an independent noise  random variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  4: end for  5: Return a set S of k items that the maximizes the ⃗v[i]’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  tions p(z) =  1  2b · exp  �  − |z|  b  �  , ∀z ∈ R, and p(z) =  1  b · exp  �  −  � z  b + exp  �  − z  b  ���  , ∀z ∈ R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Candidates noise distributions for Zi in Algorithm 2 in-  clude Lap (1/ε) (Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2021) and Gumbel (1/ε) (Dur-  fee & Rogers, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The corresponding privacy guarantees,  are stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2 ((Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Assume that Zi ∼ Lap (1/ε),  then M is 2kε-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Given δ  ∈  [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='05], if it holds  that m ≥ 2 and 8ε  �  k log (m/δ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2, then M also  satisfies  �  8ε  �  k log (m/δ), δ  �  DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3 ((Durfee & Rogers, 2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Assume that Zi ∼  Gumbel (1/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  For each δ ∈ [0, 1], M is (ε′′, δ)-DP,  where ε′′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= min  �  kε, kε  �  eε−1  eε+1  �  + ε  �  k ln 1  δ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Next, we discuss their utility guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Given β ∈ (0, 1), if the Zi ∼ Lap (1/ε),  or Gumbel (1/ε), then with probability at least 1 − β, the  returned solution by Algorithm 2 is (α, k)-accurate, for  α ∈ O  �  ln (m/β)  ε  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Compared to Laplace noise, the Gumbel noise  allows the algorithm to return a ranked list of indices, in-  stead of a set which contains on order information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' For  consistency of presentation, we assume that Algorithm 2  returns a set for both choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Sublinear Access and Time Algorithm  In this section, we present an algorithm for top-k selec-  tion problem, which achieves optimal privacy utility trade-  off, and with high probability, has an expected access  cost O(  √  mk) and computation time O(  √  mk log log m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Our presentation follows two steps: we first present a straw-  man algorithm with sublinear access cost but only linear  computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' next we show how to improve its time com-  plexity to O(  √  mk log log m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' A Strawman Approach  A natural idea is to combine the threshold algorithm ATA  with the oneshot private top-k algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Each item i ∈ [m]  now has two attributes, namely, ⃗h[i] and Zi, where Zi is  sampled independently from Lap (1/ε) or Gumbel (1/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Since the histogram ⃗h is stored in a database management  system, which allows for two types of access: sorted access  and random access, we can think of this as a sorted array L1  of m tuples, each of the form (i, val) ∈ [m] × R, where val  equals ⃗h[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='The tuples in L1 are sorted in descending or-  der by their val’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Further, L1 has an inverted index σ1 to  support random access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Additionally, we can construct another sorted array L2 of m  tuples, each of the form (i, val) ∈ [m] × R, where val  equals Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The tuples in L2 are also sorted in descending  order by their val’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' L2 also has an inverted index σ2 to  support random access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Then we can run the algorithm ATA, with input L1, L2,  σ1, σ2, and an aggregating function f(⃗h[i], Zi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= ⃗h[i]+Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  It is easy to see that f is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The pseudo-code is in  Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Algorithm 3 Private Threshold Algorithm APrivTA  1: Let I1 = 1, I2 = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Im = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Generate m  tuples (I1, Z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , (Im, Zm), where the Zi’s are  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' random variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' sort the tuples in descend-  ing order by the values of the Zi’s, denote the sorted  sequence by  �  I(1), Z(1)  �  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ,  �  I(m), Z(m)  �  , and store  this sequence in an array L2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' construct σ2 : [m] → [m],  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', L2[σ2(i)] = (i, Zi) for each item i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  2: Run Algorithm 1 on input L1, L2, σ1, σ2, with an ag-  gregation function f  �  (⃗h[i], Zi)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= ⃗h[i] + Zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Privacy and Utility Guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The privacy and utility guar-  antee of the algorithm inherits directly from Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Access and Time Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' It is easier to first discuss  the time complexity and then the access cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Generating  the random variables takes O(m) time, and sorting them  takes O(m log m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Hence the total running time is  bounded by O(m log m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  It remains to discuss the number of accesses the algorithm  performs on L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Our analysis relies on the following impor-  tant observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The I(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , I(m) are distributed uniformly  over all possible permutations over [m], and are indepen-  dent of the random variables Z(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Z(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Intuitively, the claim holds since each Zi in Algorithm 3  follows the same distribution independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The proof of  the lemma is included in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The expected access cost of Algorithm 3  on L1, E [cost (APrivTA, L1)], is bounded by O  �√  mk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The proof of the Theorem is in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Tight Data Access Bounds for Private Top-k Selection  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' An Online Sampling Approach  Pre-generating all m noise values may be excessive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' For  problems with small values of k, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', k = 10, the Algo-  rithm 3 may need to know only a small subset of tuples in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  It is of interest whether we can also reduce the number of  noisy random variables generated to O(  √  mk log log m), by  constructing the L2 (and σ2) on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The main result of  this section is stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' There is an algorithm Aoracle, that,  does not require to pre-generate L2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  answers sorted access and random access query to L2  in O(log log m) time in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Further, the tuples returned by Aoracle have the same  marginal distribution as those generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Algorithm 4 Algorithm Aoracle  Initialization  1: J ← ∅, idx ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  2: L2[i] ← nil, σ2(i) ← nil, ∀i ∈ [m]  Sorted Access  1: idx ← idx + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  2: if idx /∈ J then  3:  Sample I(idx) uniformly from [m] \\ I(J );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  4:  Invoke Aord-stat to sample Z(idx);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  5:  σ2(I(idx)) ← idx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  6:  L2[idx] ← (I(idx), Z(idx));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' J ← J ∪ {idx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  7: end if  8: return L2[idx]  Random Access (Input: item i ∈ [m])  1: if σ2(i) = nil then  2:  Sample j uniformly from [m] \\ J ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  3:  Invoke Aord-stat to sample Z(j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  4:  σ2(i) ← j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  5:  L2[j] ← (i, Z(j));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' J ← J ∪ {j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  6: end if  7: return L2[σ2(i)]  There are two key ingredients for constructing Aoracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Sampling the I(j)’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The first ingredient is Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1,  which allows Aoracle to sample the I(j) and the Z(j) inde-  pendently according to their marginal distributions, without  changing the joint distribution of the I(j) and the Z(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The  lemma states that the I(j)’s are distributed uniformly over  all possible permutations over [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' It is not hard to sample  an I(j) on the fly: let J be the set of indexes such that the  values of I(j′), j ∈ J have been determined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' if j /∈ J ,  then I(j) just distributes uniformly over the subset of unseen  items, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', [m] \\ I(J ), where I(J ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  �  I(j′) : j′ ∈ J  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Cor-  respondingly, we can also construct the inverted index σ2 on  the fly: given an item i ∈ [m], if it has not been encountered,  then σ2(i) should equal one of the undetermined indexes,  namely [m] \\ J , uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Sampling the Z(j)’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The second ingredient is an al-  gorithm Aord-stat which generates the Z(j)’s on the  fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='Formally, for each J  ⊆  [m], define Z(J )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  (Z(j), j ∈ J ), and let z(J ) refer to a vector (z(j), j ∈ J ) ∈  R|J |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Denote by pZ(J )(·) the marginal density of Z(J ), in-  duced by the generating procedure of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Call z(J )  a feasible realization of Z(J ), if pZ(J )(z(J )) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Given  such a feasible realization, let pZ(j)|Z(J )(· | z(J )) be the  density function of Z(j), conditioned on Z(J ) = z(J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The  property of Aord-stat is stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' For each J ⊆ [m] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', J ̸= [m], each j ∈  [m] \\ J , and each feasible realization z(J ) of Z(J ),  Aord-stat samples a random variable with the conditional  density pZ(j)|Z(J )(· | z(J )) in O(log log m) expected time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The proof of the Lemma is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Now,  we return to the construction of Aoracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The algorithm is  described in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The algorithm creates an empty array L2 and  an empty inverted index σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Further, it creates a set J , to  record the positions of L2 which are already sampled, and  a variable idx, to record the last visited position by sorted  access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' In practice, L2 and σ2 need not to be physically ini-  tialized, and can be implemented by hash sets with constant  initialization time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Sorted Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Indeed, it is trivial to handle the sorted ac-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We just maintain an index, idx ∈ N, of last tuple  returned by sorted access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' When a new request of sorted  access arrives, we increase idx by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let J be the set of  indexes of tuples in L2 that have already been generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  If idx ∈ J , then Aoracle returns L2[idx] directly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' other-  wise, it generates L2[idx] before returning it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Random Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' A random access request comes with a  reference to an item i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We need to identify the  tuple L2[j] = (I(j), Z(j)) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', I(j) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' There are two cases:  if item i has been encountered previously (σ2(i) ̸= nil),  then Aoracle returns L2[σ2(i)] directly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' otherwise, Aoracle  randomly pick an index j from [m] \\ J , and set I(j) ←  i, σ2(i) ← j, and calls Aord-stat to generate Z(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Finally,  it returns L2[j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' SAMPLING ORDERED NOISES  In this section, we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Deciding the condi-  tional distribution of the Z(j)’s and sampling them directly  from such distribution can be non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' As a result, we  follow the three-step approach outlined below:  First,  we  show  that  Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Zm  can  be  con-  verted  from  a  sequence  of  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  uniform  ran-  dom variables U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Um, and that the sorted se-  Tight Data Access Bounds for Private Top-k Selection  6  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' A pictorial illustration for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  quence Z(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Z(m) of Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Zm can therefore be  transferred from the sorted sequence U(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , U(m)  of U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Next, to avoid generating the entire sequence of random  variables, we study the distribution of U(j), conditioned  on a set of U(j′) which have already been sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Finally, we show how to sample an U(j) from such dis-  tribution in O(log log m) expected time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Transform U(j) to Z(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Since all potential noise distribu-  tions (Lap (1/ε) or Gumbel (1/ε)) of the Zi’s (the noise  random variables, before sorting) have continuous cumula-  tive distribution function, we can sample them indirectly via  uniform random variables, based on the inversion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5 (Inversion Method (Devroye, 1986)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let F be  a continuous cumulative distribution function on R with  inverse F −1 defined by  F −1(u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= inf {x : F(x) = u, 0 < u < 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  If U is a uniform [0, 1] random variable, then F −1(U) has  distribution function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Let F be the cumulative distribution function of the Zi,  and U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Um be independent uniform random variables  on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We compare the two post-processing procedures:  By Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5, we can obtain Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Zm by com-  puting  F −1(U1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , F −1(Um);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  then  we  ob-  tain Z(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Z(m)  by sorting this sequence in  descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Alternatively, we first sort U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Um in descending  order, to obtain a sequence U(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , U(m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' then we  compute F −1(U(1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , F −1(U(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The two procedures output the same sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The proof of the lemma is included in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Based on the lemma, we can sample U(j) first then com-  pute F −1(U(j)) to generate Z(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' It reduces to study the  distribution of U(j) (conditioned on the subset of U(j′)’s  which have been sampled), and algorithm for sampling a  random variable efficiently from such distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Distribution of U(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Recall that J is the set of in-  dexes which have been queried already .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Denote U(J ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  �  U(j′) : j′ ∈ J  �  a shorthand for the order statistics  that have been sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Further,  write u(J )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  �  u(j′) : j′ ∈ J  �  ∈  [0, 1]|J | as a set of numbers  within [0, 1], indexed by J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Call u(J ) a feasible realiza-  tion of U(J ), if for each j, j′ ∈ J s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' j ≤ j′, it holds  that u(j) ≥ u(j′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Given a new query index j ∈ [m] \\ J , we are interested in  the conditional probability density, pU(j)|U(J )(u(j) | u(J )),  of U(j), given the occurrence of a feasible realization u(J )  of U(J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' For ease of reading, we omit the subscripts of the  conditional probability densities, whenever their meaning  can be unambiguously determined from their parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Depending on the relative position of j w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' the indexes  in J , we consider the following three cases:  J is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' It reduces to study the un-conditional proba-  bility density p(u(j)) of U(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  J is not empty, and j is greater than the largest index  in J ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' in this case, j has a predecessor (the largest index  that is smaller than j), denoted by ℓ, in J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  J is not empty, and j is smaller than the largest index  in J ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' in this case, j has both a predecessor, denoted by ℓ,  and a successor (the smallest index that is larger than j),  denoted by r, in J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Hereafter, if J ̸= ∅, we consider only feasible realiza-  tion u(J ) of U(J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (1) If J is empty, then the density p(u(j))  of U(j) is given by: ∀ u(j) ∈ [0, 1],  p(u(j)) =  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (j−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='(m−j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' · (1 − u(j))j−1(u(j))m−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (2)  (2) If J is not empty, and j is greater than the largest  index in J , then given U(ℓ) = u(ℓ), U(j) is indepen-  dent of all other random variables U(j′) for all j′ ∈  J \\ {ℓ}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', p  �  u(j) | u(J )  �  = p  �  u(j) | u(ℓ)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' further, for  each u(j) ∈ [0, u(ℓ)],  p  �  u(j) | u(ℓ)  �  =  (m−ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (j−ℓ−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (m−j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='·  � u(ℓ)−u(j)  u(ℓ)  �j−ℓ−1� u(j)  u(ℓ)  �m−j  1  u(ℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (3)  (3) If J is not empty, and j is smaller than the largest in-  dex in J , then given U(ℓ) = u(ℓ) and U(r) = u(r), U(j) is  independent of all other random variables U(j′) for all j′ ∈  J \\ {ℓ, r}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', p  �  u(j) | u(J )  �  = p  �  u(j) | u(ℓ), u(r)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' fur-  ther, for each u(j) ∈ [u(r), u(ℓ)],  p  �  u(j) | u(ℓ), u(r)  �  =  (r−ℓ−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (j−ℓ−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (r−j−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='·  � u(ℓ)−u(j)  u(ℓ)−u(r)  �j−ℓ−1� u(j)−u(r)  u(ℓ)−u(r)  �r−j−1  1  u(ℓ)−u(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (4)  The theorem removes the dependency of U(j) from all but  at most two variables in U(J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The detailed proof is in  the Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We provide an informal but intuitive  sort  sort  F-1()Tight Data Access Bounds for Private Top-k Selection  7  explanation of the conditional probability densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Take  the Equation 4 for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Conditioned on U(r) = u(r)  and U(ℓ) = u(ℓ), r − ℓ − 1 uniform random variables fall  into the interval [u(r), u(ℓ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Of these r − ℓ − 1 random  variables, j − ℓ − 1 of them are ≥ ui, and r − j − 1 of them  are < ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The number of possible combinations is given  by  (r−ℓ−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (j−ℓ−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='(r−j−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='. For a fixed combination, the former  happens with probability  � u(ℓ)−u(j)  u(ℓ)−u(r)  �j−ℓ−1  , the latter hap-  pens with probability  � u(j)−u(r)  u(ℓ)−u(r)  �r−j−1  , and the probability  density of U(j) = u(j) is  1  u(ℓ)−u(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Sampling U(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We now discuss how to sample the U(j)  efficiently from their conditional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' First, note  that determining the conditional distributions may need to  find the index j’s predecessor or successor in J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' This can  be done by Van Emde Boas tree (van Emde Boas, 1975)  in O(log log m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Next, we show that sampling from  such conditional distributions takes O(1) expected time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Specifically, we will sample random variables with Beta  distributions, then convert them into ones which follow  desired conditional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='8 (Beta Distribution (Ross, 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The beta  distribution Beta (α, β) is a distribution defined on [0, 1]  whose density is given by  f(x) = xα−1(1−x)β−1  B(α,β)  ,  ∀x ∈ [0, 1],  (5)  where α, β  >  0 are shape parameters, B(α, β)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  � 1  0 xα−1(1 − x)β−1 dx is a normalisation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  It is known that, when α ≥ 1, β ≥ 1, a random vari-  able X ∼ Beta (α, β) can be generated in O(1) expected  time (Devroye, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Gentle, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Assume that ℓ < j < r ≤ m, and 1 ≥  u(ℓ) > u(r) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Then  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' If X ∼ Beta (m − j + 1, j), then the density function  of X is the same as Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' If X ∼ Beta (m − j + 1, j − ℓ), then the density func-  tion of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= u(ℓ) · X is the same as Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' If X ∼ Beta (r − j, j − ℓ), then the density function  of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= (u(ℓ) − u(r)) · X is the same as Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The proof of the Theorem is included in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Lower Bounds  In this section, we generate the lower bounds for the prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Following the setting in Section 2, since⃗h is the sum of  voting vectors of n clients, we have ||⃗h||∞ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' It follows  that an arbitrary subset S ⊆ [m] of size k is (n, k)-accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  All lower bounds in this section hold for algorithms that  are (n − O(1), k)-accurate, which is just slightly better than  the trivial error guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Random Access  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Assume that 0 ≤ β < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let A be an  algorithm that has only random access to ⃗h, does not return  items which it has not seen , and for each input, with prob-  ability at least 1 − β, returns a solution that is (n − 1, k)-  accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Then there exists a family of histograms H, and  a distribution µ on H, if ⃗h is sampled from H according to  distribution µ, it holds that  E  ⃗h  �  cost  �  A,⃗h  ��  ∈ Ω(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (6)  Note that the theorem does not even require A to be a (ε, δ)-  DP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The proof of the theorem is in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  At a high level, our construction focuses on a family of  histograms the values of whose entries are either n or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Further, if an ⃗h is sampled from H, there are roughly 2k  entries of ⃗h that have value n, and those entries appear at  random positions of ⃗h, so that A is unlikely to identify more  than k of them, before learning the values of Ω(m) entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Sorted Access  The lower bound for the sorted access case relies on the  following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let A be an (ε, δ)-DP algorithm, which for  each input histogram, with probability at least 1−β, returns  a solution that is (n − 2, k)-accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let S ⊆ [m], and ⃗hS  be a histogram, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  ⃗hS[i] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  �  n − 1,  ∀ i ∈ S,  0,  ∀ i ∈ ¯S,  (7)  where ¯S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= [m] \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let SL be a subset of S of size |S|/k  sampled uniformly at random (without replacement), SH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  S \\ SL, and ⃗hSH,SL be a histogram neighboring to ⃗h s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  ⃗hSH,SL[i] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  �  �  �  �  �  n,  ∀ i ∈ SH,  n − 1,  ∀ i ∈ SL,  0,  ∀ i /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (8)  Then,  PrSL,A  �  A(⃗hSH,SL) ∩ SL ̸= ∅  �  ≥ 1−β−δ−e−1  eε  ,  (9)  where the randomness is first over the choice of SL then  over the output of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The proof of the Lemma is included in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Note  that, for each i ∈ SL,⃗hSH,SL[i] is among the |S|/k smallest  entries of the |S| largest entries in ⃗hSH,SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Informally, the  lemma states that the probability that output of A contains  some item i ∈ SL is not too “small”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let ε, δ, β be non-negative parameters,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', ε ∈ O(1), δ+β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let A be an (ε, δ)-DP algorithm  that has only sorted access, does not return items which it  has not seen, and for each input histogram, with probability  Tight Data Access Bounds for Private Top-k Selection  8  at least 1 − β, returns a solution that is (n − 2, k)-accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Then there exists a family of histograms H so that, if ⃗h is  sampled uniformly at random from H, it holds that  E  ⃗h  �  cost  �  A,⃗h  ��  ∈ Ω(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (10)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let S = [m/2], SL be a uniform random subset  of S of size |S|/k, and ⃗hSH,SL be a histogram built as  outlined in Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let H be the collection of all  possible outcomes of ⃗hSH,SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2,  PrSL,A  �  A(⃗hSH,SL) ∩ SL ̸= ∅  �  ≥ 1−β−δ−e−1  eε  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  But for each i  ∈  SL, ⃗hSH,SL[i] is among the  top (|S| − |S|/k + 1)(th)  to |S|(th)  largest numbers  in ⃗hSH,SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Since A has only sorted access to ⃗hSH,SL and  does not return an item which it has not seen, if it returns an  item in SL, it needs to invoke at least |S| − |S|/k ∈ Ω(m)  sorted accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' It follows that the expected access cost  of A is at least 1−β−δ−e−1  eε  Ω(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Inequality (10) follows  from the assumption that ε ∈ O(1), and β + δ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Random and Sorted Access  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let ε, δ, β be non-negative parameters,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', ε ≤ 1, δ + β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let A be an algorithm that has  both sorted access and random access to ⃗h, does not return  items which it has not seen, and for each input, with prob-  ability at least 1 − β, returns a solution that is (n − 2, k)-  accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Then there exists a family of histograms H, if ⃗h is  sampled uniformly at random from H, it holds that  E  ⃗h  �  cost  �  A,⃗h  ��  ∈ Ω  �√  mk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (11)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let H be the collection of all possible ⃗hSH,SL gen-  erated as follows:  First, sample a subset S ⊆ [m] of size τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  √  mk without  replacement uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Then we randomly sample a uniform subset SL of S  of size |S|/k, and construct a histogram ⃗hSH,SL as de-  scribed by Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Since Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2 holds for each S ⊆ [m], we have  PrS,SL,A  �  A  �  ⃗hSH,SL  �  ∩ SL ̸= ∅  �  ≥  1  eε  �  1 − β − δ − 1  e  �  (a)  ≥ e−1�  1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='05 − e−1�  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='21,  where the randomness is first over the choice of S, then  over the choice of SL, and finally over the output of A,  and inequality (a) follows from the assumption that ε ≤ 1  and β + δ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' In what follows, we omit the sub-  scripts S, SL, A from the probability notations, when the  source of randomness is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Consider the event E : A accesses some item i ∈ SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' As A  does not return an item which it has not seen, the event E is  a necessary condition for A  �  ⃗hSH,SL  �  ∩ SL ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Hence,  Pr [E] ≥ Pr  �  A  �  ⃗hSH,SL  �  ∩ SL ̸= ∅  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Let η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= τ/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We decompose E into two mutually ex-  clusive events: E1 : A accesses some item i ∈ SL for the  first time within η access operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' E2 : A accesses some  item i ∈ SL for the first time after η access operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Then Pr [E] = Pr [E1] + Pr [E2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The probability that A accesses some item i ∈  SL for the first time within η access operations, denoted  by Pr [E1], is upper bounded by Pr [E1] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5 is omitted here, and is included  in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Intuitively, the lemma holds since: 1) A  can not access some i ∈ SL within η sorted accesses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' 2)  because of the way it is generated, SL is a random subset  from [m] of size  √  mk/k, hence it is also unlikely for A  to come across some i ∈ SL with at most η =  √  mk/20  random access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' To conclude the justification of (11), and  hence prove the Theorem, we apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Pr [E2]  = Pr [E] − Pr [E1]  ≥ Pr  �  A  �  ⃗hSH,SL  �  ∩ SL ̸= ∅  �  − Pr [E1]  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='21 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='19 ∈ Ω(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  But when E2 happens, the access cost is Ω(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Therefore,  the expected access cost of A is lower bounded by Ω(η) =  Ω(  √  mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Related Work  Private Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The private top-1 selection problem is a  special case of the private top-k problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The latter has been  studied extensively, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', the exponential mechanism (Mc-  Sherry & Talwar, 2007), report noisy max (Dwork & Roth,  2014), permute-and-flip (McKenna & Sheldon, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Ding  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Of interest is the permute-and-flip mecha-  nism: when the largest score of the items is known a prior ,  the mechanism can potentially terminate without iterating  over all m items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' However, in this scenario, an asymptotic  upper bound for the number of items evaluated remains  unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Private Top-k Mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Bhaskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (2010) were the  first to apply the “peeling exponential mechanism”, which  iteratively invoked the exponential mechanism to select the  item with highest score, then remove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' They also proposed  an oneshot Laplace mechanism for private top-k selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Bhaskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (2010) analyzed the pure differential privacy  guarantees of both algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Subsequently, Durfee &  Rogers (2019) showed that the peeling exponential mech-  anism has an equivalent oneshot implementation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', Al-  gorithm 2 with Gumbel noise), and studied its approximate  Tight Data Access Bounds for Private Top-k Selection  9  privacy guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (2021) provided the approxi-  mate privacy guarantee for the oneshot Laplace mechanism,  without the help of the composition theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Both Bhaskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (2010) and Durfee & Rogers (2019) have  proposed private algorithms which estimate top-k based on  the true top ¯k items for some ¯k ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Given an integer k,  both algorithm may need to set ¯k = m, in order to return k  items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Accuracy Lower Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Bafna & Ullman (2017)  and Steinke & Ullman (2017) show that, for approximate  private algorithms, the error guarantees of existing algo-  rithms (McSherry & Talwar, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Dwork & Roth, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Durfee & Rogers, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', 2021) are essentially  optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  References  Bafna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' and Ullman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The price of selection  in differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  In Kale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' and Shamir, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ), Proceedings of the 30th Conference on Learn-  ing Theory, COLT 2017, Amsterdam, The Nether-  lands, 7-10 July 2017, volume 65 of Proceedings  of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' 151–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' PMLR,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' URL http://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='mlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='press/  v65/bafna17a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Bhaskar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', Laxman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', Smith, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', and Thakurta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Dis-  covering frequent patterns in sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' In Proceed-  ings of the 16th ACM SIGKDD International Confer-  ence on Knowledge Discovery and Data Mining, KDD  ’10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' 503–512, New York, NY, USA, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Associa-  tion for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ISBN 9781450300551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1145/1835804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1835869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content=' Computational Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Statistics and Com-  puting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Springer, New York, NY, December 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content=' Probability Inequalities for sums of Bounded  Random Variables, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' 409–426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Springer New York,  New York, NY, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content='  Ilyas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content='  org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content='  Matousek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content=' ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Oxford University Press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ISBN  978-0-19-857042-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content='  Permute-and-flip:  A  new  mechanism  for  differentially  private  selec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content='  In 48th Annual IEEE Symposium  on Foundations of Computer Science (FOCS 2007),  October 20-23, 2007, Providence, RI, USA, Proceed-  ings, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content=' IEEE Computer Society, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content=' Probability and computing:  Randomization and probabilistic techniques in algorithms  and data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content='  Oneshot differen-  tially private top-k selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content=' The Ameri-  can Mathematical Monthly, 62(1):26–29, 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ISSN  00029890, 19300972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content=' Pearson, Upper  Saddle River, NJ, 10 edition, November 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content=' ),  58th IEEE Annual Symposium on Foundations of Com-  puter Science, FOCS 2017, Berkeley, CA, USA, Octo-  ber 15-17, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content='  The complexity of differential pri-  vacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  In Lindell, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content=' Springer  International Publishing,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
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+page_content='  van Emde Boas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Preserving order in a forest in less  than logarithmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' In 16th Annual Symposium on  Foundations of Computer Science, Berkeley, Califor-  nia, USA, October 13-15, 1975, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' 75–84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' IEEE Com-  puter Society, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1109/SFCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' URL  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1109/SFCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', Mironov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', and Hejazinia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Wide  network learning with differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  CoRR,  abs/2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='01294, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='org/  abs/2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='01294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Tight Data Access Bounds for Private Top-k Selection  11  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Probability Inequalities  Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1 (Chernoff Bound, (Mitzenmacher & Upfal 2017, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Xn be independent  Poisson trials such that, for i ∈ [m], Pr [Xi = 1] = pi, where 0 < pi < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Then, for X = �n  i=1 Xi, µ = E [X],  Pr [X ≥ (1 + λ)µ] ≤  �  eλ  (1 + λ)1+λ  �µ  ,  ∀λ > 0,  (12)  Pr [X ≥ (1 + λ)µ] ≤  �  eλ  (1 + λ)1+λ  �µ  ≤ e−λ2µ/3,  ∀λ ∈ (0, 1],  (13)  Pr [X ≤ (1 − λ)µ] ≤  �  e−λ  (1 − λ)1−λ  �µ  ≤ e−λ2µ/2,  ∀λ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (14)  The known concentration inequalities for sampling with replacement can be transferred to the case of sampling without  replacement, based on a notable reduction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2 ((Hoeffding, 1994) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , xN) be a finite population of N real points, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Yn denote a random  sample without replacement from X and X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Xn denote a random sample with replacement from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' If f : R → R is  continuous and convex, then  E  �  f  � n  �  i=1  Yi  ��  ≤ E  �  f  � n  �  i=1  Xi  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  In particular, the lower bound presented in Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1 can be converted into the following one, by combining its proof  in (Mitzenmacher & Upfal, 2017) and Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3 (Chernoff bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , xN) ∈ {0, 1}N be a finite population of N binary points and  Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Yn be a random sample drawn without replacement from the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Then, for Y = �n  i=1 Yi,  Pr  � n  �  i=1  Yi ≤ (1 − λ)µ  �  ≤  �  e−λ  (1 − λ)1−λ  �µ  ≤ e−λ2µ/2,  ∀λ ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (15)  where µ = np is the expectation of �n  i=1 Yi, and p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= 1  N  �N  i=1 xi is the mean of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Note that, in Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3, we also extend the range of λ from {0, 1}, to [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' When λ = 0,  �  e−λ  (1−λ)1−λ  �µ  = e−λ2µ/2 = 1,  and Inequality (15) holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='4 ((Blitzstein & Hwang, 2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let Γ(a)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  � ∞  0  xa−1e−x dx, ∀a > 0 be the Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  It holds  that Γ(a + 1) = aΓ(a), and Γ( 1  2) = √π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Then for each k ∈ N+, we have Γ(k) = (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' and  Γ  �1  2 + k  �  =  �  k − 1  2  ��  k − 3  2  �  · · 1  2Γ  �1  2  �  = 2k − 1  2  2k − 3  2  · · 1  2  √π = (2k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  4kk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  √π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (16)  Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5 ((Ross, 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Given shape parameters α, β > 0, define beta function B(α, β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  � 1  0 xα−1(1 − x)β−1 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Then  B(α, β) = Γ(α)Γ(β)  Γ(α+β) , In particular,  B(α, β) = (α − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (β − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (α + β − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  ,  ∀α, β ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (17)  The factorials can be estimated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='6 (Stirling’s Approximation (Robbins, 1955;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Matousek & Nesetril, 2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' For k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  √  2πk  �k  e  �k  exp  �  1  12k + 1  �  ≤ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ≤  √  2πk  �k  e  �k  exp  � 1  12k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (18)  Tight Data Access Bounds for Private Top-k Selection  12  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Proofs for Section 4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Proofs for Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1  Proof for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' For each i ∈ [m], let pZi(·) be the density function of random variable Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Since the Zi’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  random variables, they share the same density function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', pZ1(·) = · · · = pZm(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Let Sm be the collection of all possible permutations over [m], and s refer to a permutation in Sm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Further, let z(1) ≥  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ≥ z(m) denote a possible realization of Z(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Z(m), and z(1:m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= (z(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , z(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We write I(1:m) = s, if I(j) =  s(j), ∀j ∈ [m], and Z(1:m) = z(1:m), when Z(j) = z(j), ∀j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Let pI(1:m),Z(1:m)(s, z(1:m)) be the probability density when I(1:m) = s and Z(1:m) = z(1:m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The probability den-  sity, pZ(1:m)(z(1:m)) of Z(1:m) = z(1:m) is given by  pZ(1:m)(z(1:m)) =  �  s∈Sm  pI(1:m),Z(1:m)(s, z(1:m))  =  �  s∈Sm  �  j∈[m]  pZs(j)(z(j))  =  �  s∈Sm  �  j∈[m]  pZ1(z(j))  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  �  j∈[m]  pZ1(z(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Hence, for a given s ∈ Sm, the probability density of I(1:m) = s, conditioned on Z(1:m) = z(1:m), is given by  pI(1:m)|Z(1:m)(s | z(1:m)) =  pI(1:m),Z(1:m)(s, z(1:m))  pZ(1:m)(z(1:m))  =  �  j∈[m] pZ1(z(j))  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  j∈[m] pZ1(z(j)) = 1  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ,  which is independent of the values of the Z(1:m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Finally,  pI(1:m)(s) =  E  Z(1:m)  �  pI(1:m)|Z(1:m)(s | Z(1:m))  �  = 1  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='.  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let cost (APrivTA, L1) be the access cost of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Pr [cost (APrivTA, L1) ≥ 2 · r] ≤ ek  kk · (r2/m)  k  er2/m ,  ∀r ∈ N, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', r ≥  √  mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (19)  Proof for Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let i(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , i(r) ∈ [m] be the top-r items with highest scores in ⃗h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Also consider the items in the  first r tuples in the array L2, denoted by I(1:r) =  �  I(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , I(r)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Consider the following event:  E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  ���  i(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , i(r)  �  ∩ I(1:r)  �� > k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  We claim that  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' When event E happens, cost (APrivTA, L1) < 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The complement of E, denoted by ¯E, happens with probability Pr  � ¯E  �  ≤ ek  kk · (r2/m)k  er2/m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Combing both claims, we get  Pr [cost (APrivTA, L1) ≥ 2r] ≤ Pr  � ¯E  �  ≤ ek  kk · (r2/m)  k  er2/m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Tight Data Access Bounds for Private Top-k Selection  13  Claim One.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' It suffices to show that, when event E happens, Algorithm 3 stops before r rounds of parallel sorted access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' In  such a case, the number of random accesses incurred is less than r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Therefore, the total number of access cost to ⃗h is less  than 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Assume that Algorithm 1 invokes r rounds of parallel sorted accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Since i(r) and I(r) are the last encountered items, the  corresponding threshold (Algorithm 1, Line 1) is given by τr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= f(⃗h[i(r)], ZI(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  But for each i ∈  �  i(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , i(r)  �  ∩ I(1:r), it holds that ⃗h[i] ≥ ⃗h[i(r)], and Zi ≥ ZI(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Since f is monotone, the score of  item i, f(⃗h[i], Zi), is at least τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Since event E happens, at least k items encountered by the algorithm have score at least τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Therefore, the stopping condition of Algorithm 1 should be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Claim Two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Via Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1, the I(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , I(m) distribute uniformly over all permutations of [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Therefore, the set I(1:r)  can be viewed as a uniform sample (without replacement) of r elements from [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Hence, for each j ∈ [r], the probability  that i(j) belongs to I(1:r) is given by  Pr  �  i(j) ∈ I(1:r)  �  = r/m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Let 1[i(j)∈I(1:r)] be the indicator for the event that i(j) ∈ I(1:r), and S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= �  j∈[r] 1[i(j)∈I(1:r)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The event ¯E is equivalent  to S ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Observe that  Pr  �  1[i(j)∈I(1:r)] = 1  �  = Pr  �  i ∈ I(1:r)  �  = r/m,  µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= E [S] =  �  j∈[r]  E  �  1[i(j)∈I(1:r)]  �  = r2/m ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (20)  Since k/µ ∈ (0, 1], λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= 1 − k/µ ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' By the Chernoff bound for sampling without replacement (Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3), we have  Pr [S ≤ k] = Pr [S ≤ (k/µ) · µ] ≤  �  e−λ  (1 − λ)1−λ  �µ  =  �  ek/µ−1  (k/µ)k/µ  �µ  =  ek−µ  (k/µ)k = ekµk  kkeµ = ek  kk · (r2/m)  k  er2/m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The expectation of cost (APrivTA, L1) is given by  E [cost (APrivTA, L1)] =  � ∞  0  Pr [cost (APrivTA, L1) ≥ s] ds  = 2 ·  � ∞  0  Pr [cost (APrivTA, L1) ≥ 2r] dr  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= s/2  = 2 ·  � √  mk+1  0  Pr [cost (APrivTA, L1) ≥ 2r] dr + 2 ·  � ∞  √  mk+1  Pr [cost (APrivTA, L1) ≥ 2r] dr  (a)  ≤ 2 ·  � √  mk+1  0  Pr [cost (APrivTA, L1) ≥ r] dr + 2 ·  � ∞  √  mk+1  Pr [cost (APrivTA, L1) ≥ 2⌊r⌋] dr  (b)  ≤ 2 ·  √  mk + 2 + 2 ·  � ∞  √  mk+1  ek  kk · (⌊r⌋2/m)  k  e⌊r⌋2/m  dr  (c)  ≤ 2 ·  √  mk + 2 + 2 ·  � ∞  √  mk+1  ek  kk · ((r − 1)2/m)  k  e(r−1)2/m  dr  = 2 ·  √  mk + 2 + 2 · ek  kk ·  � ∞  4k  tke−k √m dt  2  √  t  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= (r − 1)2/m  ≤ 2 ·  √  mk + 2 + √m ek  kk ·  � ∞  0  tk−1/2e−k dt  (d)  ≤ 2 ·  √  mk + 2 + √m ek  kk · Γ(k + 1/2)  (e)  ≤ 2 ·  √  mk + 2 + √m ek  kk · (2k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  4kk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (f)  ≤  �  2 +  √  2  � √  mk + 2,  Tight Data Access Bounds for Private Top-k Selection  14  where Inequality (a) follows since Pr [cost (APrivTA, L1) ≥ 2r] ≤ Pr [cost (APrivTA, L1) ≥ 2⌊r⌋] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Inequality (b) follows  from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Inequality (c) follows since the function y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= xk/ex (with y′ = kxk−1ex−xkex  e2x  ) is decreasing when x ≥ k,  and that (r − 1)2/m ≥ k, when r ≥  √  mk + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Inequalities (d) and (e) follows from Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Inequalities (f) follows  Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='6 and that  ek  kk · (2k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  4kk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ≤ ek  kk ·  √  2π · 2k  � 2k  e  �2k exp  �  1  12·2k  �  4k√  2π · k  � k  e  �k exp  �  1  12·k+1  � ≤  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (21)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Proofs for Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' This is a directly consequence of the facts that  Based on a technique by Brassard & Kannan (1988) for sampling a random perturbation on the fly, and on the  discussions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2, it holds that each I(j), j ∈ [m], and each value of inverted index σ2(i), i ∈ [m] can be  sampled on the fly in O(1) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Based on Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='4, each Z(j), j ∈ [m] can be sampled on the fly with O(log log m) expected time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' First, observe that the following two multisets are the same  �  F −1(U1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , F −1(Um)  �  and  �  F −1(U(1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , F −1(U(m))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' By construction, Z(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Z(m) is a sorted sequence of the multiset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' To prove  the claim, it suffices to show that F −1(U(1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , F −1(U(m)) is also a sorted sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' This is true: since F is a cumulative  distribution function on R, it is non-decreasing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' it follows that for every two 1 ≤ j < j′ ≤ m, U(j) ≥ U(j′) implies that  F −1(U(j)) ≥ F −1(U(j′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' If X ∼ Beta (m − j + 1, j), then its density function f(x) satisfies f(x) = (x)m−j(1−x)j−1  B(m−j+1,j)  , ∀x ∈  [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Based on Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5, B(m − j + 1, j) = (m−j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (j−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  , which proves the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  If X ∼ Beta (m − j + 1, j − ℓ), then the density of Y , denoted by ˜f(y), is given by ∀y ∈ [0, u(ℓ)],  ˜f(y) = f  � y  u(ℓ)  �  dX  dY =  1  B(m − j + 1, j − ℓ) ·  � y  u(ℓ)  �m−j�  1 −  � y  u(ℓ)  ��j−ℓ−1 1  u(ℓ)  ,  Noting that B(m − j + 1, j − ℓ) = (m−j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (j−ℓ−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (m−ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  proves the second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  The third claim follows from similar argument as the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Proof for Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='7  We need the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Fact B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2 ((Guntuboyina, 2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The joint density of U(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , U(m) is given by  p(u(1), u(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , u(m)) = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ,  (22)  where 1 ≥ u(1) ≥ u(2) · · · ≥ u(m) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  As a sanity check, note that the simplex ∆∗  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  �  u(1), u(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , u(m) ∈ Rm : 1 ≥ u(1) ≥ u(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ≥ u(m) ≥ 0  �  has vol-  ume 1/m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='. Since U(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , U(m) distribute uniformly over ∆∗  n, each point of the simplex has density m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='. Based on this fact,  we can derive the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Tight Data Access Bounds for Private Top-k Selection  15  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , jt ∈ [m] be an increasing sequence of indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Then the joint distribution of U(j1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , U(jt) is  given by  p  �  u(j1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , u(jt)  �  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(jt)  �m−jt  (m − jt)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  1 − u(j1)  �j1−1  (j1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' t�  s=2  �  u(js−1) − u(js)  �js−js−1−1  (js − js−1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (23)  Proof of Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' p  �  u(j1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , u(jt)  �  is given by the following integration:  � 1  0  · ·  � 1  0  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' · 1[0≤u(m)≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='≤u(1)≤1] du(m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1+jt) du(−1+jt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1+jt−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(−1+j1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Integrating with respect to u(m) over the range [0, u(m−1)], we have  � 1  0  · ·  � 1  0  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' · u(m−1) · 1[0≤u(m−1)≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='≤u(1)≤1] du(m−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1+jt) du(−1+jt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1+jt−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(−1+j1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Then integrate u(m−1) over the range [0, u(m−2)], all the way down to the integral with respect to u(1+jt) over the  range [0, u(jt)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We obtain  � 1  0  · ·  � 1  0  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(jt)  �m−jt  (m − jt)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  1[0≤u(jt)≤u(−1+jt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='≤u(1)≤1] du(−1+jt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1+jt−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(−1+j1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Integrating with respect to u(−1+jt) over the range [u(jt), u(−2+jt)], we have  � 1  0  · ·  � 1  0  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  �  u(jt)  �m−jt  (m − jt)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  �  u(−2+jt) − u(jt)  �  1[0≤u(jt)≤u(−2+jt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='≤u(1)≤1] du(−2+jt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1+jt−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(−1+j1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Then integrate u(−2+jt) over the range [u(jt), u(−3+jt)], all the way down to u(1+jt−1) over the range [u(jt), u(jt−1)]:  � 1  0  · ·  � 1  0  �  �  �  �  �  �  �  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='·  �  u(jt)  �m−jt  (m − jt)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  u(jt−1) − u(jt)  �jt−jt−1−1  (jt − jt−1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  1[0≤u(jt−1)≤u(−1+jt−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='≤u(1)≤1]  �  �  �  �  �  �  �  du(−1+jt−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1+jt−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(−1+j1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' du(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Repeating the above efforts proves Equation (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Claim One: Applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3 directly, we have  p  �  u(j)  �  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(j)  �m−j  (m − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  1 − u(j)  �j−1  (j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  ,  ∀u(j) ∈ [0, 1],  which proves the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Claim Two: let J = {ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , ζc}, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', ζ1 < · · · < ζc < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Following the notation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='7, we have ℓ = ζc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' By  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3,  p  �  u(J )  �  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(ζc)  �m−ζc  (m − ζc)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  1 − u(ζ1)  �ζ1−1  (ζ1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' c  �  s=2  �  u(ζs−1) − u(ζj)  �ζs−ζs−1−1  (ζs − ζs−1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  ,  p  �  u(J ∪{j})  �  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(j)  �m−j  (m − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(ζc) − u(j)  �j−ζc−1  (j − ζc − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  1 − u(ζ1)  �ζ1−1  (ζ1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' c  �  s=2  �  u(ζs−1) − u(ζs)  �ζs−ζs−1−1  (ζs − ζs−1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Tight Data Access Bounds for Private Top-k Selection  16  Hence,  p  �  u(j) | u(J )  �  = p  �  u(J ∪{j})  �  p  �  u(J )  �  =  (m − ζc)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (j − ζc − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (m − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  �u(ζc) − u(j)  u(ζc)  �j−ζc−1� u(j)  u(ζc)  �m−j  1  u(ζc)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Similarly, the densities of p  �  u(ζc)  �  , p  �  u(ζc), u(j)  �  are given by  p  �  u(ζc)  �  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(ζc)  �m−j  (m − ζc)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  1 − u(ζc)  �ζc−1  (ζc − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  ,  p  �  u(ζc), u(j)  �  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(j)  �m−j  (m − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(ζc) − u(j)  �j−ζc−1  (j − ζc − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  1 − u(ζc)  �ζc−1  (ζc − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  It is easy to see that  p  �  u(j) | u(ζc)  �  = p  �  u(ζc), u(j)  �  p  �  u(ζc)  �  = p  �  u(j) | u(J )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Claim Three: let J = {ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , ζc}, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', ζ1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' < ζc, and there exists c′ ∈ [c − 1] for which ζc′ < j < ζc′+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Following  the notation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='7, we have ℓ = ζc′, r = ζc′+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3,  p  �  u(J )  �  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(ζc)  �m−ζc  (m − ζc)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  1 − u(ζ1)  �ζ1−1  (ζ1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' c  �  s=2  �  u(ζs−1) − u(ζs)  �ζs−ζs−1−1  (ζs − ζs−1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  ,  p  �  u(J ∪{j})  �  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(ζc)  �m−ζc  (m − ζc)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  1 − u(ζ1)  �ζ1−1  (ζ1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  u(ζc′) − u(j)  �j−ζc′−1  (j − ζc′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  u(j) − u(ζc′+1)  �ζc′+1−j−1  (ζc′+1 − j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' c′  �  j=2  �  u(ζs−1) − u(ζs)  �ζs−ζs−1−1  (ζs − ζs−1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' c  �  j=c′+2  �  u(ζs−1) − u(ζs)  �ζs−ζs−1−1  (ζs − ζs−1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Hence,  p  �  u(j) | u(J )  �  = p  �  u(J ∪{j})  �  p  �  u(J )  �  =  (ζc′+1 − ζc′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (j − ζc′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' (ζc′+1 − j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  u(ζc′) − u(j)  u(ζc′) − u(ζc′+1)  �j−ζc′−1�  u(j) − u(ζc′+1)  u(ζc′) − u(ζc′+1)  �ζc′+1−j−1  1  u(ζc′) − u(ζc′+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Similarly, the densities of p  �  u(ζc′), u(ζc′+1)  �  , p  �  u(ζc′), u(j), u(ζc′+1)  �  is given by  p  �  u(ζc′), u(ζc′+1)  �  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(ζc′+1)  �m−ζc′+1  (m − ζc′+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  u(ζc′) − u(ζc′+1)  �ζc′+1−ζc′−1  (ζc′+1 − ζc′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  1 − u(ζc′)  �ζc′−1  (ζc′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  ,  p  �  u(ζc′), u(j), u(ζc′+1)  �  = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ·  �  u(ζc′+1)  �m−ζc′+1  (m − ζc′+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  u(ζc′) − u(j)  �j−ζc′−1  (j − ζc′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  u(j) − u(ζc′+1)  �ζc′+1−j−1  (ζc′+1 − j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' �  1 − u(ζc′)  �ζc′−1  (ζc′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  It is easy to see that  p  �  u(j) | u(ζc′), u(ζc′+1)  �  =  p  �  u(ζc′), u(j), u(ζc′+1)  �  p  �  u(ζc′), u(ζc′+1)  �  = p  �  u(j) | u(J )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Tight Data Access Bounds for Private Top-k Selection  17  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Proofs for Section 5  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Proofs for Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' We do not specify the family H and the distribution µ on H directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Instead, we show how we can  sample a ⃗h from H according to µ: for each i ∈ [m], independently set  ⃗h[i] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  �  n,  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' 2k  m ,  0,  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' 1 − 2k  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (24)  This can also be understood as, for each item i ∈ [m], with probability 2k/m all of the n clients votes for i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' and with  probability 1 − 2k/m, none of the clients votes for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Assume that: 1) A retrieves entries from ⃗h (via random access) without repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' This only decreases its access cost, since  accessing a previously encountered entry does not provide additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' 2) If A terminates before retrieving all  entries in ⃗h, it continues to read the remaining entries without being charged for the additional accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' This enables A  to obtain more entries for free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Now, let J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Jm ∈ [m] be the order in which A accesses the entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' The sequence  constitutes a permutation of [m], and for each t ∈ [m], the choice of Jt can depend on previous choices J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Jt−1 and  outcomes ⃗h[J1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ,⃗h[Jt−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' However, whatever the choice of Jt is, the distribution of ⃗h[Jt] (conditioned on previous  choices and outcomes) is till given by Equation (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Hence, ⃗h[J1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ,⃗h[Jm] can be viewed as independent random  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Consider the following events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Event E1 : ⃗h has at least k non-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Since each entry of ⃗h is generated independently,  via Chernoff bound (Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1),  Pr  � ¯  E1  �  ≤ exp  �  −(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5)2 · 2k  2  �  ≤ exp (−(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5)2) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='78 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Event E2 : the number of non-zero entries among ⃗h[J1] , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ,⃗h[Jm/50] is less than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Since ⃗h[J1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' ,⃗h[Jm/50] are  independent, via Chernoff bound (Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1), and noting that k ≥ 1,  Pr  � ¯  E2  �  ≤  � e24  2525  �k/25  ≤  � e24  2525  �1/25  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Event E3 : A returns an (n − 1, k)-accurate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' By assumption, we have Pr  � ¯  E3  �  ≤ β < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Hence, Pr [E1 ∩ E2 ∩ E3] ≥ 1 − Pr  � ¯  E1  �  − Pr  � ¯  E2  �  − Pr  � ¯  E3  �  ∈ Ω(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Observe that, when E1 and E3 happens, each  item i returned by A must have frequency ⃗h[i] = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' However, when E2 happens, A cannot see k items with non-zero  frequency, from its first m/50 retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  It follows that , with probability at least Pr [E1 ∩ E2 ∩ E3], A has access cost at least m/50, which proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Proofs for Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Let O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= {o ⊆ [m] : |o| = k} is the collection of all possible outputs of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Observe that, given an  input histogram ⃗hS or ⃗hSH,SL, if A’s output is (n − 2, k)-accurate, then it must be a subset of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Therefore, define G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  {o ⊆ S : |o| = k} to be the collection of all (n − 2, k)-accurate outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Via the assumption that A outputs an (n − 2, k)-  accurate solution with probability at least 1 − β, it holds that  Pr  �  A(⃗h) ∈ G  �  ≥ 1 − β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Next, let F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= {sℓ ⊆ S : |sℓ| = |S|/k} be the collection all possible outcomes of SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' For each sℓ ∈ F, define Osℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='=  {o ∈ O : o ∩ sℓ ̸= ∅}, the collection of sets in O that has nonempty intersection with sℓ, and Gsℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='= {o ∈ G : o ∩ sℓ ̸= ∅},  the collection of sets in G that has nonempty intersection with sℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Conditioned on SL = sℓ,  Pr  �  A(⃗hSH,SL) ∩ SL ̸= ∅ | SL = sℓ  �  = Pr  �  A(⃗hSH,SL) ∈ Osℓ | SL = sℓ  �  ≥ Pr  �  A(⃗hSH,SL) ∈ Gsℓ | SL = sℓ  �  ,  Tight Data Access Bounds for Private Top-k Selection  18  where the inequality holds as Osℓ ⊇ Gsℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Moreover, since A is (ε, δ)-DP, it holds that  Pr  �  A(⃗hSH,SL) ∈ Gsℓ | SL = sℓ  �  ≥ 1  eε (Pr [A(h) ∈ Gsℓ] − δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (25)  Further, since the events that A(h) = o are mutually exclusive for different values of o ∈ G, we have  Pr [A(h) ∈ G] =  �  o∈G  Pr [A(h) = o] ≥ 1 − β ,  (26)  Pr [A(h) ∈ Gsℓ] =  �  o∈G,o∩sℓ̸=∅  Pr [A(h) = o] =  �  o∈G  1[o∩sℓ̸=∅] · Pr [A(h) = o] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  (27)  Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Pr  SL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='A  �  A(⃗hSH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='SL) ∩ SL ̸= ∅  �  =  �  sℓ∈F  Pr [SL = sℓ] · Pr  �  A(⃗hSH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='SL) ∈ Osℓ | SL = sℓ  �  ≥  �  sℓ∈F  Pr [SL = sℓ] · Pr  �  A(⃗hSH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='SL) ∈ Gsℓ | SL = sℓ  �  ≥  �  sℓ∈F  Pr [SL = sℓ] · e−ε  ��  o∈G  1[o∩sℓ̸=∅] · Pr [A(h) = o] − δ  �  = −e−εδ + e−ε �  o∈G  Pr [A(h) = o]  � �  sℓ∈F  1[o∩sℓ̸=∅] · Pr [SL = sℓ]  �  (a)  = −e−ε · δ + e−ε �  o∈G  Pr [A(h) = o]  �  1 −  �|S|−k  |S|/k  �  � |S|  |S|/k  �  �  (b)  ≥ −e−ε · δ + e−ε �  o∈G  Pr [A(h) = o] (1 − exp (−1))  ≥ −e−ε · δ + e−ε(1 − β)  �  1 − e−1�  ≥ e−ε�  1 − β − δ − e−1�  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  where equation (a) follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' since �  sℓ∈F 1[o∩sℓ̸=∅]·Pr [SL = sℓ] can interpreted as the probability that given a subset o ⊆ S  of size |k|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' the sampled subset SL has nonempty intersection with o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' and inequality (b) follows, since  �|S|−k  |S|/k  �  � |S|  |S|/k  � = |S| − k  |S|  · · |S| − k − |S|/k + 1  |S| − |S|/k + 1  ≤  �|S| − k  |S|  �|S|/k  ≤ exp  �  − k  |S| · |S|  k  �  = exp (−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Proofs for Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='3  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Assume that: 1) if A terminates before performing η accesses operations, it continues to perform more  until it reaches η, and will not be charged for any additional access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Now, let J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Jη represents the first η operations  of A: each Jt is either a character ‘s’, implying that the t(th) operation is a sorted access, or an integer in [m], implying  that the t(th) operation is a random access to the entry ⃗hSH,SL[Jt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Further, let J′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , J′  t be the subsequence of all random  access in J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , Jη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Tight Data Access Bounds for Private Top-k Selection  19  Consider an alternative algorithm A′ that operates as follows: it first performs 9η sorted accesses to obtain all frequencies of  all items from SH, followed by random accesses J′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , J′  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' It is clear that, A′ retrieves a greater number of entries than the  first η operations of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  We bound the probability that A′ does not retrieve an entry from SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' First consider random access J′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' If J′  1 had already  been retrieved by sorted access, then clearly J′  1 /∈ SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Otherwise, conditioned on the fact that the entries SH have been  determined, according to the manner S and SL are generated, each item in [m] \\ (SH) belongs to SL with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Hence,  Pr [J′  1 /∈ SL] ≥ 1 −  τ/k  m − (k − 1)τ/k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  In general, for each 1 < ℓ ≤ t, suppose that J′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , J′  ℓ−1 does not belong to SL, and has revealed mℓ ≤ ℓ − 1 distinct items  from [m] \\ (SH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Whatever the choice of J′  ℓ is, and the items J′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , J′  ℓ−1 are, each of the remaining items that have not  been queried belong to SL with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' Hence  Pr [J′  ℓ /∈ SL] ≥ 1 −  τ/k  m − (k − 1)τ/k − mℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  Noting that mℓ < t ≤ η, we have  Pr [J′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=' , J′  t /∈ SL] ≥  t�  ℓ=1  �  1 −  τ/k  m − (k − 1)τ/k − mℓ  �  ≥  t�  ℓ=1  �  1 −  τ/k  m − (k − 1)τ/k − η  �  (a)  ≥  t�  ℓ=1  exp  �  −  2τ/k  m − (k − 1)τ/k − η  �  ≥ exp  �  −  τ 2/(10k)  m − (k − 1)τ/k − η  �  ≥ exp  �  −  m/10  m − 21  √  mk/20  �  ≥ exp  �  −  m/10  m − 21m/40  �  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='  where inequality (a) holds, since 1 − x ≥ e−2x, ∀x ∈ [0, 3/4], and  τ/k ≤ 3  4(m − (k − 1)τ/k − η)  (28)  ⇐⇒ τ  �3  4 + 1  4k + 3  40  �  ≤ 3  4m  (29)  The last inequality holds, since 3  4 +  1  4k + 3  40 ≤ 3/2 for k ∈ N+, and 2τ = 2  √  mk ≤ m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
+page_content=', k ≤ m/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FQT4oBgHgl3EQfhzZX/content/2301.13347v1.pdf'}
diff --git a/bdE2T4oBgHgl3EQfaAfc/content/tmp_files/2301.03871v1.pdf.txt b/bdE2T4oBgHgl3EQfaAfc/content/tmp_files/2301.03871v1.pdf.txt
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@@ -0,0 +1,778 @@
+MS-TP-22-56
+Sigma terms of the baryon octet in 𝑵f = 2 + 1 QCD
+with Wilson quarks
+Pia Leonie Jones Petrak,𝑎,∗ Gunnar Bali,𝑏 Sara Collins,𝑏 Jochen Heitger,𝑎 Daniel
+Jenkins𝑏 and Simon Weishäupl𝑏
+𝑎Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster,
+Wilhelm-Klemm-Straße 9, 48149 Münster, Germany
+𝑏Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany
+E-mail: p_petr04@uni-muenster.de, gunnar.bali@ur.de,sara.collins@ur.de,
+heitger@uni-muenster.de, Daniel.Jenkins@ur.de, Simon.Weishaeupl@ur.de
+A lot of progress has been made in the direct determination of nucleon sigma terms. Using similar
+methods, we consider the sigma terms of the other octet baryons as well. These are determined on
+CLS gauge field ensembles employing the Lüscher-Weisz gluon action and the Sheikholeslami-
+Wohlert fermion action with 𝑁f = 2 + 1. The ensembles have pion masses ranging from 410 MeV
+down to the physical value and lattice spacings covering a range between 0.098 fm and 0.039 fm.
+We present some preliminary results for the pion and strange sigma terms and compare to indirect
+determinations. To do so, we discuss multi-state fits to tackle the well-known problem of excited
+state contamination comparing the ratio and summation methods also including priors.
+The 39th International Symposium on Lattice Field Theory,
+8th-13th August, 2022,
+Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.03871v1  [hep-lat]  10 Jan 2023
+
+Sigma terms of the octet baryons
+Pia Leonie Jones Petrak
+1.
+Introduction
+The sigma terms encode the quark contributions to the mass of a baryon. They are defined as
+the matrix element of a scalar current 𝐽 times the quark mass such that
+𝜎𝑞𝐵 = 𝑚𝑞⟨𝐵|𝐽|𝐵⟩ ,
+(1)
+where 𝐽 = ¯𝑞 1 𝑞 and the quark flavour 𝑞 ∈ {𝑢, 𝑑, 𝑠}. In order to compare with phenomenological
+determinations, the pion-baryon sigma terms are usually constructed, 𝜎𝜋𝐵 = 𝜎𝑢𝐵 + 𝜎𝑑𝐵. In the
+matrix element, 𝐵 refers to the ground state of a baryon 𝐵. Of particular interest are the nucleon
+sigma terms (𝐵 = 𝑁) which appear in the expressions for WIMP-nucleon scattering cross-sections
+and are relevant for comparing model predictions to the exclusion bounds obtained from direct
+detection dark matter experiments (such as the XENON1T experiment).
+In the analysis, we make use of and adjust methods established for the nucleon (see ref. [1]
+for a review). We determine the sigma terms of the baryon octet, i.e. of the lambda Λ, sigma Σ
+and cascade Ξ baryons. This enables us to investigate SU(3) flavour symmetry breaking and to
+determine the SU(3) low energy constants (LECs), that are currently not well known. There have
+been relatively few determinations so far, see, for example, refs. [2] and [3], with most previous
+studies focusing on the nucleon sigma terms (see ref. [4] for a review).
+In addition, discrepancies between results for the pion-nucleon sigma term from Lattice QCD
+and phenomenology are still to be resolved (see [4], and e.g., [5–7]). In a recent paper, results more
+consistent with phenomenology were obtained by explicitly including 𝑁𝜋 and 𝑁𝜋𝜋 excited states
+in the analysis [8, 9]. By considering baryons other than the nucleon, we aim to understand excited
+state contributions and other issues in more detail so as to help solve this puzzle.
+2.
+Excited state analysis
+In order to be able to extract the ground-state matrix element, we need to take care of the
+excited state contamination. Two possible approaches are the ratio method and the summation
+method (reviewed in [1, 10]) that are both based on spectral decompositions. We consider the
+two- and three-point functions of a baryon (from the octet) at rest in the initial and final state. The
+spectral decomposition of the two-point function reads
+𝐶2pt(𝑡f) =
+∑︁
+�𝑥
+�
+Osnk(�𝑥, 𝑡f) ¯Osrc(�0, 0)
+�
+=
+∑︁
+𝑛
+|𝑍𝑛|2𝑒−𝐸𝑛𝑡f ,
+(2)
+where 𝑍𝑛 ∝ ⟨Ω|Osnk|𝑛⟩ is the overlap of the interpolator Osnk onto the state 𝑛 (and Ω the vacuum
+state) and 𝑡f the source-sink separation. Summation over spin and colour indices and projection
+onto positive parity are implied. These indices become apparent when writing down the operators
+explicitly. The interpolators for the four octet baryons are set to
+O 𝛼,N
+snk = 𝜖 𝑎𝑏𝑐𝑢𝛼
+𝑎
+�
+𝑢𝛽
+𝑏(𝐶𝛾5)𝛽𝛾𝑑𝛾
+𝑐
+�
+and
+O 𝛼,Λ
+snk = 𝜖 𝑎𝑏𝑐𝑠𝛼
+𝑎
+�
+𝑢𝛽
+𝑏(𝐶𝛾5)𝛽𝛾𝑑𝛾
+𝑐
+�
+,
+O 𝛼,Σ
+snk = 𝜖 𝑎𝑏𝑐𝑢𝛼
+𝑎
+�
+𝑢𝛽
+𝑏(𝐶𝛾5)𝛽𝛾𝑠𝛾
+𝑐
+�
+and
+O 𝛼,Ξ
+snk = 𝜖 𝑎𝑏𝑐𝑠𝛼
+𝑎
+�
+𝑠𝛽
+𝑏(𝐶𝛾5)𝛽𝛾𝑢𝛾
+𝑐
+�
+.
+(3)
+2
+
+Sigma terms of the octet baryons
+Pia Leonie Jones Petrak
+𝑎, 𝑏, 𝑐 are colour indices, 𝛼, 𝛽, 𝛾 are spin indices and O 𝛼
+src = O 𝛼
+snk and ¯Osrc = O†
+src𝛾4. 𝐶 stands for
+the charge conjugation operator. Note that for the Λ we use a naive interpolator that also has overlap
+with the (heavier) Σ0. Turning to the three-point function, its spectral decomposition reads
+𝐶3pt(𝑡f, 𝑡) =
+∑︁
+�𝑥,�𝑦
+�
+Osnk(�𝑥, 𝑡f)𝐽(�𝑦, 𝑡) ¯Osrc(�0, 0)
+�
+−
+∑︁
+�𝑥,�𝑦
+⟨𝐽(�𝑦, 𝑡)⟩
+�
+Osnk(�𝑥, 𝑡f) ¯Osrc(�0, 0)
+�
+=
+∑︁
+𝑛,𝑛′
+𝑍𝑛′𝑍∗
+𝑛⟨𝑛′|𝐽|𝑛⟩𝑒−𝐸𝑛𝑡𝑒−𝐸𝑛′ (𝑡f−𝑡),
+(4)
+where 𝑡 is the insertion time of the scalar current, 𝐽 = ¯𝑞 1 𝑞. As 𝐽 has the same quantum numbers
+as the vacuum, the vacuum expectation value needs to be subtracted, see the second term in the first
+line of eq. (4). Note that depending on the type of baryon and current, different Wick contractions
+(e.g. different currents) contribute that result in connected and disconnected quark-line diagrams.
+Ratio method:
+Taking the ratio of the two spectral decompositions and truncating after the first excited state
+contribution leads to
+𝑅(𝑡f, 𝑡) = 𝐶3pt(𝑡f, 𝑡)
+𝐶2pt(𝑡f)
+= 𝑔𝑞
+𝑆 + 𝑐01e−Δ · 𝑡 + 𝑐10e−Δ · (𝑡f−𝑡) + 𝑐11e−Δ · 𝑡f + ...
+(5)
+where 𝑔𝑞
+𝑆 = ⟨𝐵| ¯𝑞 1 𝑞|𝐵⟩ is the ground-state matrix element of interest. Δ = 𝐸1 − 𝐸0 is the energy
+gap between the ground and first excited state. The coefficients 𝑐01, 𝑐10, 𝑐11 are made up of matrix
+elements for different transitions, 𝐵0 → 𝐵1, 𝐵1 → 𝐵0 , and 𝐵1 → 𝐵1, respectively. Here, 𝐵0 refers
+to the ground state of the baryon while 𝐵1 denotes the first excited state (single- or multi-particle
+state). As the baryon is at rest, 𝑐01 = 𝑐10 ≡ 𝑐0↔1. Note that we labelled 𝑐0↔1 as 𝑐1 in [11].
+Summation method:
+Summing over the ratio (5) for a range of insertion times leads to
+𝑡f/𝑎−𝑐
+∑︁
+𝑡/𝑎=𝑐
+𝑅(𝑡f, 𝑡) = 𝑔𝑞
+𝑆(𝑡f/𝑎 − 2𝑐 + 1) + 2𝑐0↔1
+1 − e𝑎Δ
+�
+eΔ(𝑐−𝑡f) − eΔ(𝑎−𝑐)�
++ 𝑐11(𝑡f/𝑎 − 2𝑐 + 1)e−Δ · 𝑡f + ...
+(6)
+where 𝑎 refers to the lattice spacing and 𝑐 > 0 to preserve reflection positivity (we set 𝑐 = 2). An
+advantage of this method is that, in principle, the slope,
+d
+d(𝑡f/𝑎)
+�𝑡f/𝑎−𝑐
+𝑡/𝑎=𝑐 𝑅(𝑡f, 𝑡) = 𝑔𝑞
+𝑆 + O(𝑡f/𝑎 · e−Δ · 𝑡f) ,
+approaches the asymptotic value faster compared to the ratio method. However, a large number of
+source-sink separations is required and due to the the numerical setup we use (discussed in the next
+section) we can only employ this approach for the disconnected three-point functions.
+3.
+Numerical Setup
+We utilise the CLS gauge field ensembles [12], which are generated employing the Lüscher-
+Weisz gluon action and the Sheikholeslami-Wohlert fermion action with 𝑁f = 2 + 1 (𝑚𝑙 = 𝑚𝑢 =
+𝑚𝑑 ≤ 𝑚𝑠). The pion-baryon and strange sigma terms are determined on the seven ensembles
+highlighted in red (diamonds) in fig. 1.
+This includes five lattice spacings in the range from
+3
+
+Sigma terms of the octet baryons
+Pia Leonie Jones Petrak
+0.0392 0.052
+0.0642
+0.0762
+0.0862
+0.0982
+a2 [fm2]
+150
+200
+250
+300
+350
+400
+Mπ [MeV]
+A653
+A654
+U103
+H101
+H102
+H105
+C101
+B450
+S400
+N401
+D450
+D200
+N300
+N302
+J303
+E300
+J500
+J501
+J500
+J303
+N202
+N203
+N200
+S400
+C101
+physical point
+3.55
+3.85
+3.46
+3.34
+3.7
+3.4
+β
+Figure 1: Overview of a subset of the CLS
+ensembles which lie on a trajectory which starts
+at a symmetric point (𝑚𝑙 = 𝑚𝑠, 𝑀𝜋 ∼ 410 MeV)
+and extends to the physical point, along which
+the flavour average of the quark masses is held
+constant. So far the ensembles highlighted in
+red (diamonds) have been analysed.
+𝑎 = 0.08528(49) fm down to 𝑎 = 0.03875(22) fm. The pion masses vary from 411 MeV down to
+222 MeV, see also [13] 1.
+To compute the connected three-point correlation functions on the ensembles with 𝑚𝑙 =
+𝑚𝑠 and 𝑀𝜋 = 410 MeV, we used the standard sequential source method [15].
+On the other
+ensembles we employed the stochastic method described in [16, 17] (see also [18–21]).
+This
+approach enables us to obtain measurements for all baryons of interest at multiple source and
+insertion positions, simultaneously. Four different source-sink separations, corresponding to 𝑡f ≈
+[0.71 fm, 0.9 fm, 1.03 fm, 1.22 fm], are employed. Two measurements are performed for each 𝑡f on
+every configuration except for the 𝑚𝑙 = 𝑚𝑠 ensembles (that we used the sequential source method
+for) where ten measurements are typically performed (one, two, three and four measurements for
+𝑡f ≈ [0.71 fm, 0.9 fm, 1.03 fm, 1.22 fm], respectively).
+The disconnected three-point functions are constructed by correlating a quark loop with a
+baryon two-point function. The loop is estimated stochastically on every timeslice (within the bulk)
+leading to additional noise on top of the Monte-Carlo gauge sampling. In order to reduce the
+noise, the truncated solver method [22], the hopping parameter expansion technique [23] and time
+partitioning [24] are utilised. Between twenty and thirty measurements of the two-point functions
+(at different temporal source positions) are performed on each configuration. A reasonable signal for
+the disconnected three-point function is obtained for a source-sink separation of up to 𝑡f ≈ 1.22 fm.
+For the analysis of the statistical errors we employ the Γ-method [25] (that is based on autocorrelation
+functions) using the pyerrors python package [26].
+When constructing the sigma terms for Wilson-type fermions, one needs to take the mixing
+between different quark flavours into account, which arises due to the singlet and non-singlet
+combinations of the scalar current renormalising differently as a result of chiral symmetry breaking
+(see , e.g., [11] for the renormalisation pattern).
+For the ratio of renormalisation factors, 𝑟m,
+we use 𝑟m(𝛽 = 3.4) = 2.335(31), 𝑟m(𝛽 = 3.46) = 1.869(19), 𝑟m(𝛽 = 3.55) = 1.523(14),
+𝑟m(𝛽 = 3.7) = 1.267(16) and 𝑟m(𝛽 = 3.85) = 1.149(18)(27)[33], determined non-perturbatively
+in [27].
+1In [11] we investigated the quark mass dependence by performing a preliminary chiral extrapolation at 𝑎 ≈ 0.064 fm
+(including preliminary SU(3) LECs from [14]).
+4
+
+Sigma terms of the octet baryons
+Pia Leonie Jones Petrak
+−8
+−6
+−4
+−2
+0
+2
+4
+6
+8
+insertion time t/a − tf/(2a)
+2.2
+2.4
+2.6
+2.8
+3.0
+3.2
+3.4
+Rcon
+gcon
+s
+= 3.31(12), ccon
+0↔1 = -1.31(12)
+tf/a = 9
+tf/a = 11
+tf/a = 14
+tf/a = 16
+−8
+−6
+−4
+−2
+0
+2
+4
+6
+8
+insertion time t/a − tf/(2a)
+3.50
+3.75
+4.00
+4.25
+4.50
+4.75
+5.00
+5.25
+Rcon
+gcon
+s
+= 5.16(14), ccon
+0↔1 = -1.80(8)
+tf/a = 9
+tf/a = 11
+tf/a = 14
+tf/a = 16
+4
+6
+8
+10
+12
+14
+16
+source-sink separation tf/a
+0
+10
+20
+30
+40
+50
+60
+70
+�
+t Rdis
+gdis
+s
+= 5.66(72), cdis
+0↔1 = -4.01(1.11), cdis
+11 = 4.91(2.18)
+4
+6
+8
+10
+12
+14
+16
+source-sink separation tf/a
+0
+10
+20
+30
+40
+50
+60
+�
+t Rdis
+gdis
+s
+= 5.22(66), cdis
+0↔1 = -3.42(1.01), cdis
+11 = 4.15(2.03)
+Figure 2: The connected and (summed) disconnected ratios contributing to the unrenormalised scalar charges
+of the Ξ baryon on ensemble S400 (𝑀𝜋 ≈ 350 MeV) for the ¯𝑢𝑢 current (left) and the ¯𝑠𝑠 current (right):
+Simultaneous fits to the connected and summed disconnected ratios are indicated by the coloured shaded
+regions. The resulting connected ground state scalar matrix elements are displayed as horizontal grey bands.
+4.
+Fitting Analysis
+In order to extract the ground-state matrix elements of interest while taking the excited state
+contamination into account we perform multi-state fits, applying the ratio method, eq. (5) and, for
+comparison, for the disconnected contribution also the summation method, eq. (6) (as noted in
+sect. 2, there are not enough source-sink separations available to apply this method to the connected
+contributions).
+For each baryon, we fit the connected and disconnected ratios for all currents
+simultaneously, with the energy gap to the first excited state Δ as the common fit parameter. As an
+example, the ratios (and fits) relevant for determining the sigma terms of the Ξ baryon on the S400
+ensemble (𝑀𝜋 ≈ 350 MeV) are displayed in fig. 2 (see fig. 2 in [11] for an example of the ratio
+method being used for both the connected and disconnected contributions). While we were able
+to resolve the first excited state term, with the coefficient 𝑐0↔1, with both methods, the 𝐵1 → 𝐵1
+transition term (see sect. 2) was not resolved when employing the ratio method and we set 𝑐11 = 0
+in the analysis when using this method. The 𝜒2/𝜒2
+expected were mostly below 1.5 (and always below
+two), where 𝜒2
+expected provides an estimate of the effective degrees of freedom expected taking into
+account autocorrelations, see [28].
+To investigate whether the excited state contributions are sufficiently controlled when extracting
+the ground state matrix elements, we vary the fit range for both the connected and disconnected
+contributions and for the latter also the fit method. In addition, we consider the impact on the fit
+of a narrow-width prior for Δ. The fit range is varied by removing insertion times symmetrically
+5
+
+Sigma terms of the octet baryons
+Pia Leonie Jones Petrak
+4
+6
+8
+4
+6
+8
+4
+6
+8
+tmin
+f
+/a (tmax
+f
+/a = 16; dis)
+0.15
+0.20
+0.25
+0.30
+a∆
+symm. cutoff in t/a = 1 (con)
+symm. cutoff in t/a = 2 (con)
+symm. cutoff in t/a = 3 (con)
+Ξ(0)π(0)π(0)
+Ξ(1)π(−1)
+ratio : gS, a∆, c0↔1 (con)
+summation : gS, a∆, c0↔1 (dis)
+summation : gS, a∆, c0↔1, c11 (dis)
+ratio : gS, a∆, c0↔1 (dis)
+ratio : gS, a∆, c0↔1, a∆ prior (dis)
+4
+6
+8
+4
+6
+8
+4
+6
+8
+tmin
+f
+/a (tmax
+f
+/a = 16; dis)
+4.0
+4.5
+5.0
+5.5
+6.0
+gdis,S
+S
+symm. cutoff in t/a = 1 (con)
+symm. cutoff in t/a = 2 (con)
+symm. cutoff in t/a = 3 (con)
+Figure 3: Effect of the fit form and range variation on the energy gap Δ = 𝐸1 − 𝐸0 (top) and the disconnected
+scalar charge 𝑔dis
+𝑆 for 𝐽 = ¯𝑠𝑠 (bottom) for the Ξ baryon on the ensemble S400 (𝑀𝜋 ≈ 350 MeV). The blue
+and red-dashed horizontal lines indicate an energy gap corresponding to a Ξ(1)𝜋(−1) and Ξ(0)𝜋(0)𝜋(0)
+excited state, respectively. The numbers in brackets stand for the momentum as a multiple of the lowest
+lattice momentum 2𝜋/𝐿. Each of the three panels displays results of simultaneous fits to the connnected and
+disconnected contributions where the fit range is fixed for the former and varied for the latter, while across
+the different panels, the fit range for the connected ratio is varied. The insertion times included are always
+symmetric w.r.t. the source and sink. This leads to a minimal 𝑡f for the fits to the disconnected ratios (for
+which all 𝑡f are, in principle, available). For the connected contributions only the ratio method (with 𝑐11 = 0)
+is employed, while for the disconnected results both the ratio and the summation method are shown. Results
+for the ratio method, using a prior (Gaussian with 𝜎 = 1%) for the energy gap corresponding to a Ξ(1)𝜋(−1)
+(non-interacting) excited state, are also displayed. The first (black) pentagon corresponds to the fit in fig. 2.
+for each source-sink separation. The results for the example of S400 are displayed in fig. 3. We
+find that the energy gap is mostly determined by the connected data: in the first row of fig. 3 the
+error on Δ increases as more connected data points are excluded from the fit. We also observe
+that the energy gap Δ = 𝐸1 − 𝐸0 is compatible with that for a Ξ(1)𝜋(−1) or Ξ(0)𝜋(0)𝜋(0) first
+excited state so that a narrow-width prior corresponding to a Ξ(1)𝜋(−1) excited state has almost
+no effect on the ground-state matrix elements (second row of fig. 3). This is also true for the other
+members of the baryon octet (when comparing to the energy of a 𝐵(1)𝜋(−1) or 𝐵(0)𝜋(0)𝜋(0)
+excited state). However, as the pion mass decreases, with the exception of the nucleon, we find
+𝐸1 to be larger than the energies of these levels. This may be related to the fact that the spectrum
+(for the nucleon) is denser at smaller pion masses [10] making it more difficult to separate the first
+excited state contributions from others. However, we can enforce the first excitation to have the
+energy of a non-interacting 𝐵(1)𝜋(−1) state by using a narrow-width prior and the sigma terms are
+usually only mildly affected, see fig. 4.
+6
+
+Sigma terms of the octet baryons
+Pia Leonie Jones Petrak
+phys.
+6
+8
+10
+12
+14
+16
+18
+m2
+π[(100 MeV)2]
+0
+50
+100
+150
+200
+250
+σB
+π [MeV]
+S400
+N203
+N200
+N202
+C101
+J500
+J303
+BMW 2012 [3]
+RATIO
+SUM
+RATIO + PRIOR Bπ
+N
+Λ
+Σ
+Ξ
+N
+Λ
+Σ
+Ξ
+N
+Λ
+Σ
+Ξ
+phys.
+6
+8
+10
+12
+14
+16
+18
+m2
+π[(100 MeV)2]
+0
+50
+100
+150
+200
+250
+300
+350
+σB
+s [MeV]
+S400
+N203
+N200
+N202
+C101
+J500
+J303
+BMW 2012 [3]
+RATIO
+SUM
+RATIO + PRIOR Bπ
+N
+Λ
+Σ
+Ξ
+N
+Λ
+Σ
+Ξ
+N
+Λ
+Σ
+Ξ
+Figure 4: Preliminary results for the pion-baryon and strange sigma terms as a function of the pion mass
+squared obtained utilising the ratio method (with 𝑐11 = 0) for both contributions (with and without a prior
+for the energy gap) and utilising the ratio method for the connected contribution and the summation method
+(with 𝑐11 ≠ 0) for the disconnected contribution. The prior width is set to 1% centered around an energy gap
+compatible with a 𝐵(1)𝜋(−1) excited state. We also compare our results to those of BMW 2012 [3].
+5.
+Preliminary results and consistency with indirect determinations
+The ground-state matrix elements of interest are extracted from the fits presented in the previous
+section and combined with the corresponding quark masses and renormalisation factors (see [11]) to
+form the pion-baryon and strange sigma terms for all four octet baryons. Our preliminary results are
+displayed as a function of the pion mass squared in fig. 4. We see that the different fitting methods
+employed lead to compatible results for the sigma terms. In addition, we compare our results
+from the ratio method (extracted at finite lattice spacing) in fig. 5 to the quark mass dependence
+7
+
+Sigma terms of the octet baryons
+Pia Leonie Jones Petrak
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+48
+96
+145
+193
+241
+290
+1352
+2002
+2502
+3002
+3502
+4002
+[MeV2]
+N
+Λ
+Σ
+Ξ
+√8t0σπB
+[MeV]
+8t0M 2
+π
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+48
+96
+145
+193
+241
+290
+338
+386
+1352
+2002
+2502
+3002
+3502
+4002
+[MeV2]
+N
+Λ
+Σ
+Ξ
+√8t0σsB
+[MeV]
+8t0M 2
+π
+Figure 5: Results for the pion-baryon and strange sigma terms as a function of the pion mass squared
+obtained using the ratio method (blue squares). The shaded regions show the quark mass dependence of 𝜎𝐵
+𝜋
+and 𝜎𝐵
+𝑠 determined via the Feynman-Hellmann theorem utilising a fit to the octet baryon masses evaluated
+on 47 CLS ensembles [13]. Finite-volume effects are accounted for and NNLO baryon chiral perturbation
+theory is used to describe the quark mass dependence.
+in the continuum limit obtained from an indirect determination via the Feyman-Hellmann theorem
+and a fit to the octet baryon masses, see [13]. Almost all our preliminary results (from our direct
+determination) lie within the error bands of the indirect determination suggesting that, after having
+taken the continuum and infinite volume limits, the results will remain consistent.
+6.
+Conclusion and outlook
+Our aim is to determine the strange and pion sigma terms for the baryon octet.
+In these
+proceedings we reported on the progress made in controlling excited state contaminations. We find
+that the ratio and summation methods give compatible results for the sigma terms. Although these
+methods do not always result in a first excited state compatible with a 𝐵(1)𝜋(−1) or 𝐵(0)𝜋(0)𝜋(0)
+state (for octet baryons other than the nucleon), when a narrow-width prior is used for the energy
+gap, consistent results for the sigma terms are, nonetheless, obtained. The next steps will include
+analysing additional ensembles so as to allow for an extrapolation to the physical point and an
+investigation of cut-off and finite-volume effects.
+Acknowledgments
+This work is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research
+Training Group “GRK 2149: Strong and Weak Interactions – from Hadrons to Dark Matter” (P.
+L. J. P. and J. H.). G. B., S. C., D. J. and S. W. were supported by the European Union’s Horizon
+2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement
+no. 813942 (ITN EuroPLEx) and grant agreement no. 824093 (STRONG-2020). S.C. and S.W.
+received support through the German Research Foundation (DFG) grant CO 758/1-1.
+We gratefully acknowledge computing time granted by the John von Neumann Institute for
+Computing (NIC), provided on the Booster partition of the supercomputer JURECA [29] at Jülich
+8
+
+Sigma terms of the octet baryons
+Pia Leonie Jones Petrak
+Supercomputing Centre (JSC). Additional simulations were carried out at the QPACE 3 Xeon Phi
+cluster of SFB/TRR 55. The authors also gratefully acknowledge the Helmholtz Data Federation
+(HDF) for funding this work by providing services and computing time on the HDF Cloud cluster
+at the Jülich Supercomputing Centre (JSC) [30].
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+10
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf,len=510
+page_content='MS-TP-22-56  Sigma terms of the baryon octet in 𝑵f = 2 + 1 QCD  with Wilson quarks  Pia Leonie Jones Petrak,𝑎,∗ Gunnar Bali,𝑏 Sara Collins,𝑏 Jochen Heitger,𝑎 Daniel  Jenkins𝑏 and Simon Weishäupl𝑏  𝑎Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster,  Wilhelm-Klemm-Straße 9, 48149 Münster, Germany  𝑏Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany  E-mail: p_petr04@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='de, gunnar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='bali@ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='de,sara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='collins@ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='de,  heitger@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='de, Daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Jenkins@ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='de, Simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Weishaeupl@ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='de  A lot of progress has been made in the direct determination of nucleon sigma terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Using similar  methods, we consider the sigma terms of the other octet baryons as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' These are determined on  CLS gauge field ensembles employing the Lüscher-Weisz gluon action and the Sheikholeslami-  Wohlert fermion action with 𝑁f = 2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The ensembles have pion masses ranging from 410 MeV  down to the physical value and lattice spacings covering a range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='098 fm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='039 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  We present some preliminary results for the pion and strange sigma terms and compare to indirect  determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' To do so, we discuss multi-state fits to tackle the well-known problem of excited  state contamination comparing the ratio and summation methods also including priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  The 39th International Symposium on Lattice Field Theory,  8th-13th August, 2022,  Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='03871v1  [hep-lat]  10 Jan 2023  Sigma terms of the octet baryons  Pia Leonie Jones Petrak  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  Introduction  The sigma terms encode the quark contributions to the mass of a baryon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' They are defined as  the matrix element of a scalar current 𝐽 times the quark mass such that  𝜎𝑞𝐵 = 𝑚𝑞⟨𝐵|𝐽|𝐵⟩ ,  (1)  where 𝐽 = ¯𝑞 1 𝑞 and the quark flavour 𝑞 ∈ {𝑢, 𝑑, 𝑠}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' In order to compare with phenomenological  determinations, the pion-baryon sigma terms are usually constructed, 𝜎𝜋𝐵 = 𝜎𝑢𝐵 + 𝜎𝑑𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' In the  matrix element, 𝐵 refers to the ground state of a baryon 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Of particular interest are the nucleon  sigma terms (𝐵 = 𝑁) which appear in the expressions for WIMP-nucleon scattering cross-sections  and are relevant for comparing model predictions to the exclusion bounds obtained from direct  detection dark matter experiments (such as the XENON1T experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  In the analysis, we make use of and adjust methods established for the nucleon (see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' [1]  for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' We determine the sigma terms of the baryon octet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' of the lambda Λ, sigma Σ  and cascade Ξ baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' This enables us to investigate SU(3) flavour symmetry breaking and to  determine the SU(3) low energy constants (LECs), that are currently not well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' There have  been relatively few determinations so far, see, for example, refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' [2] and [3], with most previous  studies focusing on the nucleon sigma terms (see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' [4] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  In addition, discrepancies between results for the pion-nucleon sigma term from Lattice QCD  and phenomenology are still to be resolved (see [4], and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=', [5–7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' In a recent paper, results more  consistent with phenomenology were obtained by explicitly including 𝑁𝜋 and 𝑁𝜋𝜋 excited states  in the analysis [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' By considering baryons other than the nucleon, we aim to understand excited  state contributions and other issues in more detail so as to help solve this puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  Excited state analysis  In order to be able to extract the ground-state matrix element, we need to take care of the  excited state contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Two possible approaches are the ratio method and the summation  method (reviewed in [1, 10]) that are both based on spectral decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' We consider the  two- and three-point functions of a baryon (from the octet) at rest in the initial and final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The  spectral decomposition of the two-point function reads  𝐶2pt(𝑡f) =  ∑︁  �𝑥  �  Osnk(�𝑥, 𝑡f) ¯Osrc(�0, 0)  �  =  ∑︁  𝑛  |𝑍𝑛|2𝑒−𝐸𝑛𝑡f ,  (2)  where 𝑍𝑛 ∝ ⟨Ω|Osnk|𝑛⟩ is the overlap of the interpolator Osnk onto the state 𝑛 (and Ω the vacuum  state) and 𝑡f the source-sink separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Summation over spin and colour indices and projection  onto positive parity are implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' These indices become apparent when writing down the operators  explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The interpolators for the four octet baryons are set to  O 𝛼,N  snk = 𝜖 𝑎𝑏𝑐𝑢𝛼  𝑎  �  𝑢𝛽  𝑏(𝐶𝛾5)𝛽𝛾𝑑𝛾  𝑐  �  and  O 𝛼,Λ  snk = 𝜖 𝑎𝑏𝑐𝑠𝛼  𝑎  �  𝑢𝛽  𝑏(𝐶𝛾5)𝛽𝛾𝑑𝛾  𝑐  �  ,  O 𝛼,Σ  snk = 𝜖 𝑎𝑏𝑐𝑢𝛼  𝑎  �  𝑢𝛽  𝑏(𝐶𝛾5)𝛽𝛾𝑠𝛾  𝑐  �  and  O 𝛼,Ξ  snk = 𝜖 𝑎𝑏𝑐𝑠𝛼  𝑎  �  𝑠𝛽  𝑏(𝐶𝛾5)𝛽𝛾𝑢𝛾  𝑐  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  (3)  2  Sigma terms of the octet baryons  Pia Leonie Jones Petrak  𝑎, 𝑏, 𝑐 are colour indices, 𝛼, 𝛽, 𝛾 are spin indices and O 𝛼  src = O 𝛼  snk and ¯Osrc = O†  src𝛾4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 𝐶 stands for  the charge conjugation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Note that for the Λ we use a naive interpolator that also has overlap  with the (heavier) Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Turning to the three-point function, its spectral decomposition reads  𝐶3pt(𝑡f, 𝑡) =  ∑︁  �𝑥,�𝑦  �  Osnk(�𝑥, 𝑡f)𝐽(�𝑦, 𝑡) ¯Osrc(�0, 0)  �  −  ∑︁  �𝑥,�𝑦  ⟨𝐽(�𝑦, 𝑡)⟩  �  Osnk(�𝑥, 𝑡f) ¯Osrc(�0, 0)  �  =  ∑︁  𝑛,𝑛′  𝑍𝑛′𝑍∗  𝑛⟨𝑛′|𝐽|𝑛⟩𝑒−𝐸𝑛𝑡𝑒−𝐸𝑛′ (𝑡f−𝑡),  (4)  where 𝑡 is the insertion time of the scalar current, 𝐽 = ¯𝑞 1 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' As 𝐽 has the same quantum numbers  as the vacuum, the vacuum expectation value needs to be subtracted, see the second term in the first  line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Note that depending on the type of baryon and current, different Wick contractions  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' different currents) contribute that result in connected and disconnected quark-line diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  Ratio method:  Taking the ratio of the two spectral decompositions and truncating after the first excited state  contribution leads to  𝑅(𝑡f, 𝑡) = 𝐶3pt(𝑡f, 𝑡)  𝐶2pt(𝑡f)  = 𝑔𝑞  𝑆 + 𝑐01e−Δ · 𝑡 + 𝑐10e−Δ · (𝑡f−𝑡) + 𝑐11e−Δ · 𝑡f + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  (5)  where 𝑔𝑞  𝑆 = ⟨𝐵| ¯𝑞 1 𝑞|𝐵⟩ is the ground-state matrix element of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Δ = 𝐸1 − 𝐸0 is the energy  gap between the ground and first excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The coefficients 𝑐01, 𝑐10, 𝑐11 are made up of matrix  elements for different transitions, 𝐵0 → 𝐵1, 𝐵1 → 𝐵0 , and 𝐵1 → 𝐵1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Here, 𝐵0 refers  to the ground state of the baryon while 𝐵1 denotes the first excited state (single- or multi-particle  state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' As the baryon is at rest, 𝑐01 = 𝑐10 ≡ 𝑐0↔1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Note that we labelled 𝑐0↔1 as 𝑐1 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  Summation method:  Summing over the ratio (5) for a range of insertion times leads to  𝑡f/𝑎−𝑐  ∑︁  𝑡/𝑎=𝑐  𝑅(𝑡f, 𝑡) = 𝑔𝑞  𝑆(𝑡f/𝑎 − 2𝑐 + 1) + 2𝑐0↔1  1 − e𝑎Δ  �  eΔ(𝑐−𝑡f) − eΔ(𝑎−𝑐)�  + 𝑐11(𝑡f/𝑎 − 2𝑐 + 1)e−Δ · 𝑡f + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  (6)  where 𝑎 refers to the lattice spacing and 𝑐 > 0 to preserve reflection positivity (we set 𝑐 = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' An  advantage of this method is that, in principle, the slope,  d  d(𝑡f/𝑎)  �𝑡f/𝑎−𝑐  𝑡/𝑎=𝑐 𝑅(𝑡f, 𝑡) = 𝑔𝑞  𝑆 + O(𝑡f/𝑎 · e−Δ · 𝑡f) ,  approaches the asymptotic value faster compared to the ratio method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' However, a large number of  source-sink separations is required and due to the the numerical setup we use (discussed in the next  section) we can only employ this approach for the disconnected three-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  Numerical Setup  We utilise the CLS gauge field ensembles [12], which are generated employing the Lüscher-  Weisz gluon action and the Sheikholeslami-Wohlert fermion action with 𝑁f = 2 + 1 (𝑚𝑙 = 𝑚𝑢 =  𝑚𝑑 ≤ 𝑚𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The pion-baryon and strange sigma terms are determined on the seven ensembles  highlighted in red (diamonds) in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  This includes five lattice spacings in the range from  3  Sigma terms of the octet baryons  Pia Leonie Jones Petrak  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='052  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0642  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0762  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0862  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0982  a2 [fm2]  150  200  250  300  350  400  Mπ [MeV]  A653  A654  U103  H101  H102  H105  C101  B450  S400  N401  D450  D200  N300  N302  J303  E300  J500  J501  J500  J303  N202  N203  N200  S400  C101  physical point  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='55  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='85  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='46  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='34  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='4  β  Figure 1: Overview of a subset of the CLS  ensembles which lie on a trajectory which starts  at a symmetric point (𝑚𝑙 = 𝑚𝑠, 𝑀𝜋 ∼ 410 MeV)  and extends to the physical point, along which  the flavour average of the quark masses is held  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' So far the ensembles highlighted in  red (diamonds) have been analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='08528(49) fm down to 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='03875(22) fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The pion masses vary from 411 MeV down to  222 MeV, see also [13] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  To compute the connected three-point correlation functions on the ensembles with 𝑚𝑙 =  𝑚𝑠 and 𝑀𝜋 = 410 MeV, we used the standard sequential source method [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  On the other  ensembles we employed the stochastic method described in [16, 17] (see also [18–21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  This  approach enables us to obtain measurements for all baryons of interest at multiple source and  insertion positions, simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Four different source-sink separations, corresponding to 𝑡f ≈  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='71 fm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='9 fm, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='03 fm, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='22 fm], are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Two measurements are performed for each 𝑡f on  every configuration except for the 𝑚𝑙 = 𝑚𝑠 ensembles (that we used the sequential source method  for) where ten measurements are typically performed (one, two, three and four measurements for  𝑡f ≈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='71 fm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='9 fm, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='03 fm, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='22 fm], respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  The disconnected three-point functions are constructed by correlating a quark loop with a  baryon two-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The loop is estimated stochastically on every timeslice (within the bulk)  leading to additional noise on top of the Monte-Carlo gauge sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' In order to reduce the  noise, the truncated solver method [22], the hopping parameter expansion technique [23] and time  partitioning [24] are utilised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Between twenty and thirty measurements of the two-point functions  (at different temporal source positions) are performed on each configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' A reasonable signal for  the disconnected three-point function is obtained for a source-sink separation of up to 𝑡f ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='22 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  For the analysis of the statistical errors we employ the Γ-method [25] (that is based on autocorrelation  functions) using the pyerrors python package [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  When constructing the sigma terms for Wilson-type fermions, one needs to take the mixing  between different quark flavours into account, which arises due to the singlet and non-singlet  combinations of the scalar current renormalising differently as a result of chiral symmetry breaking  (see , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=', [11] for the renormalisation pattern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  For the ratio of renormalisation factors, 𝑟m,  we use 𝑟m(𝛽 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='4) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='335(31), 𝑟m(𝛽 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='46) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='869(19), 𝑟m(𝛽 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='55) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='523(14),  𝑟m(𝛽 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='7) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='267(16) and 𝑟m(𝛽 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='85) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='149(18)(27)[33], determined non-perturbatively  in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  1In [11] we investigated the quark mass dependence by performing a preliminary chiral extrapolation at 𝑎 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='064 fm  (including preliminary SU(3) LECs from [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  4  Sigma terms of the octet baryons  Pia Leonie Jones Petrak  −8  −6  −4  −2  0  2  4  6  8  insertion time t/a − tf/(2a)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='4  Rcon  gcon  s  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='31(12), ccon  0↔1 = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='31(12)  tf/a = 9  tf/a = 11  tf/a = 14  tf/a = 16  −8  −6  −4  −2  0  2  4  6  8  insertion time t/a − tf/(2a)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='50  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='75  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='00  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='50  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='75  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='00  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='25  Rcon  gcon  s  = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='16(14), ccon  0↔1 = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='80(8)  tf/a = 9  tf/a = 11  tf/a = 14  tf/a = 16  4  6  8  10  12  14  16  source-sink separation tf/a  0  10  20  30  40  50  60  70  �  t Rdis  gdis  s  = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='66(72), cdis  0↔1 = -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='01(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='11), cdis  11 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='91(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='18)  4  6  8  10  12  14  16  source-sink separation tf/a  0  10  20  30  40  50  60  �  t Rdis  gdis  s  = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='22(66), cdis  0↔1 = -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='42(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='01), cdis  11 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='15(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='03)  Figure 2: The connected and (summed) disconnected ratios contributing to the unrenormalised scalar charges  of the Ξ baryon on ensemble S400 (𝑀𝜋 ≈ 350 MeV) for the ¯𝑢𝑢 current (left) and the ¯𝑠𝑠 current (right):  Simultaneous fits to the connected and summed disconnected ratios are indicated by the coloured shaded  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The resulting connected ground state scalar matrix elements are displayed as horizontal grey bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  Fitting Analysis  In order to extract the ground-state matrix elements of interest while taking the excited state  contamination into account we perform multi-state fits, applying the ratio method, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' (5) and, for  comparison, for the disconnected contribution also the summation method, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' (6) (as noted in  sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 2, there are not enough source-sink separations available to apply this method to the connected  contributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  For each baryon, we fit the connected and disconnected ratios for all currents  simultaneously, with the energy gap to the first excited state Δ as the common fit parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' As an  example, the ratios (and fits) relevant for determining the sigma terms of the Ξ baryon on the S400  ensemble (𝑀𝜋 ≈ 350 MeV) are displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 2 (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 2 in [11] for an example of the ratio  method being used for both the connected and disconnected contributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' While we were able  to resolve the first excited state term, with the coefficient 𝑐0↔1, with both methods, the 𝐵1 → 𝐵1  transition term (see sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 2) was not resolved when employing the ratio method and we set 𝑐11 = 0  in the analysis when using this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The 𝜒2/𝜒2  expected were mostly below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='5 (and always below  two), where 𝜒2  expected provides an estimate of the effective degrees of freedom expected taking into  account autocorrelations, see [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  To investigate whether the excited state contributions are sufficiently controlled when extracting  the ground state matrix elements, we vary the fit range for both the connected and disconnected  contributions and for the latter also the fit method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' In addition, we consider the impact on the fit  of a narrow-width prior for Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The fit range is varied by removing insertion times symmetrically  5  Sigma terms of the octet baryons  Pia Leonie Jones Petrak  4  6  8  4  6  8  4  6  8  tmin  f  /a (tmax  f  /a = 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' dis)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='30  a∆  symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' cutoff in t/a = 1 (con)  symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' cutoff in t/a = 2 (con)  symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' cutoff in t/a = 3 (con)  Ξ(0)π(0)π(0)  Ξ(1)π(−1)  ratio : gS, a∆, c0↔1 (con)  summation : gS, a∆, c0↔1 (dis)  summation : gS, a∆, c0↔1, c11 (dis)  ratio : gS, a∆, c0↔1 (dis)  ratio : gS, a∆, c0↔1, a∆ prior (dis)  4  6  8  4  6  8  4  6  8  tmin  f  /a (tmax  f  /a = 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' dis)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0  gdis,S  S  symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' cutoff in t/a = 1 (con)  symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' cutoff in t/a = 2 (con)  symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' cutoff in t/a = 3 (con)  Figure 3: Effect of the fit form and range variation on the energy gap Δ = 𝐸1 − 𝐸0 (top) and the disconnected  scalar charge 𝑔dis  𝑆 for 𝐽 = ¯𝑠𝑠 (bottom) for the Ξ baryon on the ensemble S400 (𝑀𝜋 ≈ 350 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The blue  and red-dashed horizontal lines indicate an energy gap corresponding to a Ξ(1)𝜋(−1) and Ξ(0)𝜋(0)𝜋(0)  excited state, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The numbers in brackets stand for the momentum as a multiple of the lowest  lattice momentum 2𝜋/𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Each of the three panels displays results of simultaneous fits to the connnected and  disconnected contributions where the fit range is fixed for the former and varied for the latter, while across  the different panels, the fit range for the connected ratio is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The insertion times included are always  symmetric w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' the source and sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' This leads to a minimal 𝑡f for the fits to the disconnected ratios (for  which all 𝑡f are, in principle, available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' For the connected contributions only the ratio method (with 𝑐11 = 0)  is employed, while for the disconnected results both the ratio and the summation method are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Results  for the ratio method, using a prior (Gaussian with 𝜎 = 1%) for the energy gap corresponding to a Ξ(1)𝜋(−1)  (non-interacting) excited state, are also displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The first (black) pentagon corresponds to the fit in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  for each source-sink separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The results for the example of S400 are displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' We  find that the energy gap is mostly determined by the connected data: in the first row of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 3 the  error on Δ increases as more connected data points are excluded from the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' We also observe  that the energy gap Δ = 𝐸1 − 𝐸0 is compatible with that for a Ξ(1)𝜋(−1) or Ξ(0)𝜋(0)𝜋(0) first  excited state so that a narrow-width prior corresponding to a Ξ(1)𝜋(−1) excited state has almost  no effect on the ground-state matrix elements (second row of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' This is also true for the other  members of the baryon octet (when comparing to the energy of a 𝐵(1)𝜋(−1) or 𝐵(0)𝜋(0)𝜋(0)  excited state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' However, as the pion mass decreases, with the exception of the nucleon, we find  𝐸1 to be larger than the energies of these levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' This may be related to the fact that the spectrum  (for the nucleon) is denser at smaller pion masses [10] making it more difficult to separate the first  excited state contributions from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' However, we can enforce the first excitation to have the  energy of a non-interacting 𝐵(1)𝜋(−1) state by using a narrow-width prior and the sigma terms are  usually only mildly affected, see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  6  Sigma terms of the octet baryons  Pia Leonie Jones Petrak  phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  6  8  10  12  14  16  18  m2  π[(100 MeV)2]  0  50  100  150  200  250  σB  π [MeV]  S400  N203  N200  N202  C101  J500  J303  BMW 2012 [3]  RATIO  SUM  RATIO + PRIOR Bπ  N  Λ  Σ  Ξ  N  Λ  Σ  Ξ  N  Λ  Σ  Ξ  phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='π[(100 MeV)2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='250  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='350  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='σB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='s [MeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='S400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='N203  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='N200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='N202  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='C101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='J500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='J303  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='BMW 2012 [3]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='RATIO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='SUM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='RATIO + PRIOR Bπ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Σ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Ξ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Σ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Ξ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Σ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Ξ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='Figure 4: Preliminary results for the pion-baryon and strange sigma terms as a function of the pion mass  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='squared obtained utilising the ratio method (with 𝑐11 = 0) for both contributions (with and without a prior  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='for the energy gap) and utilising the ratio method for the connected contribution and the summation method  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='(with 𝑐11 ≠ 0) for the disconnected contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The prior width is set to 1% centered around an energy gap  compatible with a 𝐵(1)𝜋(−1) excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' We also compare our results to those of BMW 2012 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  Preliminary results and consistency with indirect determinations  The ground-state matrix elements of interest are extracted from the fits presented in the previous  section and combined with the corresponding quark masses and renormalisation factors (see [11]) to  form the pion-baryon and strange sigma terms for all four octet baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Our preliminary results are  displayed as a function of the pion mass squared in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' We see that the different fitting methods  employed lead to compatible results for the sigma terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' In addition, we compare our results  from the ratio method (extracted at finite lattice spacing) in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 5 to the quark mass dependence  7  Sigma terms of the octet baryons  Pia Leonie Jones Petrak  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
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+page_content='8  0  48  96  145  193  241  290  1352  2002  2502  3002  3502  4002  [MeV2]  N  Λ  Σ  Ξ  √8t0σπB  [MeV]  8t0M 2  π  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
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+page_content='8  0  48  96  145  193  241  290  338  386  1352  2002  2502  3002  3502  4002  [MeV2]  N  Λ  Σ  Ξ  √8t0σsB  [MeV]  8t0M 2  π  Figure 5: Results for the pion-baryon and strange sigma terms as a function of the pion mass squared  obtained using the ratio method (blue squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The shaded regions show the quark mass dependence of 𝜎𝐵  𝜋  and 𝜎𝐵  𝑠 determined via the Feynman-Hellmann theorem utilising a fit to the octet baryon masses evaluated  on 47 CLS ensembles [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Finite-volume effects are accounted for and NNLO baryon chiral perturbation  theory is used to describe the quark mass dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  in the continuum limit obtained from an indirect determination via the Feyman-Hellmann theorem  and a fit to the octet baryon masses, see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Almost all our preliminary results (from our direct  determination) lie within the error bands of the indirect determination suggesting that, after having  taken the continuum and infinite volume limits, the results will remain consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  Conclusion and outlook  Our aim is to determine the strange and pion sigma terms for the baryon octet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  In these  proceedings we reported on the progress made in controlling excited state contaminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' We find  that the ratio and summation methods give compatible results for the sigma terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Although these  methods do not always result in a first excited state compatible with a 𝐵(1)𝜋(−1) or 𝐵(0)𝜋(0)𝜋(0)  state (for octet baryons other than the nucleon), when a narrow-width prior is used for the energy  gap, consistent results for the sigma terms are, nonetheless, obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The next steps will include  analysing additional ensembles so as to allow for an extrapolation to the physical point and an  investigation of cut-off and finite-volume effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  Acknowledgments  This work is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research  Training Group “GRK 2149: Strong and Weak Interactions – from Hadrons to Dark Matter” (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' were supported by the European Union’s Horizon  2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 813942 (ITN EuroPLEx) and grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' 824093 (STRONG-2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  received support through the German Research Foundation (DFG) grant CO 758/1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content='  We gratefully acknowledge computing time granted by the John von Neumann Institute for  Computing (NIC), provided on the Booster partition of the supercomputer JURECA [29] at Jülich  8  Sigma terms of the octet baryons  Pia Leonie Jones Petrak  Supercomputing Centre (JSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' Additional simulations were carried out at the QPACE 3 Xeon Phi  cluster of SFB/TRR 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
+page_content=' The authors also gratefully acknowledge the Helmholtz Data Federation  (HDF) for funding this work by providing services and computing time on the HDF Cloud cluster  at the Jülich Supercomputing Centre (JSC) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE2T4oBgHgl3EQfaAfc/content/2301.03871v1.pdf'}
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+ 
+ 
+ 
+ 
+ 
+ 
+FUTURE OF SUPPLY CHAIN: CHALLENGES, 
+TRENDS, AND PROSPECTS 
+Cristiana L. Lara1*, John Wassick2 
+1Modeling and Optimization, Amazon.com, Bellevue WA 98004 
+2Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh PA 
+15213 
+Abstract 
+This paper discusses the broad challenges shared by e-commerce and the process industries operating global supply 
+chains. Specifically, we discuss how process industries and e-commerce differ in many aspects but have similar challenges 
+ahead of them in order to remain competitive, keep up with the always increasing requirements of the customers and 
+stakeholders, and gain efficiency. While both industries have been early adopters of decision support tools based on machine 
+intelligence, both share unresolved challenges related to scalability, integration of decision-making over different time 
+horizons (e.g. strategic, tactical and execution-level decisions) and across internal business units, and orchestration of human 
+and computer-based decision-makers.  We discuss future trends and research opportunities in the area of supply chain, and 
+suggest that the methods of multi-agent systems supported by rigorous treatment of human decision-making in combination 
+with machine intelligence is a great contender to address these critical challenges. 
+Keywords 
+Supply chain, logistics, process industry, e-commerce, artificial intelligence. 
+Introduction
+The concept of supply chain as an interconnected, 
+dynamic physical system to deliver goods and services to 
+consumers has become a popular topic for the public in 
+recent years.   This increased awareness is due to the 
+recognition by the public of the connection between supply 
+chain problems and the lived experience of disappoints 
+concerning 
+product 
+availability 
+and 
+price.  
+Past 
+globalization of trade, and the enterprises that engage in it, 
+has 
+increased 
+their 
+scope 
+and 
+their 
+complexity.  Consequently, supply chains are more 
+challenging 
+to 
+operate 
+and 
+more 
+vulnerable 
+to 
+disruptions.  The recent pandemic, high profile logistical 
+disruptions, and war have exposed the vulnerability of 
+global supply chains.   The corresponding swings in 
+consumer preferences and demand have stressed the 
+management 
+practices 
+surrounding 
+supply 
+chain 
+ 
+* To whom all correspondence should be addressed. This work does not relate to the author’s position at Amazon, regardless of their 
+affiliation. 
+operations.  Both companies serving end consumers and 
+those farther upstream have been focused on modernizing 
+and digitizing their supply chains to retain and grow their 
+customer base and their bottom line.   
+The ongoing activities by the many diverse operating 
+companies pursuing improvements to their supply chains 
+raises the question of shared ideals for their future supply 
+chains.  It is safe to say that all companies seek to be more 
+flexible and agile in the face of disruptions, more adaptable 
+to evolving markets, and more aligned to suppliers and 
+customers in general. However, they also share specific 
+objectives at an operational level.  This can be demonstrated 
+by examining the shared challenges of supply chains of e-
+commerce and the process industries.  Both are challenged 
+with the scalability and integration needed to deliver end-
+to-end performance.  Both operate enterprise scale 
+
+ 
+ 
+ 
+ 2 
+workflows that require coordination of decision-making 
+over different time granularities and geographic areas, i.e., 
+closing the gap between long-term planning, tactical 
+planning, and execution.  Both are deploying advanced 
+analytical methods to work in concert with the human 
+decision makers. Considered together, the aspirations for 
+supply chain in e-commerce and the process industries 
+provides a consistent view of the future of supply chain and 
+research to support it. 
+This paper is structured as follows.  The difference in 
+the nature of the supply chains for the process industry and 
+e-commerce are first described.  Their shared challenges are 
+discussed next.   Then future trends and research 
+opportunities are identified to address the shared 
+challenges.  Finally, in a future where artificial intelligence 
+is pervasively used in supply chain operations, the place for 
+the use of human intelligence is indicated. 
+Different Industries, similar challenges 
+Process Industries 
+Within the broad process industries, the chemical 
+industry operates some of the most complex and 
+challenging supply chains.   Key reasons include level of 
+vertical integration of product lines, diversity of markets 
+served and position in value chains, energy intensity, and 
+capital intensity.   Large chemical companies have complex, 
+globally distributed manufacturing sites that are themselves 
+tightly coupled, internal supply chains (Wassick, 2009).   
+Recent figures from The Dow Chemical Company 
+provide a good example of the scale and complexity of a 
+global chemical company (SupplyChainWorld 2015).  
+Manufacturing more than 6,000 product families at 201 
+sites in 35 countries, Dow manages roughly 6,000 
+shipments per day to serve 45,000 customer locations.  The 
+company’s annual shipment volume is about 130 billion 
+pounds and uses 450 warehouses and 150 contract 
+terminals. The company works with 650 service providers 
+in 160 countries and receives more than 110 billion pounds 
+of raw materials and intermediates each year from 4,000 
+suppliers. 
+ We will use the Supply Chain Operations Reference 
+(SCOR) model, commonly used by supply chain 
+professionals as a paradigm for discussing supply chains, 
+and comprised of the business processes Plan, Source, 
+Make, Deliver, and Return, to briefly discuss the nature of 
+chemical industry supply chains (Delipinar and Kocaoglu, 
+2016).  
+Central to the Plan process are strategic and tactical 
+supply and demand balancing, strategies to support a 
+business financial plan, and long-term asset strategy.  These 
+decisions become very challenging when vertical 
+integration requires planning across multiple businesses, 
+each with its own profit and loss metric, serving diverse 
+markets.  Business supply chains are highly interdependent 
+with each echelon serving both external markets and 
+internal downstream businesses.  Inventory management is 
+confounded by the various dimensions of the value of 
+upstream inventory. 
+The Source process, responsible for raw material 
+acquisition and the associated supplier management, has 
+several complexities in the chemical industry.  For one, 
+external sources can be used to supplement the internal 
+sources found in the vertical integration of a company.  Raw 
+materials can include globally traded commodities like oil, 
+naphtha, and natural gas for which long term hedging and 
+contracts can be used for procurement while also exploiting 
+spot market prices.  Large manufacturing sites can operate 
+cogeneration electrical power plants that serve the onsite 
+production units but also can supply to the local electrical 
+grid, or power can be purchased from the grid when 
+advantageous. 
+The Make process focuses on the broad set of 
+manufacturing activities which includes production, 
+production scheduling, packaging, and site logistics.  Safety 
+and environmental protection are top priority in the 
+chemical industry and the nature of the products and 
+processes that make them leads to complex business rules 
+and constraints. Here too, vertical integration at a 
+manufacturing site adds another layer of constraints.  
+Logistics operations can include massive railyards, tank 
+farms, and marine ports. 
+Within the Deliver process are activities associated 
+with order management, warehousing, and transportation.  
+Most of the commerce in the chemical industry is made up 
+of business-to-business transactions which can involve long 
+lead time orders of massive volumes and complicated sales 
+contracts.  Transportation for specific orders may be 
+supplied by ocean going vessels, barges, railcars, and 
+dedicated tank trucks, and combinations of these.  
+International shipments to customers are common which 
+requires sophisticated international trade operations to 
+manage compliance with duties and tariffs and shipping 
+requirements. 
+E-commerce 
+Online retailers have come of age with the dot-com 
+boom, and since then, have been competing with other 
+retailers within the broader retailing industry. Xu (2005) 
+summarizes some characteristics of online retailing, 
+reinterpreted in the list below: 
+• 
+Large Selection: Not limited by physical space in the 
+store front, online retailers are able to offer hundreds of 
+thousands of Stock-Keeping Units (SKU) in stock.  
+• 
+Logistics as a Matter of Trust: Online customers con-
+sider the timely delivery of products to be a significant 
+component of trust. Thus, the reliability and efficiency 
+of the supply chain are even more crucial than for 
+traditional brick-and-mortar retailers. 
+• 
+Data-driven decision making: These companies rely on 
+rigorous data analytics to make decisions at every stage 
+
+ 
+ 
+ 
+3 
+of supply chain, from selection of products and 
+inventory placement to delivery.  
+• 
+Flexibility when fulfilling customer orders: In online 
+retailing, there is typically a time delay between when 
+a demand occurs (customer order) and when inventory 
+is physically deployed and transported to meet a 
+customer order. By delaying the decisions on how to 
+fulfill customer orders, these companies can make 
+better operational decisions to utilize their resources 
+and information more efficiently. For instance, e-tailers 
+can profit from bundling multiple items ordered by the 
+same customer into a single shipment. By moving 
+products after orders are placed, an online retailer has 
+flexibility to decide from which of its warehouses or 
+fulfillment centers it is going to serve a customer 
+demand, providing new opportunities to minimize 
+operating costs (Janis, 2019). 
+ 
+E-commerce supply chain structure can be divided into 
+three main pieces: first-mile, middle-mile and last-mile: 
+• 
+First-mile refers to the transportation of goods across 
+the first leg of the supply chain, which in the case of e-
+commerce is the inbound transportation between 
+vendors/merchants and the distribution centers or 
+fulfillment center. The inbound network has global 
+scale, uses a combination of transportation modes (e.g., 
+ships, airplanes and trucks), and deals with time 
+horizon of the order of weeks to months. The main 
+technical challenges in the first-mile space are related 
+to inbound network design, and inventory placement 
+and replenishment (Chen, 2017, Govindarajan et al, 
+2020, DeValve et al, 2021). 
+• 
+Middle-mile encompasses the outbound transportation 
+network between the fulfillment center and the delivery 
+nodes. The outbound network has national/continental 
+scale, uses a combination of transportation modes (e.g., 
+trucks and airplanes). As customers got accustomed to 
+faster delivery timelines (from days to hours), new 
+challenges have arisen in the middle-mile space to be 
+able to design a transportation network that allows 
+aggressive delivery promises while keeping operations 
+cost-effective (Lara et al, 2022). 
+• 
+Last-mile is the last leg of the supply chain to deliver 
+packages to customers’ doors. It accounts for a 
+significant part of the total fulfillment cost, but deals 
+with much smaller scales: it focuses on city/county 
+level, and the time-horizon is of the order of hours. The 
+main technical challenge in this space is related to 
+vehicle routing (Delling et al, 2017).  
+ 
+Traditionally, each of these steps was performed by 
+different companies: e-commerce companies would rely on 
+third party logistics to transport between vendors and the 
+fulfillment center and between the fulfillment center and the 
+final customers. However, more recently, some players 
+have invested in verticalization as a way to be less 
+susceptive of global disruptions such as the 2021 supply 
+chain crisis (Schoolov, 2021). 
+ 
+ 
+Figure 1.   E-commerce supply chain structure 
+To exemplify the complexity of e-commerce 
+operations and decisions, following are some recent figures 
+from Amazon.com. Their logistics network includes more 
+than 110 aircrafts, 50,000 trailers, 400 fulfillment centers, 
+150 sortation centers, and 1,000 delivery stations globally 
+(Amazon, 2022), to store and transport hundreds of millions 
+of products sold across multiple geographies (Amazon, 
+2021).  
+Using the SCOR model as reference again, the main 
+difference between e-commerce and the process industry 
+lies in the Source and Make processes. E-commerce 
+retailers buy ready-to-sell goods from vendors; hence, the 
+manufacturing of the product is not included as part of their 
+supply chain. The Source process becomes responsible for 
+demand forecasting and inventory planning. The Make 
+process skips production and focuses on packaging and site 
+logistics. For e-commerce, the Deliver process is the core of 
+the business, i.e., fulfilling customer orders and delivering 
+to their home.  
+Challenges 
+While the Process and E-commerce industries differ in 
+structure, end-customer and overall processes, they share 
+similar issues, and have similar challenges to overcome in 
+order to remain competitive, keep up with the always 
+increasing requirements of the customers and stakeholders, 
+and move towards a more digitalized and efficient supply 
+chain. 
+It is well understood that analytical methods support a 
+mathematically rigorous description of the problem to be 
+solved, which provides for precise definition of the role of 
+the variables involved, the analysis of the current situation, 
+and numerical assessment of the alternative actions.  Thus, 
+analytical methods enforce a discipline and consistency to 
+decision-making that is not guaranteed by humans, but 
+which is generally sought.  Efficiency is also gained as 
+computers can execute the decision steps much faster than 
+
+ 
+ 
+ 
+ 4 
+humans.  The decision domain is also expanded as 
+analytical methods are capable of handling problems whose 
+scope and complexity exceeds the capacity of human 
+intelligence.  Thus, the application of the strengths of 
+machine intelligence to supply chain decision-making has 
+been the subject of a tremendous amount of research and the 
+basis of sophisticated commercial software (Wenzel, 2019, 
+Ni, 2020).  
+The process industries use many advanced analytical 
+methods, embodied in commercial software, in supply chain 
+operations. Optimal design or redesign of a supply chain 
+network using mathematical optimization has gained wide 
+acceptance and is routinely practiced (supports Plan) 
+(Bassett, 2018). Realtime optimization of world scale 
+process plants based on the solution of flowsheet models 
+has widespread applications (supports Make) (AspenTech, 
+2022). AI powered predictive analytics is providing real 
+time transportation visibility to a variety of industries, 
+including 
+the 
+process 
+industry 
+(supports 
+Deliver) 
+(FourKites, 2021).   For problems not suitably addressed by 
+commercial offerings, operating companies have been 
+active in developing inhouse solutions based on advanced 
+methods.   For the most part, success with machine 
+intelligence has occurred at specific decision points in the 
+enterprise scale workflow of supply chain operations.  This 
+leaves integration gaps that need to be addressed both 
+horizontally and vertically.   
+Similarly, e-commerce companies heavily rely on 
+machine 
+learning, 
+mathematical 
+optimization 
+and 
+simulation to plan, manage and execute their supply chains. 
+For example, at Amazon, forecasting models predict the 
+demand for every product, buying systems determine the 
+right level of product to purchase from different suppliers, 
+while large-scale placement systems determine the optimal 
+location for products across hundreds of fulfillment centers 
+(Amazon, 2021). Inside the fulfillment centers, computer 
+vision systems keep track of where every product is.  When 
+an order is placed, it triggers multiple systems that have to 
+solve large-scale optimization problems in real time to 
+decide how to pick, pack, load, and move packages to the 
+customer’s house (Karlinsky, 2019). The real-time 
+decisions rely on previously made tactical (e.g., truck, flight 
+and labor scheduling) and strategic decisions (e.g., where to 
+build new fulfillment centers, sortation center and delivery 
+stations and for what capacity) that are also supported by 
+analytical tools.  
+While there have been significant advancements in 
+computing power (Hao, 2019), machine learning (Stanford 
+University, 2021), and optimization solvers (Bixby, 2012), 
+the size and complexity of supply chains grew in even faster 
+pace, making it intractable to optimize the entire network 
+and operations as a single problem. The question becomes: 
+how to best break down these problems in a way that makes 
+business coordination easier, and how to use modeling to 
+help coordinate between multiple organizations and 
+reconcile between their conflicting objectives. 
+This problem decomposition in the modeling space 
+tends to follow the same structure as the business division 
+in the organizational space. And as it happens in the 
+modeling space, it becomes challenging to make sure that 
+all organizations use a consistent set of data and 
+assumptions and are optimizing for the overall business 
+(global optimal) instead of their own specific metrics (local 
+optimal). In the case of e-commerce, it seems intuitive to 
+break the problem down between first-mile, middle-mile 
+and last-mile, as they deal with different scales and tackle 
+different parts of the network. However, the decisions made 
+in each problem deeply impacts the others. For example, the 
+inventory placement decision (first-mile) is critical for the 
+design of the middle-mile network. If one optimizes for the 
+first-mile alone, it may seem like a good idea to have a huge 
+fulfillment center that stores all products and is close to a 
+shipping port. The inbound transportation cost to the 
+fulfillment center is minimal, and this structure eliminates 
+the complexity of trying to decide which fulfillment center 
+should carry which product. However, such solution 
+significantly hurts the middle-mile. Most orders would have 
+to travel long distances between the fulfillment center and 
+customers’ homes, increasing outbound transportation cost, 
+hurting delivery speed and, in turn, customer satisfaction.  
+Another crucial challenge is how to better coordinate 
+decision-making across different time scales. Traditionally, 
+long-term strategic decisions, mid-term tactical decisions 
+and short-term execution decisions are made by different 
+teams. The closer the timeline the more information 
+available 
+but 
+the 
+less 
+flexibility 
+for 
+significant 
+modifications to the plan. There is a large body of literature 
+on how to coordinate between strategic, tactical and 
+execution-level decisions (Brunaud, 2019), but in practice 
+there is still a lot to be done in this area to reduce the gap 
+between plan and execution. Planning should be better at 
+taking variability and uncertainty into account so it can 
+actually be executed, and the real-time systems should be 
+better at adhering to the original plan. 
+Additionally, data issues are often a significant barrier 
+to the successful application of advanced analytical 
+methods. The data involved in an enterprise workflow such 
+as supply chain operations spans many diverse data 
+repositories which are not well connected for data 
+exchange, lack consistency in naming conventions, and 
+have poorly defined data relationships. To be able to have 
+more automated processes and take full advantage of 
+computer-based analytical processes, companies need to 
+ensure that: (i) everyone is making the same assumptions 
+and using the same source of truth for data; (ii) data can be 
+automatically 
+aggregated/disaggregated 
+to 
+support 
+consistency between multiple time scales; and (iii) data 
+relationships can be modeled to support end-to-end process 
+and business coordination. 
+Finally, the role of the human intelligence in supply 
+chain decisions needs more rigorous treatment. With very 
+few exceptions, advanced analytical methods applied in 
+supply chain operations rely on some form of human 
+
+ 
+ 
+ 
+5 
+intervention or assistance in carrying out the intended 
+decision-making role.  It is important to understand where 
+human reasoning is more effective versus computer-based 
+analytical methods and how to blend both cognitive 
+approaches along decision points in the end-to-end process 
+for the most effective solution. 
+ 
+Future trends and opportunities 
+Future Supply Chain Management 
+In the near future, decisions of supply chain 
+professionals will be supported by virtual assistants 
+facilitating 
+the 
+use 
+of 
+advanced 
+analytical 
+methods. Alternative courses of action, derived from an 
+understanding of event propagation or the correlation of 
+multiple events across the supply chain through time, will 
+be provided in a manner that fits within a decision maker’s 
+personal workflow.  Logic and machine learning will derive 
+insights from trends, identify deviations from normal 
+behavior, or derive other patterns that signify or predict 
+actionable business interventions such as opportunities to 
+reduce costs or working capital.  Supply chain professionals 
+at all levels will have quantitative, financial measures to 
+assess the alternatives involved in their decisions, providing 
+them with complete visibility of the impact of decisions on 
+financial results.  
+In the long-term, there is ever increasing scope of 
+decision-making automation and a diminishing reliance on 
+manual intervention.  Reminiscent of the level of 
+automation achieved in manufacturing, automation of 
+short-term, operational level supply chain decisions will be 
+commonplace.  Supply chain professionals will manage the 
+process of their role as carried out by one or more intelligent 
+software agents, rather than exercise direct decision-
+making.    Thus, decision makers are released from day-to-
+day decision-making to focus on more strategic decisions 
+and value-added activities. 
+This future system of agents, both human and 
+computer-based, will be supported by a common supply 
+chain digital twin tightly aligning the decision processes of 
+all decision-makers. The digital twin spanning the entire 
+supply chain, and fed by operational, transactional, and cash 
+flow data, will allow the system to operate with fully 
+integrated strategic, tactical, and operational decisions. 
+There is no siloed decision making. Myopic and chaotic 
+decisions are avoided as decision makers have full upstream 
+and downstream awareness of decision implications.  The 
+resulting 
+digital 
+supply 
+chain 
+will 
+operate 
+with 
+competitively advantaged agility and adaptability while 
+balancing cost and other financial measures across the 
+supply chain to achieve alignment among all stakeholders. 
+Promising Research Directions 
+The properties associated with the human dimension of 
+supply chain decisions and the multi-agent nature of supply 
+chain operations have not received widespread attention in 
+the PSE literature.   We believe both these streams of 
+research will have profound impact on the challenges of 
+scalability, business coordination, multiple time scales, and 
+the prevalence of human decision-making.  
+ Research on design of analytical methods that 
+rigorously account for human input, response, and 
+intervention is needed to balance the tremendous amount of 
+current research developing model-based solutions.  The 
+emerging area of Cognitive Engineering is developing 
+mathematical models that describe human decision-making, 
+as well as analytical methods that incorporate human 
+intuition and judgement.  Engineering human-in-the-loop 
+interactions as carried out in cyber-physical systems is 
+another good source of ideas for supply chain operations 
+(Gil et al, 2020).  Developing machine learning methods to 
+extract from historical records policies practiced by humans 
+is another fruitful area of research. Ultimately PSE research 
+should lead to holistic solution designs where mathematical 
+model is just one component and hybrid intelligence is 
+achieved. 
+The field of multi-agent systems provides the 
+foundation for achieving the future supply chain described 
+in the previous section (Jones, 2018).  As a collection of 
+multiple decision-making agents which interact in a shared 
+environment to achieve common or conflicting goals, a 
+multi-agent system is an excellent paradigm to analyze and 
+synthesize an end-to-end process like supply chain.  The 
+definition of an agent and the structure of the system are 
+highly flexible.  Human or artificial agents are 
+accommodated, robotic warehouses and autonomous 
+trucks/ships can be modeled as agents, and agents can 
+reflect existing organizational roles, or organizations 
+themselves. Multi-agent systems perform distributed 
+decision-making 
+offering 
+a 
+natural 
+approach 
+to 
+decomposing problems that are intractable when taken as a 
+whole.  Traditional decomposition methods used in 
+optimization can be viewed as multi-agent systems.  
+Because agent connections are flexible and dynamic to 
+create solutions as needs arise, multi-agent systems are 
+naturally more agile than hierarchical methods, and 
+therefore more resilient. 
+Final Thoughts 
+One motivation for our proposed research directions is 
+that we agree with other researchers on the complimentary 
+nature of human intelligence and machine intelligence 
+(human-AI symbiosis) (Jarrahi, 2018).  We consider 
+machine intelligence to include artificial intelligence and 
+machine learning, as well as the broad spectrum of 
+traditional process systems engineering methods like 
+optimization modeling, simulation modeling, and general 
+regression modeling.  Given the strengths of these methods 
+as cited earlier, machine intelligence will clearly be central 
+to the future supply chain.   
+However, the operation of massive and evolving supply 
+chains as found in e-commerce and the process industries 
+
+ 
+ 
+ 
+ 6 
+involves 
+problems 
+where 
+human 
+intelligence 
+has 
+advantages.  The external connections found in supply 
+chains create decision domains with limited information for 
+which human intuition is well suited.  Machine intelligence 
+can produce spurious results or impractical solutions and 
+can be subjected to situations where assumptions do not 
+hold.  Under these conditions, common sense practiced by 
+humans is important in reaching effective decisions.  
+Human creativity provides solutions to problems not 
+previously encountered such as occur in supply chain 
+disruptions. Supply chain networks are also social networks 
+comprised of suppliers and customers, enterprise-scale 
+workflows involving many employees, and a variety of 
+external stakeholders.  Humans, as social creatures, have the 
+natural ability to account for the tendencies, priorities, 
+motivations of others and the tendency to engage with 
+others for mutual success.  Notwithstanding the tremendous 
+progress of machine intelligence, we see human intelligence 
+as critical to the success of the future supply chain.  When 
+both are used to their greatest advantage the future supply 
+chain will meet the challenges of today, adapt to the 
+challenges of the future, satisfy the needs of all 
+stakeholders, and recede from the headlines. 
+Conclusions 
+Modern, global industries, such as e-commerce and 
+chemical manufacturing, operate massive supply chains 
+with complex enterprise-scale workflows.    While both 
+industries have been early adopters of decision support tools 
+based on machine intelligence, both share unresolved 
+challenges, such as those related to the scale of their supply 
+chain operations, the integration of strategic, tactical, and 
+operational decisions, harmonizing across internal business 
+units, and the effective orchestration of the many decision-
+makers, both human and computer-based.  We believe the 
+methods of multi-agent systems supported by rigorous 
+treatment of human decision-making has excellent potential 
+to address these critical challenges and complement the 
+existing contributions of the Process Systems Engineering 
+community. 
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf,len=333
+page_content='FUTURE OF SUPPLY CHAIN: CHALLENGES,  TRENDS, AND PROSPECTS  Cristiana L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Lara1*, John Wassick2  1Modeling and Optimization, Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='com, Bellevue WA 98004  2Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh PA  15213  Abstract  This paper discusses the broad challenges shared by e-commerce and the process industries operating global supply  chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Specifically, we discuss how process industries and e-commerce differ in many aspects but have similar challenges  ahead of them in order to remain competitive, keep up with the always increasing requirements of the customers and  stakeholders, and gain efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' While both industries have been early adopters of decision support tools based on machine  intelligence, both share unresolved challenges related to scalability, integration of decision-making over different time  horizons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' strategic, tactical and execution-level decisions) and across internal business units, and orchestration of human  and computer-based decision-makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' We discuss future trends and research opportunities in the area of supply chain, and  suggest that the methods of multi-agent systems supported by rigorous treatment of human decision-making in combination  with machine intelligence is a great contender to address these critical challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Keywords  Supply chain, logistics, process industry, e-commerce, artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Introduction  The concept of supply chain as an interconnected,  dynamic physical system to deliver goods and services to  consumers has become a popular topic for the public in  recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' This increased awareness is due to the  recognition by the public of the connection between supply  chain problems and the lived experience of disappoints  concerning  product  availability  and  price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Past  globalization of trade, and the enterprises that engage in it,  has  increased  their  scope  and  their  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Consequently, supply chains are more  challenging  to  operate  and  more  vulnerable  to  disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The recent pandemic, high profile logistical  disruptions, and war have exposed the vulnerability of  global supply chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The corresponding swings in  consumer preferences and demand have stressed the  management  practices  surrounding  supply  chain  To whom all correspondence should be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' This work does not relate to the author’s position at Amazon, regardless of their  affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Both companies serving end consumers and  those farther upstream have been focused on modernizing  and digitizing their supply chains to retain and grow their  customer base and their bottom line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  The ongoing activities by the many diverse operating  companies pursuing improvements to their supply chains  raises the question of shared ideals for their future supply  chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' It is safe to say that all companies seek to be more  flexible and agile in the face of disruptions, more adaptable  to evolving markets, and more aligned to suppliers and  customers in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' However, they also share specific  objectives at an operational level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' This can be demonstrated  by examining the shared challenges of supply chains of e-  commerce and the process industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Both are challenged  with the scalability and integration needed to deliver end-  to-end performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Both operate enterprise scale  2  workflows that require coordination of decision-making  over different time granularities and geographic areas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=',  closing the gap between long-term planning, tactical  planning, and execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Both are deploying advanced  analytical methods to work in concert with the human  decision makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Considered together, the aspirations for  supply chain in e-commerce and the process industries  provides a consistent view of the future of supply chain and  research to support it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The difference in  the nature of the supply chains for the process industry and  e-commerce are first described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Their shared challenges are  discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Then future trends and research  opportunities are identified to address the shared  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Finally, in a future where artificial intelligence  is pervasively used in supply chain operations, the place for  the use of human intelligence is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Different Industries, similar challenges  Process Industries  Within the broad process industries, the chemical  industry operates some of the most complex and  challenging supply chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Key reasons include level of  vertical integration of product lines, diversity of markets  served and position in value chains, energy intensity, and  capital intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Large chemical companies have complex,  globally distributed manufacturing sites that are themselves  tightly coupled, internal supply chains (Wassick, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Recent figures from The Dow Chemical Company  provide a good example of the scale and complexity of a  global chemical company (SupplyChainWorld 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Manufacturing more than 6,000 product families at 201  sites in 35 countries, Dow manages roughly 6,000  shipments per day to serve 45,000 customer locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The  company’s annual shipment volume is about 130 billion  pounds and uses 450 warehouses and 150 contract  terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The company works with 650 service providers  in 160 countries and receives more than 110 billion pounds  of raw materials and intermediates each year from 4,000  suppliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  We will use the Supply Chain Operations Reference  (SCOR) model, commonly used by supply chain  professionals as a paradigm for discussing supply chains,  and comprised of the business processes Plan, Source,  Make, Deliver, and Return, to briefly discuss the nature of  chemical industry supply chains (Delipinar and Kocaoglu,  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Central to the Plan process are strategic and tactical  supply and demand balancing, strategies to support a  business financial plan, and long-term asset strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' These  decisions become very challenging when vertical  integration requires planning across multiple businesses,  each with its own profit and loss metric, serving diverse  markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Business supply chains are highly interdependent  with each echelon serving both external markets and  internal downstream businesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Inventory management is  confounded by the various dimensions of the value of  upstream inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  The Source process, responsible for raw material  acquisition and the associated supplier management, has  several complexities in the chemical industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' For one,  external sources can be used to supplement the internal  sources found in the vertical integration of a company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Raw  materials can include globally traded commodities like oil,  naphtha, and natural gas for which long term hedging and  contracts can be used for procurement while also exploiting  spot market prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Large manufacturing sites can operate  cogeneration electrical power plants that serve the onsite  production units but also can supply to the local electrical  grid, or power can be purchased from the grid when  advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  The Make process focuses on the broad set of  manufacturing activities which includes production,  production scheduling, packaging, and site logistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Safety  and environmental protection are top priority in the  chemical industry and the nature of the products and  processes that make them leads to complex business rules  and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Here too, vertical integration at a  manufacturing site adds another layer of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Logistics operations can include massive railyards, tank  farms, and marine ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Within the Deliver process are activities associated  with order management, warehousing, and transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Most of the commerce in the chemical industry is made up  of business-to-business transactions which can involve long  lead time orders of massive volumes and complicated sales  contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Transportation for specific orders may be  supplied by ocean going vessels, barges, railcars, and  dedicated tank trucks, and combinations of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  International shipments to customers are common which  requires sophisticated international trade operations to  manage compliance with duties and tariffs and shipping  requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  E-commerce  Online retailers have come of age with the dot-com  boom, and since then, have been competing with other  retailers within the broader retailing industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Xu (2005)  summarizes some characteristics of online retailing,  reinterpreted in the list below: Large Selection: Not limited by physical space in the  store front, online retailers are able to offer hundreds of  thousands of Stock-Keeping Units (SKU) in stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Logistics as a Matter of Trust: Online customers con-  sider the timely delivery of products to be a significant  component of trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Thus, the reliability and efficiency  of the supply chain are even more crucial than for  traditional brick-and-mortar retailers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Data-driven decision making: These companies rely on  rigorous data analytics to make decisions at every stage  3  of supply chain, from selection of products and  inventory placement to delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Flexibility when fulfilling customer orders: In online  retailing, there is typically a time delay between when  a demand occurs (customer order) and when inventory  is physically deployed and transported to meet a  customer order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' By delaying the decisions on how to  fulfill customer orders, these companies can make  better operational decisions to utilize their resources  and information more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' For instance, e-tailers  can profit from bundling multiple items ordered by the  same customer into a single shipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' By moving  products after orders are placed, an online retailer has  flexibility to decide from which of its warehouses or  fulfillment centers it is going to serve a customer  demand, providing new opportunities to minimize  operating costs (Janis, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  E-commerce supply chain structure can be divided into  three main pieces: first-mile, middle-mile and last-mile: First-mile refers to the transportation of goods across  the first leg of the supply chain, which in the case of e-  commerce is the inbound transportation between  vendors/merchants and the distribution centers or  fulfillment center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The inbound network has global  scale, uses a combination of transportation modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=',  ships, airplanes and trucks), and deals with time  horizon of the order of weeks to months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The main  technical challenges in the first-mile space are related  to inbound network design, and inventory placement  and replenishment (Chen, 2017, Govindarajan et al,  2020, DeValve et al, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Middle-mile encompasses the outbound transportation  network between the fulfillment center and the delivery  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The outbound network has national/continental  scale, uses a combination of transportation modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=',  trucks and airplanes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' As customers got accustomed to  faster delivery timelines (from days to hours), new  challenges have arisen in the middle-mile space to be  able to design a transportation network that allows  aggressive delivery promises while keeping operations  cost-effective (Lara et al, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Last-mile is the last leg of the supply chain to deliver  packages to customers’ doors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' It accounts for a  significant part of the total fulfillment cost, but deals  with much smaller scales: it focuses on city/county  level, and the time-horizon is of the order of hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The  main technical challenge in this space is related to  vehicle routing (Delling et al, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Traditionally, each of these steps was performed by  different companies: e-commerce companies would rely on  third party logistics to transport between vendors and the  fulfillment center and between the fulfillment center and the  final customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' However, more recently, some players  have invested in verticalization as a way to be less  susceptive of global disruptions such as the 2021 supply  chain crisis (Schoolov, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' E-commerce supply chain structure  To exemplify the complexity of e-commerce  operations and decisions, following are some recent figures  from Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Their logistics network includes more  than 110 aircrafts, 50,000 trailers, 400 fulfillment centers,  150 sortation centers, and 1,000 delivery stations globally  (Amazon, 2022), to store and transport hundreds of millions  of products sold across multiple geographies (Amazon,  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Using the SCOR model as reference again, the main  difference between e-commerce and the process industry  lies in the Source and Make processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' E-commerce  retailers buy ready-to-sell goods from vendors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' hence, the  manufacturing of the product is not included as part of their  supply chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The Source process becomes responsible for  demand forecasting and inventory planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The Make  process skips production and focuses on packaging and site  logistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' For e-commerce, the Deliver process is the core of  the business, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=', fulfilling customer orders and delivering  to their home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Challenges  While the Process and E-commerce industries differ in  structure, end-customer and overall processes, they share  similar issues, and have similar challenges to overcome in  order to remain competitive, keep up with the always  increasing requirements of the customers and stakeholders,  and move towards a more digitalized and efficient supply  chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  It is well understood that analytical methods support a  mathematically rigorous description of the problem to be  solved, which provides for precise definition of the role of  the variables involved, the analysis of the current situation,  and numerical assessment of the alternative actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Thus,  analytical methods enforce a discipline and consistency to  decision-making that is not guaranteed by humans, but  which is generally sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Efficiency is also gained as  computers can execute the decision steps much faster than  4  humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The decision domain is also expanded as  analytical methods are capable of handling problems whose  scope and complexity exceeds the capacity of human  intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Thus, the application of the strengths of  machine intelligence to supply chain decision-making has  been the subject of a tremendous amount of research and the  basis of sophisticated commercial software (Wenzel, 2019,  Ni, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  The process industries use many advanced analytical  methods, embodied in commercial software, in supply chain  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Optimal design or redesign of a supply chain  network using mathematical optimization has gained wide  acceptance and is routinely practiced (supports Plan)  (Bassett, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Realtime optimization of world scale  process plants based on the solution of flowsheet models  has widespread applications (supports Make) (AspenTech,  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' AI powered predictive analytics is providing real  time transportation visibility to a variety of industries,  including  the  process  industry  (supports  Deliver)  (FourKites, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' For problems not suitably addressed by  commercial offerings, operating companies have been  active in developing inhouse solutions based on advanced  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' For the most part, success with machine  intelligence has occurred at specific decision points in the  enterprise scale workflow of supply chain operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' This  leaves integration gaps that need to be addressed both  horizontally and vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Similarly, e-commerce companies heavily rely on  machine  learning,  mathematical  optimization  and  simulation to plan, manage and execute their supply chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  For example, at Amazon, forecasting models predict the  demand for every product, buying systems determine the  right level of product to purchase from different suppliers,  while large-scale placement systems determine the optimal  location for products across hundreds of fulfillment centers  (Amazon, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Inside the fulfillment centers, computer  vision systems keep track of where every product is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' When  an order is placed, it triggers multiple systems that have to  solve large-scale optimization problems in real time to  decide how to pick, pack, load, and move packages to the  customer’s house (Karlinsky, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The real-time  decisions rely on previously made tactical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=', truck, flight  and labor scheduling) and strategic decisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=', where to  build new fulfillment centers, sortation center and delivery  stations and for what capacity) that are also supported by  analytical tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  While there have been significant advancements in  computing power (Hao, 2019), machine learning (Stanford  University, 2021), and optimization solvers (Bixby, 2012),  the size and complexity of supply chains grew in even faster  pace, making it intractable to optimize the entire network  and operations as a single problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The question becomes:  how to best break down these problems in a way that makes  business coordination easier, and how to use modeling to  help coordinate between multiple organizations and  reconcile between their conflicting objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  This problem decomposition in the modeling space  tends to follow the same structure as the business division  in the organizational space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' And as it happens in the  modeling space, it becomes challenging to make sure that  all organizations use a consistent set of data and  assumptions and are optimizing for the overall business  (global optimal) instead of their own specific metrics (local  optimal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' In the case of e-commerce, it seems intuitive to  break the problem down between first-mile, middle-mile  and last-mile, as they deal with different scales and tackle  different parts of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' However, the decisions made  in each problem deeply impacts the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' For example, the  inventory placement decision (first-mile) is critical for the  design of the middle-mile network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' If one optimizes for the  first-mile alone, it may seem like a good idea to have a huge  fulfillment center that stores all products and is close to a  shipping port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The inbound transportation cost to the  fulfillment center is minimal, and this structure eliminates  the complexity of trying to decide which fulfillment center  should carry which product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' However, such solution  significantly hurts the middle-mile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Most orders would have  to travel long distances between the fulfillment center and  customers’ homes, increasing outbound transportation cost,  hurting delivery speed and, in turn, customer satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Another crucial challenge is how to better coordinate  decision-making across different time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Traditionally,  long-term strategic decisions, mid-term tactical decisions  and short-term execution decisions are made by different  teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The closer the timeline the more information  available  but  the  less  flexibility  for  significant  modifications to the plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' There is a large body of literature  on how to coordinate between strategic, tactical and  execution-level decisions (Brunaud, 2019), but in practice  there is still a lot to be done in this area to reduce the gap  between plan and execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Planning should be better at  taking variability and uncertainty into account so it can  actually be executed, and the real-time systems should be  better at adhering to the original plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Additionally, data issues are often a significant barrier  to the successful application of advanced analytical  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The data involved in an enterprise workflow such  as supply chain operations spans many diverse data  repositories which are not well connected for data  exchange, lack consistency in naming conventions, and  have poorly defined data relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' To be able to have  more automated processes and take full advantage of  computer-based analytical processes, companies need to  ensure that: (i) everyone is making the same assumptions  and using the same source of truth for data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' (ii) data can be  automatically  aggregated/disaggregated  to  support  consistency between multiple time scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' and (iii) data  relationships can be modeled to support end-to-end process  and business coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Finally, the role of the human intelligence in supply  chain decisions needs more rigorous treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' With very  few exceptions, advanced analytical methods applied in  supply chain operations rely on some form of human  5  intervention or assistance in carrying out the intended  decision-making role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' It is important to understand where  human reasoning is more effective versus computer-based  analytical methods and how to blend both cognitive  approaches along decision points in the end-to-end process  for the most effective solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Future trends and opportunities  Future Supply Chain Management  In the near future, decisions of supply chain  professionals will be supported by virtual assistants  facilitating  the  use  of  advanced  analytical  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Alternative courses of action, derived from an  understanding of event propagation or the correlation of  multiple events across the supply chain through time, will  be provided in a manner that fits within a decision maker’s  personal workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Logic and machine learning will derive  insights from trends, identify deviations from normal  behavior, or derive other patterns that signify or predict  actionable business interventions such as opportunities to  reduce costs or working capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Supply chain professionals  at all levels will have quantitative, financial measures to  assess the alternatives involved in their decisions, providing  them with complete visibility of the impact of decisions on  financial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  In the long-term, there is ever increasing scope of  decision-making automation and a diminishing reliance on  manual intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Reminiscent of the level of  automation achieved in manufacturing, automation of  short-term, operational level supply chain decisions will be  commonplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Supply chain professionals will manage the  process of their role as carried out by one or more intelligent  software agents, rather than exercise direct decision-  making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Thus, decision makers are released from day-to-  day decision-making to focus on more strategic decisions  and value-added activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  This future system of agents, both human and  computer-based, will be supported by a common supply  chain digital twin tightly aligning the decision processes of  all decision-makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The digital twin spanning the entire  supply chain, and fed by operational, transactional, and cash  flow data, will allow the system to operate with fully  integrated strategic, tactical, and operational decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  There is no siloed decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Myopic and chaotic  decisions are avoided as decision makers have full upstream  and downstream awareness of decision implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The  resulting  digital  supply  chain  will  operate  with  competitively advantaged agility and adaptability while  balancing cost and other financial measures across the  supply chain to achieve alignment among all stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Promising Research Directions  The properties associated with the human dimension of  supply chain decisions and the multi-agent nature of supply  chain operations have not received widespread attention in  the PSE literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' We believe both these streams of  research will have profound impact on the challenges of  scalability, business coordination, multiple time scales, and  the prevalence of human decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Research on design of analytical methods that  rigorously account for human input, response, and  intervention is needed to balance the tremendous amount of  current research developing model-based solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The  emerging area of Cognitive Engineering is developing  mathematical models that describe human decision-making,  as well as analytical methods that incorporate human  intuition and judgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Engineering human-in-the-loop  interactions as carried out in cyber-physical systems is  another good source of ideas for supply chain operations  (Gil et al, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Developing machine learning methods to  extract from historical records policies practiced by humans  is another fruitful area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Ultimately PSE research  should lead to holistic solution designs where mathematical  model is just one component and hybrid intelligence is  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  The field of multi-agent systems provides the  foundation for achieving the future supply chain described  in the previous section (Jones, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' As a collection of  multiple decision-making agents which interact in a shared  environment to achieve common or conflicting goals, a  multi-agent system is an excellent paradigm to analyze and  synthesize an end-to-end process like supply chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The  definition of an agent and the structure of the system are  highly flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Human or artificial agents are  accommodated, robotic warehouses and autonomous  trucks/ships can be modeled as agents, and agents can  reflect existing organizational roles, or organizations  themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Multi-agent systems perform distributed  decision-making  offering  a  natural  approach  to  decomposing problems that are intractable when taken as a  whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Traditional decomposition methods used in  optimization can be viewed as multi-agent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Because agent connections are flexible and dynamic to  create solutions as needs arise, multi-agent systems are  naturally more agile than hierarchical methods, and  therefore more resilient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Final Thoughts  One motivation for our proposed research directions is  that we agree with other researchers on the complimentary  nature of human intelligence and machine intelligence  (human-AI symbiosis) (Jarrahi, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' We consider  machine intelligence to include artificial intelligence and  machine learning, as well as the broad spectrum of  traditional process systems engineering methods like  optimization modeling, simulation modeling, and general  regression modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Given the strengths of these methods  as cited earlier, machine intelligence will clearly be central  to the future supply chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  However, the operation of massive and evolving supply  chains as found in e-commerce and the process industries  6  involves  problems  where  human  intelligence  has  advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The external connections found in supply  chains create decision domains with limited information for  which human intuition is well suited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Machine intelligence  can produce spurious results or impractical solutions and  can be subjected to situations where assumptions do not  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Under these conditions, common sense practiced by  humans is important in reaching effective decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Human creativity provides solutions to problems not  previously encountered such as occur in supply chain  disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Supply chain networks are also social networks  comprised of suppliers and customers, enterprise-scale  workflows involving many employees, and a variety of  external stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Humans, as social creatures, have the  natural ability to account for the tendencies, priorities,  motivations of others and the tendency to engage with  others for mutual success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Notwithstanding the tremendous  progress of machine intelligence, we see human intelligence  as critical to the success of the future supply chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' When  both are used to their greatest advantage the future supply  chain will meet the challenges of today, adapt to the  challenges of the future, satisfy the needs of all  stakeholders, and recede from the headlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Conclusions  Modern, global industries, such as e-commerce and  chemical manufacturing, operate massive supply chains  with complex enterprise-scale workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' While both  industries have been early adopters of decision support tools  based on machine intelligence, both share unresolved  challenges, such as those related to the scale of their supply  chain operations, the integration of strategic, tactical, and  operational decisions, harmonizing across internal business  units, and the effective orchestration of the many decision-  makers, both human and computer-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' We believe the  methods of multi-agent systems supported by rigorous  treatment of human decision-making has excellent potential  to address these critical challenges and complement the  existing contributions of the Process Systems Engineering  community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' thesis, MIT, Cambridge,  MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' Procedia  Social and Behavioral Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content='  Transportation Science 51(2):566-591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=', Wei, Y, Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' (2021) Understanding the  Value of Fulfillment Flexibility in an Online Retailing  Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Manufacturing & Service Operations  Management 0(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  FourKites (2022) https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='fourkites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=', Fons, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=', Pelechano, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' (2020) Engineering  human-in-the-loop  interactions  in  cyber-physical  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content='  Hao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The computing power needed to train AI is now rising  seven times faster than ever before, November 11, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  MIT  Technology  Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content='technologyreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='com/2019/11/11/13200  4/the-computing-power-needed-to-train-ai-is-now-  rising-seven-times-faster-than-ever-before/  Jasin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=', Uichanco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' In: Gallino, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' (eds) Operations in an Omnichannel World.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Springer Series in Supply Chain Management, vol 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='1007/978-3-030-  20119-7_2  Jarrahi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' Business Horizons 61(4): 577-586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content='  Manufacturing Letters 17:6-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' How artificial intelligence helps Amazon  deliver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Amazon  News,  June  5,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='aboutamazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='com/news/innovation-at-  amazon/how-artificial-intelligence-helps-amazon-  deliver  Lara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=', Koenemann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=', de Souza, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' (2022),  Scalable  Timing-Aware  Network  Design  via  Lagrangian Decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Submitted for publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Ni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' & Lim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' (2020), A systematic review of the  research trends of machine learning in supply chain  management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' & Cyber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' 11, 1463–  1482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Schoolov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Amazon Is Making Its Own Containers and  Bypassing Supply Chain Chaos with Chartered Ships  and Long-Haul Planes.” CNBC, December 4, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='cnbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='com/2021/12/04/how-amazon-beats-  supply-chain-chaos-with-ships-and-long-haul-  planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  7  Stanford University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Gathering Strength, Gathering Storms: The  One Hundred Year Study on Artificial Intelligence  (AI100) 2021 Study Panel Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Stanford University,  Stanford,  CA,  September  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  http://ai100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='edu/2021-report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  SupplyChainWorld (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' The Dow Chemical Company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  SupplyChainWorld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' https://scw-mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' Computers & Chemical  Engineering 33(12):1950-1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' Smit, Daniel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Sardesai, Saskia (2019) : A  literature review on machine learning in supply chain  management, In: Kersten, Wolfgang Blecker, Thorsten  Ringle, Christian M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' ): Artificial Intelligence and  Digital Transformation in Supply Chain Management:  Innovative Approaches for Supply Chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' Proceedings  of the Hamburg International Conference of Logistics  (HICL), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' 27, ISBN 978-3-7502-4947-9, epubli  GmbH, Berlin, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' 413-441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='  Xu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
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+page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
+page_content=' thesis, MIT, Cambridge, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FPT4oBgHgl3EQfzDWy/content/2301.13174v1.pdf'}
diff --git a/ddE0T4oBgHgl3EQfoQEX/content/tmp_files/2301.02522v1.pdf.txt b/ddE0T4oBgHgl3EQfoQEX/content/tmp_files/2301.02522v1.pdf.txt
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@@ -0,0 +1,260 @@
+1 
+ 
+STUDY ON THE HYDROGEN BEHAVIOR IN Y AND Co DOPED BARIUM-ZIRCONATE 
+ 
+M. Khalid Hossain*1,2, Kenichi Hashizume**1 
+1. Department of Advanced Energy Engineering Science, Interdisciplinary Graduate School of Engineering 
+Science, Kyushu University, Kasuga, Fukuoka 816-8580, Japan 
+2. Atomic Energy Research Establishment, Bangladesh Atomic Energy Commission, Savar, Dhaka 1349, 
+Bangladesh 
+E-mail: *khalid.baec@gmail.com, **hashi@nucl.kyushu-u.ac.jp 
+ 
+ 
+ABSTRACT 
+Hydrogen release behavior from mixed-perovskite 
+oxide, yttrium and cobalt doped barium-zirconate, 
+BaZr0.955Y0.03Co0.015O3-α (BZYC) exposed to D2 or 
+D2O atmosphere was studied by thermal desorption 
+spectroscopy (TDS) detecting deuterium gas (D2) 
+and heavy water vapor (D2O) separately. The 
+release of hydrogen appeared at around 1000 K. 
+Hydrogen solubility was higher in the BZYC 
+sample exposed to the D2O. 
+ 
+KEY WORDS 
+Barium-zirconate, perovskite oxide, hydrogen 
+dissolution, hydrogen solubility, TDS.  
+ 
+1. INTRODUCTION 
+Proton-conducting oxides are promising materials 
+for electro-chemical devices like fuel cell, hydrogen 
+pump, hydrogen sensor, and for nuclear fusion 
+reactor’s tritium purification and recovery system. 
+Therefore, the study of hydrogen behavior in such 
+oxide materials is very fundamental and important. 
+Yttrium (Y) and cobalt (Co) doped barium-
+zirconate (BaZrO3) is a proton conducting oxide has 
+mixed-perovskite structure [1]. In this study, the 
+dissolution of hydrogen in BZYC specimens were 
+done using D2 gas and D2O vapor separately. Then 
+the hydrogen release and hydrogen solubility were 
+investigated by thermal desorption spectroscopy 
+(TDS) method.  
+ 
+2. EXPERIMENTAL 
+ 
+2.1 Sample preparation  
+The disc shape specimen (diameter: 7.7mm, 
+thickness: 2.2mm) was prepared using conventional 
+powder sintering method at 1923 K for 20h, which 
+is described elsewhere [2]. The SEM and EDX 
+mapping were performed before and after polishing 
+the sample to understand its surface properties. 
+From Fig. 1 it is clear that all chemical components 
+of the specimen were uniformly distributed 
+throughout the surface after polishing. The sintered 
+disc specimen was cut using a diamond saw into 
+parallelepiped pieces (7.7 x 2.2 x 0.5 mm3) for D2 
+gas or D2O vapor exposure and TDS study. 
+ 
+2.2 D2 and D2O exposure  
+The cut sample was placed in a quartz glass tube and 
+evacuated to a low pressure (~10-6 Pa). At a 
+temperature of 1273 K, vacuum annealing was 
+performed for 30 min to remove the volatile 
+substances from the sample. Then at 873 K, 
+deuterium gas (D2, 1.3 kPa) was exposed to the 
+sample for 1 hr using the apparatus shown in Fig. 
+2(a) [2]. After the exposure, the quartz glass tube 
+was quenched with room temperature water. Finally, 
+the sample was taken out from the quartz glass tube 
+after recovering the D2 gas. The same sample was 
+used repeatedly for heavy water vapor (D2O, 
+1.3kPa) exposure and the TDS study.  
+ 
+Fig. l: SEM micrograph and its EDS mapping for 
+the sintered disc sample surface after polishing. 
+ 
+Fig. 2: (a) Schematic of D2 gas and D2O vapor exposure 
+apparatus, (b) schematic of temperature programmed 
+desorption (TDS) system. 
+
+Ba
+Zr
+10p
+BALiC
+10um
+ZrL
+Co
+0
+5gm
+OK(a)
+Diaphragm
+Gauge(≤1000Torr)
+IG
+vacuum
+system
+D,O
+heater
+specimen
+(q)
+QMS
+IG
+heater
+vacuum
+specimen
+system2 
+ 
+2.3 D2 and D2O measurement  
+The TDS apparatus used in this experiment to 
+measure the released hydrogen (D2, HD, D2O and 
+HDO) from the D2 gas and D2O vapor exposed 
+sample is shown in Fig. 2(b) [2]. First, the baking of 
+the quartz glass tube without the sample was 
+performed with an electric furnace to release and 
+remove substances accumulated in the device at 
+1273 K for 30 min. Then after cooling down the 
+glass tube to room temperature, the D2 gas exposed 
+sample was placed in the glass tube for TDS study. 
+Then after evacuating the glass tube to a low 
+pressure (~10-6 Pa), the temperature of the sample 
+was allowed to raise (at a heating rate of 0.5 K/s) up 
+to 1273 K. After 30 min stay at 1273 K, the heater 
+power was disconnected automatically by using a 
+programable temperature controller. The gas 
+released from the samples was measured using a 
+quadrupole 
+mass 
+spectrometer 
+(Q-MAS) 
+(TSPTT200, INFICON) [3] during this sample 
+heating and cooling down cycle. The above same 
+procedure was also applied for the D2O exposed 
+sample. 
+ 
+3. RESULTS AND DISCUSSION 
+The ion currents obtained from Q-MAS data for the 
+D2 gas and D2O vapor exposed samples are shown in 
+Fig 3. It is clear from the figure that in both cases of 
+exposures, release of hydrogen gas (D2 and HD) and 
+water vapor (D2O and HDO) was observed at 
+around 1000 K. D2 gas was released with an extra 
+peak at 700 K in the case of D2 exposed sample and 
+at 850 K for D2O exposed sample. Highest release 
+peak was detected as HD gas from D2 exposed 
+sample and there may appear a trend for the 
+hydrogen release as HD > HDO > D2 > D2O. On the 
+other hand, the highest peak was observed as HDO 
+vapor from D2O exposed sample and there may 
+appear a trend for the hydrogen release as HDO > 
+D2O > HD > D2.   
+ 
+Table 1: D amount released from D2 and D2O exposed 
+samples  
+ 
+The total amounts of D-containing gas released 
+from the D2 or D2O exposed samples and its ratios 
+are listed in Table 1. Total amount of deuterium (D) 
+released from D2 exposed sample was 9.0 x 10-4 
+(D/M), where released deuterium gas (~ 4.6 x 10-4 
+D/M) and water vapor (~ 4.4 x 10-4 D/M) were 
+contributed equally. On the other hand, total amount 
+of D released from D2O exposed sample was 1.8 x 
+10-4 (D/M), where three-fourth of the released D 
+was water vapor (~ 1.3 x 10-4 D/M). It is also clear 
+that total amount of deuterium released from D2O 
+exposed sample is twice than the D2 exposed sample. 
+Percentage of water vapor released from the D2O 
+exposed sample is higher than that of the D2 
+exposed sample, whereas an opposite result appears 
+for deuterium gas release. The obtained result of the 
+experiment suggests that, under D2O exposed 
+condition and at 1000 K operating temperature, Y 
+and Co doped barium-zirconate may preferable for 
+electrochemical application due to its higher 
+hydrogen solubility.    
+ 
+ 
+4. CONCLUSIONS 
+In this study, hydrogen (deuterium) was dissolved 
+in BaZr0.955Y0.03Co0.015O3-α by the exposure of 
+deuterium gas or heavy water vapor, and the amount 
+of dissolved deuterium in the sample was measured 
+by TDS method. It was observed that deuterium 
+solubility is higher in the BZYC sample under the 
+D2O exposure condition and release of deuterium 
+was appeared at around 1000 K.  
+ 
+ACKNOWLEDGEMENTS 
+We would like to thank our laboratory members for 
+their cordial support to operate the experimental 
+instruments. 
+ 
+REFERENCES 
+1. J. Melnik, J. Luo, K.T. Chuang, and A.R. Sanger, 
+Open Fuels Energy Sci. J. 1 (2008) 7.  
+2. K. Yamashita, T. Otsuka, and K. Hashizume, Solid 
+State Ionics 275 (2015) 43.  
+3. S. Maher, S.U. Syed, D.M. Hughes, J.R. Gibson, 
+and S. Taylor, J. Am. Soc. Mass Spectrom. 24 (2013) 
+1307.  
+Deuterium (D) amount 
+D2 
+exposed  
+sample 
+D2O 
+exposed  
+sample 
+D from D2 and HD (D/M)  
+D from D2O and HDO (D/M) 
+Total D (D/M) 
+Fraction of deuterium gas (%) 
+Fraction of water vapor (%) 
+4.6 x 10-4 
+4.4 x 10-4 
+9.0 x 10-4 
+51 
+49 
+4.5 x 10-4 
+1.3 x 10-3 
+1.8 x 10-3 
+25 
+75 
+ 
+Fig. 3: TDS results for (a) D2 gas, and (b) D2O 
+vapor exposure samples.  
+
+D.Exposed
+Ir-01X9
+Ion Current (A)
+2
+HD
+HDO
+4x10-11
+DO
+D
+2x10-11
+0
+400
+600
+800
+1000
+1200
+Temparature (K)
+1.5x10*
+D.O Exposed
+Ion Current (A)
+2
+HDO
+1x10-10
+D.O
+HD
+5x10-11
+D
+0
+400
+600
+800
+1000
+1200
+Temparature(K)
\ No newline at end of file
diff --git a/ddE0T4oBgHgl3EQfoQEX/content/tmp_files/load_file.txt b/ddE0T4oBgHgl3EQfoQEX/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/ddE0T4oBgHgl3EQfoQEX/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf,len=123
+page_content='1  STUDY ON THE HYDROGEN BEHAVIOR IN Y AND Co DOPED BARIUM-ZIRCONATE  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Khalid Hossain*1,2, Kenichi Hashizume**1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Department of Advanced Energy Engineering Science, Interdisciplinary Graduate School of Engineering  Science, Kyushu University, Kasuga, Fukuoka 816-8580, Japan  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Atomic Energy Research Establishment, Bangladesh Atomic Energy Commission, Savar, Dhaka 1349,  Bangladesh  E-mail: *khalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='baec@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='com, **hashi@nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='kyushu-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='jp  ABSTRACT  Hydrogen release behavior from mixed-perovskite  oxide, yttrium and cobalt doped barium-zirconate,  BaZr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='955Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='03Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='015O3-α (BZYC) exposed to D2 or  D2O atmosphere was studied by thermal desorption  spectroscopy (TDS) detecting deuterium gas (D2)  and heavy water vapor (D2O) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' The  release of hydrogen appeared at around 1000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  Hydrogen solubility was higher in the BZYC  sample exposed to the D2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  KEY WORDS  Barium-zirconate, perovskite oxide, hydrogen  dissolution, hydrogen solubility, TDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' INTRODUCTION  Proton-conducting oxides are promising materials  for electro-chemical devices like fuel cell, hydrogen  pump, hydrogen sensor, and for nuclear fusion  reactor’s tritium purification and recovery system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  Therefore, the study of hydrogen behavior in such  oxide materials is very fundamental and important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  Yttrium (Y) and cobalt (Co) doped barium-  zirconate (BaZrO3) is a proton conducting oxide has  mixed-perovskite structure [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' In this study, the  dissolution of hydrogen in BZYC specimens were  done using D2 gas and D2O vapor separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Then  the hydrogen release and hydrogen solubility were  investigated by thermal desorption spectroscopy  (TDS) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' EXPERIMENTAL  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='1 Sample preparation  The disc shape specimen (diameter: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='7mm,  thickness: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='2mm) was prepared using conventional  powder sintering method at 1923 K for 20h, which  is described elsewhere [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' The SEM and EDX  mapping were performed before and after polishing  the sample to understand its surface properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' 1 it is clear that all chemical components  of the specimen were uniformly distributed  throughout the surface after polishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' The sintered  disc specimen was cut using a diamond saw into  parallelepiped pieces (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='7 x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='2 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='5 mm3) for D2  gas or D2O vapor exposure and TDS study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='2 D2 and D2O exposure  The cut sample was placed in a quartz glass tube and  evacuated to a low pressure (~10-6 Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' At a  temperature of 1273 K, vacuum annealing was  performed for 30 min to remove the volatile  substances from the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Then at 873 K,  deuterium gas (D2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='3 kPa) was exposed to the  sample for 1 hr using the apparatus shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  2(a) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' After the exposure, the quartz glass tube  was quenched with room temperature water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Finally,  the sample was taken out from the quartz glass tube  after recovering the D2 gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' The same sample was  used repeatedly for heavy water vapor (D2O,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='3kPa) exposure and the TDS study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' l: SEM micrograph and its EDS mapping for  the sintered disc sample surface after polishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' 2: (a) Schematic of D2 gas and D2O vapor exposure  apparatus, (b) schematic of temperature programmed  desorption (TDS) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  Ba  Zr  10p  BALiC  10um  ZrL  Co  0  5gm  OK(a)  Diaphragm  Gauge(≤1000Torr)  IG  vacuum  system  D,O  heater  specimen  (q)  QMS  IG  heater  vacuum  specimen  system2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='3 D2 and D2O measurement  The TDS apparatus used in this experiment to  measure the released hydrogen (D2, HD, D2O and  HDO) from the D2 gas and D2O vapor exposed  sample is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' 2(b) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' First, the baking of  the quartz glass tube without the sample was  performed with an electric furnace to release and  remove substances accumulated in the device at  1273 K for 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Then after cooling down the  glass tube to room temperature, the D2 gas exposed  sample was placed in the glass tube for TDS study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  Then after evacuating the glass tube to a low  pressure (~10-6 Pa), the temperature of the sample  was allowed to raise (at a heating rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='5 K/s) up  to 1273 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' After 30 min stay at 1273 K, the heater  power was disconnected automatically by using a  programable temperature controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' The gas  released from the samples was measured using a  quadrupole  mass  spectrometer  (Q-MAS)  (TSPTT200, INFICON) [3] during this sample  heating and cooling down cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' The above same  procedure was also applied for the D2O exposed  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  The ion currents obtained from Q-MAS data for the  D2 gas and D2O vapor exposed samples are shown in  Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' It is clear from the figure that in both cases of  exposures, release of hydrogen gas (D2 and HD) and  water vapor (D2O and HDO) was observed at  around 1000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' D2 gas was released with an extra  peak at 700 K in the case of D2 exposed sample and  at 850 K for D2O exposed sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Highest release  peak was detected as HD gas from D2 exposed  sample and there may appear a trend for the  hydrogen release as HD > HDO > D2 > D2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' On the  other hand, the highest peak was observed as HDO  vapor from D2O exposed sample and there may  appear a trend for the hydrogen release as HDO >  D2O > HD > D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  Table 1: D amount released from D2 and D2O exposed  samples  The total amounts of D-containing gas released  from the D2 or D2O exposed samples and its ratios  are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Total amount of deuterium (D)  released from D2 exposed sample was 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='0 x 10-4  (D/M), where released deuterium gas (~ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='6 x 10-4  D/M) and water vapor (~ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='4 x 10-4 D/M) were  contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' On the other hand, total amount  of D released from D2O exposed sample was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='8 x  10-4 (D/M), where three-fourth of the released D  was water vapor (~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='3 x 10-4 D/M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' It is also clear  that total amount of deuterium released from D2O  exposed sample is twice than the D2 exposed sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  Percentage of water vapor released from the D2O  exposed sample is higher than that of the D2  exposed sample, whereas an opposite result appears  for deuterium gas release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' The obtained result of the  experiment suggests that, under D2O exposed  condition and at 1000 K operating temperature, Y  and Co doped barium-zirconate may preferable for  electrochemical application due to its higher  hydrogen solubility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' CONCLUSIONS  In this study, hydrogen (deuterium) was dissolved  in BaZr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='955Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='03Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='015O3-α by the exposure of  deuterium gas or heavy water vapor, and the amount  of dissolved deuterium in the sample was measured  by TDS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' It was observed that deuterium  solubility is higher in the BZYC sample under the  D2O exposure condition and release of deuterium  was appeared at around 1000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We would like to thank our laboratory members for  their cordial support to operate the experimental  instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  REFERENCES  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
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+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Yamashita, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Otsuka, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Hashizume, Solid  State Ionics 275 (2015) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Maher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Syed, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Hughes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Gibson,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' Mass Spectrom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' 24 (2013)  1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  Deuterium (D) amount  D2  exposed  sample  D2O  exposed  sample  D from D2 and HD (D/M)  D from D2O and HDO (D/M)  Total D (D/M)  Fraction of deuterium gas (%)  Fraction of water vapor (%)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='6 x 10-4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='4 x 10-4  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='0 x 10-4  51  49  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='5 x 10-4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='3 x 10-3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='8 x 10-3  25  75  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content=' 3: TDS results for (a) D2 gas, and (b) D2O  vapor exposure samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='Exposed  Ir-01X9  Ion Current (A)  2  HD  HDO  4x10-11  DO  D  2x10-11  0  400  600  800  1000  1200  Temparature (K)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='5x10*  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='O Exposed  Ion Current (A)  2  HDO  1x10-10  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
+page_content='O  HD  5x10-11  D  0  400  600  800  1000  1200  Temparature(K)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE0T4oBgHgl3EQfoQEX/content/2301.02522v1.pdf'}
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+Brain Tissue Segmentation Across the Human
+Lifespan via Supervised Contrastive Learning
+Xiaoyang Chen, Jinjian Wu, Wenjiao Lyu, Yicheng Zou, Kim-Han Thung,
+Siyuan Liu, Ye Wu, Sahar Ahmad, and Pew-Thian Yap(�)
+Department of Radiology and Biomedical Research Imaging Center (BRIC),
+University of North Carolina, Chapel Hill, NC, USA
+� ptyap@med.unc.edu
+Abstract. Automatic segmentation of brain MR images into white mat-
+ter (WM), gray matter (GM), and cerebrospinal fluid (CSF) is critical for
+tissue volumetric analysis and cortical surface reconstruction. Due to dra-
+matic structural and appearance changes associated with developmental
+and aging processes, existing brain tissue segmentation methods are only
+viable for specific age groups. Consequently, methods developed for one
+age group may fail for another. In this paper, we make the first attempt
+to segment brain tissues across the entire human lifespan (0 − 100 years
+of age) using a unified deep learning model. To overcome the challenges
+related to structural variability underpinned by biological processes, in-
+tensity inhomogeneity, motion artifacts, scanner-induced differences, and
+acquisition protocols, we propose to use contrastive learning to improve
+the quality of feature representations in a latent space for effective lifes-
+pan tissue segmentation. We compared our approach with commonly
+used segmentation methods on a large-scale dataset of 2,464 MR images.
+Experimental results show that our model accurately segments brain tis-
+sues across the lifespan and outperforms existing methods.
+1
+Introduction
+Segmentation of gray matter (GM), white matter (WM), and cerebrospinal fluid
+(CSF) from magnetic resonance (MR) images is a prerequisite for delineating
+brain anatomical structures and for quantifying changes in tissue volumes and
+cortical geometry in relation to development and aging. Manual brain tissue seg-
+mentation carried out by experts can be time-consuming, laborious, and sensitive
+to intra- and inter-rater variability. Therefore, effective automated segmentation
+approaches for brain tissues are highly desirable.
+Although commonly employed segmentation methods, including SPM [1],
+FreeSurfer [3], and FSL [14] are effective in delineating brain tissues, they are
+tailored for adult MR images. These methods are less effective in segmenting
+infant brain MR images due to dynamic morphological and appearance changes
+caused by developmental processes during the first few years after birth. To
+cater to the unique characteristics of infant brains, the developing human con-
+nectome project (dHCP) processing pipeline [7] and Infant FreeSurfer [15] have
+arXiv:2301.01369v1  [eess.IV]  3 Jan 2023
+
+2
+Chen et al.
+been developed. These are atlas-based segmentation methods, which warp man-
+ually segmented atlas to the target image using non-rigid registration methods.
+However, accurate segmentation of fine structures remains difficult due to large
+anatomical variability, leading to sub-optimal results. Inaccurate segmentation
+can cause problems in the reconstruction of the cortical surfaces need to measur-
+ing cortical morphology such as cortical thickness, surface area, and curvature.
+Deep neural networks (DNNs) have been successfully used for brain tissue
+segmentation. For example, Roy et al. [10] first pretrained a model with auxil-
+iary labels for a large unlabeled dataset using FreeSurfer and then fine-tuned the
+model with limited manually labeled data. Like FreeSurfer, this method is effec-
+tive only for adult images. Nie et al. [9] trained multiple models, each of which
+is designed for a different imaging modality to segment images particularly in
+isointense phase (≈ 6 − 8 months of age). These methods are tailored for specific
+age groups and are not generalizable to other age groups.
+In this paper, we propose a novel brain tissues segmentation method that is
+applicable across the entire human lifespan (0 − 100 years of age). Designing a
+unified model that takes into account the developmental and aging related struc-
+tural and contrast changes in brain MR images (Fig. 1) is challenging. Moreover,
+common imaging issues such as noise, intensity inhomogeneity, partial volume
+effects, artifacts, and acquisition protocols further complicate model design. To
+address these problems, we propose to use supervised contrastive learning to
+learn discriminative feature representations that are robust to intensity differ-
+ences and imaging imperfections.
+The key contributions of our work are as follows: (i) We propose an unprece-
+dented lifespan segmentation method based on a single deep learning model;
+(ii) We use novel supervised contrastive learning strategy to regularize network
+training to learn discriminative features that are robust to structural and tissue
+contrast changes driven by biological mechanisms, inconsistent acquisition pro-
+tocols, and imaging imperfections; and (iii) We demonstrate on a large dataset
+of 2,464 brain MR images the effectiveness of our method.
+2
+Methods
+2.1
+Supervised Contrastive Learning
+Despite architecture differences, deep learning segmentation models share two
+main components: (i) feature extractor (FE) and (ii) classifier. For medical im-
+age segmentation, the FE usually has an encoder-decoder architecture to enable
+full-resolution segmentation. The FE is used to learn voxel-wise feature repre-
+sentations. A classifier, which is usually implemented with one or a few 1 × 1 × 1
+convolutional layers, is used to perform voxel-wise classification by taking the
+feature representations learned by the FE as the input. Previous approaches typ-
+ically jointly learn the FE and the classifier via a voxel-wise loss function such
+as categorical cross-entropy.
+However, the above joint learning scheme is unable to learn discriminative
+features [13]. As a result, the margin between feature clusters with different
+
+Lifespan Brain Tissue Segmentation
+3
+(a) 10 days
+(b) 5 months
+(c) 9 months
+(d) 1 year
+(e) 32 years
+(f) 100 years
+Fig. 1: Brain morphology and tissue contrast vary across the human lifespan.
+labels is narrow, leading to error-prone classifiers and models that are sensitive
+to noise, artifacts, and intensity and contrast changes. Inspired by the success
+of Momentum Contrast (MoCo) [5], which is a self-supervised learning approach
+for instance discrimination, we adopt a similar contrastive learning strategy to
+learn discriminative features. Unlike MoCo, where contrastive learning is used
+for unsupervised pre-training, we perform contrastive learning in a supervised
+manner.
+The key idea of contrastive learning is to pull the feature representations of
+the same class close together and push those from different classes far apart in
+a latent space. To this end, we use a memory bank to store a few typical feature
+representations as proxies for each semantic category. It should be noted that
+using proxies is more feasible than making pair-wise comparisons because around
+a million of voxels are involved at each training step in our case; otherwise, the
+problem becomes intractable. To implement contrastive learning, cosine simi-
+larity between a feature representation and the proxies in the memory bank is
+computed. Then, contrastive loss (i.e., InfoNCE) is calculated voxel-wise. We
+use the label of a given voxel as guidance to select positive and negative proxies
+from the memory bank and therefore we call our training strategy supervised
+contrastive learning.
+We use a memory bank with the shape of C×M×D, where C is the number
+of semantic categories, M is number of proxies for each category, and D is the
+dimensionality of proxies. Note that D is identical to the number of channels of
+the last layer of the FE. The memory bank is trainable, adaptively learned and
+optimized via back-propagation during training. A proxy in the memory bank
+
+4
+Chen et al.
+acts as a positive proxy if a query (i.e., the feature representation of a voxel)
+shares the same label with it, otherwise it will be used as a negative proxy in
+calculating the contrastive loss.
+The contrastive loss for an arbitrary query qi is defined as:
+Li
+ctr = − 1
+M ∑
+{k+
+i}
+log
+sim(qi,k+
+i )/τ
+sim(qi,k+
+i )/τ + ∑{k−
+i} sim(qi,k−
+i )/τ ,
+(1)
+where sim(u,v) = u⊺v
+∣u∣∣v∣ is cosine similarity, τ is the temperature hyper-parameter,
+qi is the feature representation of a query voxel with index i, {k+
+i } and {k−
+i } are
+the set of positive and negative proxies for qi, respectively. The size of sets {k+
+i }
+and {k−
+i } is equal to M and M(C − 1), respectively.
+2.2
+Joint Feature Regularization and Classification
+In addition to Li
+ctr for feature regularization, categorical cross-entropy loss Li
+ce
+is used for joint training. The overall loss function Li
+total for query voxel i is a
+weighted sum of Li
+ce and Li
+ctr, balanced by λctr, and is formally defined as
+Li
+total = Li
+ce + λctrLi
+ctr,
+(2)
+where
+Li
+ce = −
+C
+∑
+c=1
+yic log(ˆpic).
+(3)
+In (3), ˆpic is the c-th element of the output probability vector for query voxel i.
+yic, taking either 0 or 1, is the ground truth value for ˆpic.
+2.3
+Learning from Imbalanced Data
+The class imbalance issue is common in brain tissue segmentation. At the image
+level, volumes of GM and WM are usually 2 − 3 times larger than that of CSF
+and all the three tissues are considerably smaller than the background region,
+i.e., the region that is not GM, WM or CSF. At the mini-batch level, the class
+imbalance issue can be even more severe, since an image patch is expected to
+contain only voxels from as few as one class.
+We solve the class imbalance issue using two different approaches. The first
+approach is to use stratified sampling. Instead of randomly taking samples from
+the whole image region, we first derive the minimum bounding box of each
+tissue type using the ground truth label map and then randomly take an equal
+number of samples (i.e., image patches) from each tissue-specific bounding box.
+For simplicity, we allow overlapped bounding boxes. The second approach is to
+select an equal number of samples from each class that is present in a mini-batch
+for loss computation.
+Due to randomness in sampling, the label set L for a given mini-batch is only
+a subset of {0,1,...,C−1} (positive integers represent foreground class labels and
+
+Lifespan Brain Tissue Segmentation
+5
+0 denotes the background). We first count the number of voxels for each category
+in L and obtain the minimum Nmin = min{Nc ∶ c ∈ L}, where Nc is the number of
+samples of class c in the mini-batch. We then randomly select Nmin samples from
+each category present in the mini-batch and compute the loss for each sample.
+The final loss is the average of the losses for the selected samples.
+2.4
+Implementation Details
+We implemented the proposed method in Keras with TensorFlow as backend. We
+used a V-Net variant as the feature extractor. In contrast to the original V-Net
+[8], we (i) removed the original head for classification; (ii) adjusted the size of the
+convolutional kernel from 5 to 3 to lower the risk of overfitting; and (iii) used
+instance normalization [11] after each convolutional layer as it is suitable for
+small batch sizes. We used a simple multi-layer perceptron (MLP), consisting of
+three 1 × 1 × 1 convolutional layers, as the classifier. In the MLP, the first two
+convolutional layers are followed by a rectified linear unit (ReLU) and the last
+one is followed by a softmax layer. We used the Adam optimizer with an initial
+learning rate of 0.0001 to update the parameters. The convolutional kernels and
+the memory bank were randomly initialized. The bias terms were initialized to
+0. The patch size was set to 80 × 80 × 80 and the batch size to 2.
+3
+Experimental Results
+3.1
+Dataset
+The dataset consists of T1- and T2-weighted image pairs from five Lifespan Hu-
+man Connectome Projects [12,4,15]: (i) Developing Human Connectome Project
+(dHCP; gestational age: 37 – 44 weeks) [7]; (ii) Baby Connectome Project (BCP,
+ages: 0 – 5 years) [6]; (iii) HCP Development (HCP-D, ages: 6 – 22 years);
+(iv) HCP Young Adult (HCP-YA, ages: 22 – 37 years); and (v) HCP Aging
+(HCP-A, ages: 36 – 100 years). Of the total 2,462 image pairs used, 682 were used
+for training, 113 for validation and the rest for testing. Among the images used
+for training, we manually annotated 62 images (high-quality set) and the tissue
+segmentation maps for the remaining 620 images (uncorrected set) were ob-
+tained using the standard HCP minimal preprocessed pipeline. The high-quality
+set includes 31 images from BCP, 20 from HCP-D, 7 from HCP-YA, and 4 from
+HCP-D. Since the number of images in the high-quality set is much smaller than
+in the uncorrected set, we take half of the training samples from the high-quality
+set and the other half from the uncorrected set to make sure the model is not bi-
+ased to uncorrected labels. For testing, we used dice similarity coefficient (DSC)
+and average surface distance (ASD) as metrics for performance evaluation.
+3.2
+Experimental Setup
+We compared the proposed method with the vanilla V-Net [8] and the V-Net
+variant (with the same MLP as the classification head). The V-Net variant and
+
+6
+Chen et al.
+Table 1: Comparison of DSC and ASD on the testing set for various methods.
+Method
+DSC (%)
+ASD (mm)
+GM
+WM
+CSF
+GM
+WM
+CSF
+V-Net [8]
+81.4±10.8
+85.5 ± 7.6
+77.3 ± 6.7
+0.26 ± 0.11
+0.45 ± 0.23
+0.28 ± 0.13
+V-Net variant
+93.2 ± 1.6
+95.4 ± 1.2
+80.1 ± 6.2
+0.08 ± 0.02
+0.07 ± 0.02
+0.22 ± 0.12
+Proposed
+94.2 ± 1.1
+95.7 ± 1.0
+88.5 ± 3.8
+0.06 ± 0.01 0.07 ± 0.02 0.15 ± 0.07
+Table 2: Comparison of DSC and ASD on the testing set with different inputs.
+Modality
+DSC (%)
+ASD (mm)
+GM
+WM
+CSF
+GM
+WM
+CSF
+T1
+92.7 ± 3.0
+95.1 ± 2.1
+87.2 ± 6.5
+0.09 ± 0.05
+0.09 ± 0.06
+0.17 ± 0.09
+T2
+92.3 ± 1.9
+94.7 ± 1.8
+88.1 ± 4.5
+0.09 ± 0.02
+0.09 ± 0.03
+0.15 ± 0.06
+the proposed method differ in that the V-Net variant was trained with categorical
+cross-entropy (CE) loss only.
+3.3
+Results
+Main results The experimental results are shown in Table 1. The vanilla V-
+Net only achieves DSC of 81.4 ± 10.8, 85.5 ± 7.6 and 77.3 ± 6.7 on GM, WM
+and CSF, respectively. Compared with vanilla V-Net, the V-Net variant signif-
+icantly improves the performance by 11.8%, 9.9% and 2.8% on GM, WM and
+CSF, respectively (p < 0.001). With the proposed supervised contrastive learn-
+ing strategy applied, our method further improves the DSC by 1.0%, 0.3% and
+8.4% on GM, WM and CSF, respectively, over the V-Net variant. Notably, the
+improvement on CSF is much larger than that of GM and WM because GM and
+WM are majority classes and can be segmented satisfactorily. However, accurate
+CSF segmentation is the key to ensure topological correctness in reconstructing
+the pial surface.
+Single modality We studied the performance of the proposed method with only
+either T1- or T2-weighted image as the input. The experimental results are shown
+in Table 2. Compared with the performance with paired T1- and T2-weighted
+images as input, the segmentation performance decreases marginally (ranging
+from −1.9% to −0.4% DSC). T1- and T2-weighted images provide complemen-
+tary information that can help improve brain tissue segmentation. However, the
+marginal decrease indicates that our models learn effectively from limited infor-
+mation.
+Growth modeling of tissue volumes We modeled the growth trajectories of pre-
+dicted tissue volumes using generalized additive mixture model (GAMM). We
+fitted GAMM to the volume of each tissue type across the lifespan with cubic
+regression spline as smooth nonlinear function of age and subject-specific ran-
+dom intercept. The growth trajectories, shown in Fig. 2, indicate that the GM
+
+Lifespan Brain Tissue Segmentation
+7
+and WM volumes exhibit an increase-then-decrease pattern; whereas the CSF
+volume increases throughout the lifespan.
+Surface reconstruction Fig. 3 shows example white/pial surfaces and tissue seg-
+mentation maps generated using our method for various time points across the
+human lifespan. In general, the results indicate good segmentation details partic-
+ularly at the cortex as evidenced by the detailed surface convolution with clear
+gryi and sulci.
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+2
+1
+5 8 16 3560 100
+Age (years; log scale)
+50
+250
+450
+650
+850
+GM Volume
+(
+)
+cm3
+R2 = 0.93
+�
+�
+�
+� ��
+��
+��
+��
+���
+��� ���
+����
+�����
+������
+������
+������
+������
+������
+������
+������
+���
+��
+Birth
+2
+1
+5 8 16 3560 100
+Age (years; log scale)
+50
+250
+450
+650
+WM Volume
+(
+)
+cm3
+R2 = 0.84
+�
+�
+�
+� ��
+��
+��
+��
+���
+��� ���
+����
+�����
+�����
+�����
+�����
+�����
+�����
+�����
+���
+���
+Birth
+2
+1
+5 8 16 3560 100
+Age (years; log scale)
+0
+200
+400
+600
+CSF Volume
+(
+)
+cm3
+R2 = 0.89
+Fig. 2: Growth trajectories of tissue volumes across the human lifespan.
+1 Year
+10 Years
+75 Years
+6 Months
+White Surface
+Pial Surface
+Tissue Overlay
+Fig. 3: Reconstructed white and pial surfaces and tissue segmentation maps at
+4 different time points.
+
+8
+Chen et al.
+3.4
+Ablation Study
+Impact of using balanced losses We studied the impact of selecting equal number
+of voxels of different categories to compute both the CE and contrastive losses.
+As shown in Table 3, balancing the contributions from different class are critical
+to better segmentation of minority classes, e.g., the CSF.
+Table 3: Impact of balanced losses on DSC.
+GM
+WM
+CSF
+Unbalanced
+93.6 ± 1.5
+95.1 ± 1.3
+83.0 ± 6.2
+Balanced
+94.2 ± 1.1
+95.7 ± 1.0
+88.5 ± 3.8
+Impact of λctr We fixed M to 10 and τ to 0.1. We tried 0.5 and 1.0 for λctr. As
+shown in Table 4, the performance is marginally changed using different λctr.
+Table 4: Impact of λctr on DSC.
+0.5
+1.0
+GM
+WM
+CSF
+GM
+WM
+CSF
+DSC
+94.0 ± 5.1
+95.5 ± 1.1
+88.6 ± 3.6
+94.2 ± 1.1
+95.7 ± 1.0
+88.5 ± 3.8
+Impact of M and τ We fixed λctr to 1 as it is found that the performance is
+insensitive to the choice of λctr. As shown in Table 5, larger M and smaller τ
+seem generate better results. This is consistent with [2], where it is found that
+a larger number of negative samples is beneficial for contrastive learning.
+Table 5: Impact of M and τ on DSC.
+τ / M
+5
+10
+0.1
+93.9 ± 5.2
+95.5 ± 1.2
+88.4 ± 4.0
+94.2 ± 1.1
+95.7 ± 1.0
+88.5 ± 3.8
+0.2
+93.6 ± 5.6
+94.8 ± 1.6
+88.1 ± 4.2
+93.9 ± 5.4
+95.3 ± 1.3
+88.2 ± 3.8
+4
+Conclusion
+We have presented a novel supervised contrastive learning method for brain
+tissue segmentation across the human lifespan. Experimental results on a large
+dataset show that our method can effectively segment the tissues with paired
+T1- and T2-weighted images or with just T1- or T2-weighted images.
+
+Lifespan Brain Tissue Segmentation
+9
+Acknowledgement
+This work was supported in part by the United States National Institutes of
+Health (NIH) under grants EB008374 and MH125479. Data were provided in
+part by the developing Human Connectome Project, KCL-Imperial-Oxford Con-
+sortium funded by the European Research Council under the European Union
+Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no.
+319456. Data were provided in part by the Human Connectome Project, WU-
+Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil;
+1U54MH091657) funded by the 16 NIH Institutes and Centers that support the
+NIH Blueprint for Neuroscience Research and by the McDonnell Center for Sys-
+tems Neuroscience at Washington University.
+References
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+(2005)
+2. Chen, T., Kornblith, S., Norouzi, M., Hinton, G.: A simple framework for con-
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+learning. pp. 1597–1607. PMLR (2020)
+3. Fischl, B., Salat, D.H., Busa, E., Albert, M., Dieterich, M., Haselgrove, C., Van
+Der Kouwe, A., Killiany, R., Kennedy, D., Klaveness, S., et al.: Whole brain seg-
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+Neuron 33(3), 341–355 (2002)
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+Computer Vision and Pattern Recognition. pp. 9729–9738 (2020)
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+modality isointense infant brain image segmentation. In: IEEE International Sym-
+posium on Biomedical Imaging (ISBI). pp. 1342–1345 (2016)
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+12. Van Essen, D., Ugurbil, K., Auerbach, E., Barch, D., Behrens, T., Bucholz, R.,
+Chang, A., Chen, L., Corbetta, M., Curtiss, S., Della Penna, S., Feinberg, D.,
+Glasser, M., Harel, N., Heath, A., Larson-Prior, L., Marcus, D., Michalareas, G.,
+Moeller, S., Oostenveld, R., Petersen, S., Prior, F., Schlaggar, B., Smith, S., Sny-
+der, A., Xu, J., Yacoub, E.: The Human Connectome Project: A data acquisition
+perspective. NeuroImage 62(4), 2222 – 2231 (2012)
+13. Wen, Y., Zhang, K., Li, Z., Qiao, Y.: A discriminative feature learning approach for
+deep face recognition. In: European Conference on Computer Vision. pp. 499–515.
+Springer (2016)
+14. Zhang, Y., Brady, M., Smith, S.: Segmentation of brain mr images through a
+hidden markov random field model and the expectation-maximization algorithm.
+IEEE Transactions on Medical Imaging 20(1), 45–57 (2001)
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+data of infants 0–2 years. NeuroImage 218, 116946 (2020)
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf,len=475
+page_content='Brain Tissue Segmentation Across the Human  Lifespan via Supervised Contrastive Learning  Xiaoyang Chen, Jinjian Wu, Wenjiao Lyu, Yicheng Zou, Kim-Han Thung,  Siyuan Liu, Ye Wu, Sahar Ahmad, and Pew-Thian Yap(�)  Department of Radiology and Biomedical Research Imaging Center (BRIC),  University of North Carolina, Chapel Hill, NC, USA  � ptyap@med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='edu  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Automatic segmentation of brain MR images into white mat-  ter (WM), gray matter (GM), and cerebrospinal fluid (CSF) is critical for  tissue volumetric analysis and cortical surface reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Due to dra-  matic structural and appearance changes associated with developmental  and aging processes, existing brain tissue segmentation methods are only  viable for specific age groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Consequently, methods developed for one  age group may fail for another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' In this paper, we make the first attempt  to segment brain tissues across the entire human lifespan (0 − 100 years  of age) using a unified deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' To overcome the challenges  related to structural variability underpinned by biological processes, in-  tensity inhomogeneity, motion artifacts, scanner-induced differences, and  acquisition protocols, we propose to use contrastive learning to improve  the quality of feature representations in a latent space for effective lifes-  pan tissue segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' We compared our approach with commonly  used segmentation methods on a large-scale dataset of 2,464 MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Experimental results show that our model accurately segments brain tis-  sues across the lifespan and outperforms existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  1  Introduction  Segmentation of gray matter (GM), white matter (WM), and cerebrospinal fluid  (CSF) from magnetic resonance (MR) images is a prerequisite for delineating  brain anatomical structures and for quantifying changes in tissue volumes and  cortical geometry in relation to development and aging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Manual brain tissue seg-  mentation carried out by experts can be time-consuming, laborious, and sensitive  to intra- and inter-rater variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Therefore, effective automated segmentation  approaches for brain tissues are highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Although commonly employed segmentation methods, including SPM [1],  FreeSurfer [3], and FSL [14] are effective in delineating brain tissues, they are  tailored for adult MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' These methods are less effective in segmenting  infant brain MR images due to dynamic morphological and appearance changes  caused by developmental processes during the first few years after birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' To  cater to the unique characteristics of infant brains, the developing human con-  nectome project (dHCP) processing pipeline [7] and Infant FreeSurfer [15] have  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='01369v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='IV]  3 Jan 2023  2  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' These are atlas-based segmentation methods, which warp man-  ually segmented atlas to the target image using non-rigid registration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  However, accurate segmentation of fine structures remains difficult due to large  anatomical variability, leading to sub-optimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Inaccurate segmentation  can cause problems in the reconstruction of the cortical surfaces need to measur-  ing cortical morphology such as cortical thickness, surface area, and curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Deep neural networks (DNNs) have been successfully used for brain tissue  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' For example, Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' [10] first pretrained a model with auxil-  iary labels for a large unlabeled dataset using FreeSurfer and then fine-tuned the  model with limited manually labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Like FreeSurfer, this method is effec-  tive only for adult images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' [9] trained multiple models, each of which  is designed for a different imaging modality to segment images particularly in  isointense phase (≈ 6 − 8 months of age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' These methods are tailored for specific  age groups and are not generalizable to other age groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  In this paper, we propose a novel brain tissues segmentation method that is  applicable across the entire human lifespan (0 − 100 years of age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Designing a  unified model that takes into account the developmental and aging related struc-  tural and contrast changes in brain MR images (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' 1) is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Moreover,  common imaging issues such as noise, intensity inhomogeneity, partial volume  effects, artifacts, and acquisition protocols further complicate model design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' To  address these problems, we propose to use supervised contrastive learning to  learn discriminative feature representations that are robust to intensity differ-  ences and imaging imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  The key contributions of our work are as follows: (i) We propose an unprece-  dented lifespan segmentation method based on a single deep learning model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  (ii) We use novel supervised contrastive learning strategy to regularize network  training to learn discriminative features that are robust to structural and tissue  contrast changes driven by biological mechanisms, inconsistent acquisition pro-  tocols, and imaging imperfections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' and (iii) We demonstrate on a large dataset  of 2,464 brain MR images the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  2  Methods  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1  Supervised Contrastive Learning  Despite architecture differences, deep learning segmentation models share two  main components: (i) feature extractor (FE) and (ii) classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' For medical im-  age segmentation, the FE usually has an encoder-decoder architecture to enable  full-resolution segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The FE is used to learn voxel-wise feature repre-  sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' A classifier, which is usually implemented with one or a few 1 × 1 × 1  convolutional layers, is used to perform voxel-wise classification by taking the  feature representations learned by the FE as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Previous approaches typ-  ically jointly learn the FE and the classifier via a voxel-wise loss function such  as categorical cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  However, the above joint learning scheme is unable to learn discriminative  features [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' As a result, the margin between feature clusters with different  Lifespan Brain Tissue Segmentation  3  (a) 10 days  (b) 5 months  (c) 9 months  (d) 1 year  (e) 32 years  (f) 100 years  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' 1: Brain morphology and tissue contrast vary across the human lifespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  labels is narrow, leading to error-prone classifiers and models that are sensitive  to noise, artifacts, and intensity and contrast changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Inspired by the success  of Momentum Contrast (MoCo) [5], which is a self-supervised learning approach  for instance discrimination, we adopt a similar contrastive learning strategy to  learn discriminative features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Unlike MoCo, where contrastive learning is used  for unsupervised pre-training, we perform contrastive learning in a supervised  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  The key idea of contrastive learning is to pull the feature representations of  the same class close together and push those from different classes far apart in  a latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' To this end, we use a memory bank to store a few typical feature  representations as proxies for each semantic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' It should be noted that  using proxies is more feasible than making pair-wise comparisons because around  a million of voxels are involved at each training step in our case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' otherwise, the  problem becomes intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' To implement contrastive learning, cosine simi-  larity between a feature representation and the proxies in the memory bank is  computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Then, contrastive loss (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', InfoNCE) is calculated voxel-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' We  use the label of a given voxel as guidance to select positive and negative proxies  from the memory bank and therefore we call our training strategy supervised  contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  We use a memory bank with the shape of C×M×D, where C is the number  of semantic categories, M is number of proxies for each category, and D is the  dimensionality of proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Note that D is identical to the number of channels of  the last layer of the FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The memory bank is trainable, adaptively learned and  optimized via back-propagation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' A proxy in the memory bank  4  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  acts as a positive proxy if a query (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', the feature representation of a voxel)  shares the same label with it, otherwise it will be used as a negative proxy in  calculating the contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  The contrastive loss for an arbitrary query qi is defined as:  Li  ctr = − 1  M ∑  {k+  i}  log  sim(qi,k+  i )/τ  sim(qi,k+  i )/τ + ∑{k−  i} sim(qi,k−  i )/τ ,  (1)  where sim(u,v) = u⊺v  ∣u∣∣v∣ is cosine similarity, τ is the temperature hyper-parameter,  qi is the feature representation of a query voxel with index i, {k+  i } and {k−  i } are  the set of positive and negative proxies for qi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The size of sets {k+  i }  and {k−  i } is equal to M and M(C − 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2  Joint Feature Regularization and Classification  In addition to Li  ctr for feature regularization, categorical cross-entropy loss Li  ce  is used for joint training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The overall loss function Li  total for query voxel i is a  weighted sum of Li  ce and Li  ctr, balanced by λctr, and is formally defined as  Li  total = Li  ce + λctrLi  ctr,  (2)  where  Li  ce = −  C  ∑  c=1  yic log(ˆpic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  (3)  In (3), ˆpic is the c-th element of the output probability vector for query voxel i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  yic, taking either 0 or 1, is the ground truth value for ˆpic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='3  Learning from Imbalanced Data  The class imbalance issue is common in brain tissue segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' At the image  level, volumes of GM and WM are usually 2 − 3 times larger than that of CSF  and all the three tissues are considerably smaller than the background region,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', the region that is not GM, WM or CSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' At the mini-batch level, the class  imbalance issue can be even more severe, since an image patch is expected to  contain only voxels from as few as one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  We solve the class imbalance issue using two different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The first  approach is to use stratified sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Instead of randomly taking samples from  the whole image region, we first derive the minimum bounding box of each  tissue type using the ground truth label map and then randomly take an equal  number of samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', image patches) from each tissue-specific bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  For simplicity, we allow overlapped bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The second approach is to  select an equal number of samples from each class that is present in a mini-batch  for loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Due to randomness in sampling, the label set L for a given mini-batch is only  a subset of {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=',C−1} (positive integers represent foreground class labels and  Lifespan Brain Tissue Segmentation  5  0 denotes the background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' We first count the number of voxels for each category  in L and obtain the minimum Nmin = min{Nc ∶ c ∈ L}, where Nc is the number of  samples of class c in the mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' We then randomly select Nmin samples from  each category present in the mini-batch and compute the loss for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  The final loss is the average of the losses for the selected samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='4  Implementation Details  We implemented the proposed method in Keras with TensorFlow as backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' We  used a V-Net variant as the feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' In contrast to the original V-Net  [8], we (i) removed the original head for classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' (ii) adjusted the size of the  convolutional kernel from 5 to 3 to lower the risk of overfitting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' and (iii) used  instance normalization [11] after each convolutional layer as it is suitable for  small batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' We used a simple multi-layer perceptron (MLP), consisting of  three 1 × 1 × 1 convolutional layers, as the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' In the MLP, the first two  convolutional layers are followed by a rectified linear unit (ReLU) and the last  one is followed by a softmax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' We used the Adam optimizer with an initial  learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0001 to update the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The convolutional kernels and  the memory bank were randomly initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The bias terms were initialized to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The patch size was set to 80 × 80 × 80 and the batch size to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  3  Experimental Results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1  Dataset  The dataset consists of T1- and T2-weighted image pairs from five Lifespan Hu-  man Connectome Projects [12,4,15]: (i) Developing Human Connectome Project  (dHCP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' gestational age: 37 – 44 weeks) [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' (ii) Baby Connectome Project (BCP,  ages: 0 – 5 years) [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' (iii) HCP Development (HCP-D, ages: 6 – 22 years);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  (iv) HCP Young Adult (HCP-YA, ages: 22 – 37 years);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' and (v) HCP Aging  (HCP-A, ages: 36 – 100 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Of the total 2,462 image pairs used, 682 were used  for training, 113 for validation and the rest for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Among the images used  for training, we manually annotated 62 images (high-quality set) and the tissue  segmentation maps for the remaining 620 images (uncorrected set) were ob-  tained using the standard HCP minimal preprocessed pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The high-quality  set includes 31 images from BCP, 20 from HCP-D, 7 from HCP-YA, and 4 from  HCP-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Since the number of images in the high-quality set is much smaller than  in the uncorrected set, we take half of the training samples from the high-quality  set and the other half from the uncorrected set to make sure the model is not bi-  ased to uncorrected labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' For testing, we used dice similarity coefficient (DSC)  and average surface distance (ASD) as metrics for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2  Experimental Setup  We compared the proposed method with the vanilla V-Net [8] and the V-Net  variant (with the same MLP as the classification head).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The V-Net variant and  6  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Table 1: Comparison of DSC and ASD on the testing set for various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Method  DSC (%)  ASD (mm)  GM  WM  CSF  GM  WM  CSF  V-Net [8]  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='4±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='6  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='3 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='13  V-Net variant  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='6  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='12  Proposed  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='07  Table 2: Comparison of DSC and ASD on the testing set with different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Modality  DSC (%)  ASD (mm)  GM  WM  CSF  GM  WM  CSF  T1  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='09  T2  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='9  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='06  the proposed method differ in that the V-Net variant was trained with categorical  cross-entropy (CE) loss only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='3  Results  Main results The experimental results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The vanilla V-  Net only achieves DSC of 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='4 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='6 and 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='3 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='7 on GM, WM  and CSF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Compared with vanilla V-Net, the V-Net variant signif-  icantly improves the performance by 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8%, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='9% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8% on GM, WM and  CSF, respectively (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' With the proposed supervised contrastive learn-  ing strategy applied, our method further improves the DSC by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='3% and  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='4% on GM, WM and CSF, respectively, over the V-Net variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Notably, the  improvement on CSF is much larger than that of GM and WM because GM and  WM are majority classes and can be segmented satisfactorily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' However, accurate  CSF segmentation is the key to ensure topological correctness in reconstructing  the pial surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Single modality We studied the performance of the proposed method with only  either T1- or T2-weighted image as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The experimental results are shown  in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Compared with the performance with paired T1- and T2-weighted  images as input, the segmentation performance decreases marginally (ranging  from −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='9% to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='4% DSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' T1- and T2-weighted images provide complemen-  tary information that can help improve brain tissue segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' However, the  marginal decrease indicates that our models learn effectively from limited infor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Growth modeling of tissue volumes We modeled the growth trajectories of pre-  dicted tissue volumes using generalized additive mixture model (GAMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' We  fitted GAMM to the volume of each tissue type across the lifespan with cubic  regression spline as smooth nonlinear function of age and subject-specific ran-  dom intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' The growth trajectories, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' 2, indicate that the GM  Lifespan Brain Tissue Segmentation  7  and WM volumes exhibit an increase-then-decrease pattern;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' whereas the CSF  volume increases throughout the lifespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Surface reconstruction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' 3 shows example white/pial surfaces and tissue seg-  mentation maps generated using our method for various time points across the  human lifespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' In general, the results indicate good segmentation details partic-  ularly at the cortex as evidenced by the detailed surface convolution with clear  gryi and sulci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  �  �  �  � ��  ��  ��  ��  ���  ��� ���  ����  �����  ������  ������  ������  ������  ������  ������  ������  ������  ������  ���  ��  Birth  2  1  5 8 16 3560 100  Age (years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' log scale)  50  250  450  650  850  GM Volume  (  )  cm3  R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='93  �  �  �  � ��  ��  ��  ��  ���  ��� ���  ����  �����  ������  ������  ������  ������  ������  ������  ������  ���  ��  Birth  2  1  5 8 16 3560 100  Age (years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' log scale)  50  250  450  650  WM Volume  (  )  cm3  R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='84  �  �  �  � ��  ��  ��  ��  ���  ��� ���  ����  �����  �����  �����  �����  �����  �����  �����  ���  ���  Birth  2  1  5 8 16 3560 100  Age (years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' log scale)  0  200  400  600  CSF Volume  (  )  cm3  R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='89  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' 2: Growth trajectories of tissue volumes across the human lifespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  1 Year  10 Years  75 Years  6 Months  White Surface  Pial Surface  Tissue Overlay  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' 3: Reconstructed white and pial surfaces and tissue segmentation maps at  4 different time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  8  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='4  Ablation Study  Impact of using balanced losses We studied the impact of selecting equal number  of voxels of different categories to compute both the CE and contrastive losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  As shown in Table 3, balancing the contributions from different class are critical  to better segmentation of minority classes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', the CSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Table 3: Impact of balanced losses on DSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  GM  WM  CSF  Unbalanced  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='3  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2  Balanced  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8  Impact of λctr We fixed M to 10 and τ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' We tried 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0 for λctr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' As  shown in Table 4, the performance is marginally changed using different λctr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Table 4: Impact of λctr on DSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0  GM  WM  CSF  GM  WM  CSF  DSC  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='6  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8  Impact of M and τ We fixed λctr to 1 as it is found that the performance is  insensitive to the choice of λctr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' As shown in Table 5, larger M and smaller τ  seem generate better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' This is consistent with [2], where it is found that  a larger number of negative samples is beneficial for contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Table 5: Impact of M and τ on DSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  τ / M  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='9 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='4 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='0  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='6 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='6  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='6  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='1 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='9 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='4  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='3  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='8  4  Conclusion  We have presented a novel supervised contrastive learning method for brain  tissue segmentation across the human lifespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Experimental results on a large  dataset show that our method can effectively segment the tissues with paired  T1- and T2-weighted images or with just T1- or T2-weighted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  Lifespan Brain Tissue Segmentation  9  Acknowledgement  This work was supported in part by the United States National Institutes of  Health (NIH) under grants EB008374 and MH125479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Data were provided in  part by the developing Human Connectome Project, KCL-Imperial-Oxford Con-  sortium funded by the European Research Council under the European Union  Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  319456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Data were provided in part by the Human Connectome Project, WU-  Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  1U54MH091657) funded by the 16 NIH Institutes and Centers that support the  NIH Blueprint for Neuroscience Research and by the McDonnell Center for Sys-  tems Neuroscience at Washington University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content=' Fischl, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content=', Killiany, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Kennedy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content='  Neuron 33(3), 341–355 (2002)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Glasser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Coalson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content=', Ander-  sson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Jbabdi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Webster, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Polimeni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' : The minimal  preprocessing pipelines for the Human Connectome Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' NeuroImage 80, 105–  124 (2013)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Fan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Girshick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=': Momentum contrast for unsupervised  visual representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.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/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' 9729–9738 (2020)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Howell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Styner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Gao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Yap, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Baluyot, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Yacoub,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Potts, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Salzwedel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' : The UNC/UMN Baby Connectome  Project (BCP): An overview of the study design and protocol development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Neu-  roImage 185, 891–905 (2019)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Makropoulos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Robinson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Schuh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Wright, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Fitzgibbon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Bozek,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Counsell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Steinweg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Vecchiato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Passerat-Palmbach, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' : The  developing human connectome project: A minimal processing pipeline for neonatal  cortical surface reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' NeuroImage 173, 88–112 (2018)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' Milletari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Navab, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=', Ahmadi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=': V-net: Fully convolutional neural networks  for volumetric medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content=' IEEE (2016)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content=' arXiv preprint arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content=' Van Essen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content=' Wen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content=': A discriminative feature learning approach for  deep face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
+page_content=' In: European Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+page_content='  Springer (2016)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfafxZ/content/2301.01369v1.pdf'}
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+arXiv:2301.08752v1  [eess.IV]  20 Jan 2023
+OPTIMIZED LEARNED ENTROPY CODING PARAMETERS FOR PRACTICAL
+NEURAL-BASED IMAGE AND VIDEO COMPRESSION
+Amir Said
+Reza Pourreza
+Hoang Le
+Qualcomm AI Research,∗ San Diego, CA, USA
+ABSTRACT
+Neural-based image and video codecs are significantly more
+power-efficient when weights and activations are quantized
+to low-precision integers. While there are general-purpose
+techniques for reducing quantization effects, large losses can
+occur when specific entropy coding properties are not con-
+sidered. This work analyzes how entropy coding is affected
+by parameter quantizations, and provides a method to min-
+imize losses.
+It is shown that, by using a certain type of
+coding parameters to be learned, uniform quantization be-
+comes practically optimal, also simplifying the minimization
+of code memory requirements. The mathematical properties
+of the new representation are presented, and its effectiveness
+is demonstrated by coding experiments, showing that good
+results can be obtained with precision as low as 4 bits per
+network output, and practically no loss with 8 bits.
+Index Terms— Learned image and video compression,
+neural network quantization, entropy coding.
+1. INTRODUCTION
+After years of research showing the advantages of neural-
+based techniques for video coding [1, 2], and recent demon-
+strations of real-time decoding in mobile devices [3, 4], there
+is growing interest in their standardization and commercial
+deployment [5], and a main practical implementation chal-
+lenge is the minimization of decoding power requirements.
+Implementing neural network (NN) computations with
+low-precision integers significantly reduces power needs,
+but requires minimizing the effects of quantizing networks
+weights and activations [6, 7, 8]. For neural image and video
+codecs, efficacy is strongly defined by the entropy coding
+stage and related networks, and can be severely degraded.
+Analyzing quantization effects in NNs can be difficult
+because they implement non-linear processes, and learned
+variables have unknown properties.
+Techniques based on
+re-training under quantization constraints help to reduce the
+losses, but have limited efficacy.
+In this work it is shown that NN outputs used for entropy
+coding have well-defined meaning and properties, which can
+be used for predicting quantization effects.
+Furthermore,
+those properties can be used for modifying the loss function
+∗Qualcomm AI Research is an initiative of Qualcomm Technologies, Inc.
+Fig. 1. Quantized values from normal distribution with std. dev. σ,
+commonly used in neural-based codecs.
+used while learning, and it is demonstrated that networks can
+be trained to be minimally sensitive to quantization.
+Further analysis shows that this optimized representa-
+tion is remarkably independent of the quantization, and thus
+networks do not need to be retrained even after significant
+changes to entropy coding design choices. Thus, in those
+networks the uniform quantization can also be extended be-
+yond the hardware requirements, and used to minimize the
+memory needed for storing entropy coding tables.
+Experimental results show that the proposed solution can
+guarantee good performance (≈ 2% redundancy) with pre-
+cision as low as 4 bits, and with relative redundancy reduc-
+tion by a factor of 4 for each additional bit, reaching less
+than 0.01% for 8 bits.
+2. ENTROPY CODING FOR NN-BASED CODECS
+Entropy coding in conventional codecs is based on estimat-
+ing the probabilities of variables to be coded. For example,
+in video standards like HEVC and VVC [9, 10, 11], adap-
+tive binary arithmetic coding contexts are used for estimating
+probabilities related to data elements.
+Neural-based codecs use a different principle: a param-
+eterized probability distribution (pdf) is chosen for training,
+and a learned scheme (table or NN) is used to determine the
+parameter value to be used for coding a quantized random
+variable. Fig. 1 shows an example of the commonly used nor-
+mal distribution, and unit-step quantization.
+The sequence of probabilities of random variable y ∼
+N(0, σ2), quantized to value n ∈ Z, is
+pn(σ) = 1
+2
+�
+erfc
+�2n − 1
+√
+8σ
+�
+− erfc
+�2n + 1
+√
+8σ
+��
+,
+(1)
+
+Fig. 2. Example of a neural-based codec architecture, including
+scale hyperprior networks for entropy coding.
+with entropy
+H(σ)
+def= −
+∞
+�
+n=−∞
+pn(σ) log2(pn(σ)) ,
+(2)
+where the complementary error function is
+erfc(x)
+def=
+2
+√π
+� ∞
+x
+e−t2 dt.
+(3)
+In practical implementations special symbols are used to
+code values in underflow and overflow infinite sets, which
+have very small but nonzero probabilities, as shown in Fig. 1.
+Within those sets the values are sub-optimally coded with, for
+example, Elias universal codes [12].
+Also shown in Fig. 1 are the elements of code vector c(σ)
+(CV), which contain the probabilities to be used for arithmetic
+coding (AC), within a given range R
+ci(σ) =
+
+
+
+
+
+�−R
+n=−∞ pn(σ),
+i = 0,
+pi−R(σ),
+i = 1, 2, . . . , 2R − 1,
+�∞
+n=R pn(σ),
+i = 2R.
+(4)
+To simplify the notation we ignore implementation de-
+tails like conversions to integer-valued cumulative sums for
+AC [13, 14]. It is assumed that R and AC precision are cho-
+sen to obtain compression very close to entropy.
+This type of entropy coding is used within a NN-based
+codec as shown in Fig. 2, corresponding to the scale hyper-
+prior architecture [15]. There are other codec configurations,
+but our main interest is in the per-element coding stages
+shown in Fig. 3, which is used with all factorized priors.
+The latent y to be coded is quantized with Qy→q, which
+is taken into account during floating-point training. The prob-
+lems addressed here are caused by the quantization stage
+Qσ→ˆσ, representing the use of low-precision network outputs
+during coding, but not during training.
+Since code vectors must be bitwise identical at the en-
+coder and decoder [16], given a range U = [σmin, σmax],
+we assume they share a monotonic sequence {tk}N
+k=0 parti-
+tioning U, and that latent y is coded using c(ρ∗
+k) whenever
+ˆσ ∈ [tk, tk+1).
+To determine the optimal values of ρ∗
+k for code vector
+computation we measure the average redundancy, defined by
+Fig. 3. Per-element entropy coding, including quantization of pdf
+parameter σ.
+Fig. 4. Example of nonuniform quantization of parameter σ, de-
+signed for constant maximum relative redundancy.
+the Kullback-Leibler divergence
+R(σ, ρ)
+def=
+∞
+�
+n=−∞
+pn(σ) log2
+�pn(σ)
+pn(ρ)
+�
+.
+(5)
+Since practical image and video codecs are meant to be
+efficient in a wide range of bit rates, it is interesting to have
+entropy coding designed to minimize relative redundancy
+L(σ, ρ)
+def= R(σ, ρ)
+H(σ) .
+(6)
+and to enable good performance for all use cases use
+ρ∗
+k = arg min
+ρ
+� tk+1
+tk
+L(σ, ρ) dσ.
+(7)
+3. NEW PDF PARAMETERIZATION
+From these definitions it is possible to optimize non-uniform
+quantization schemes as shown in the example of Fig. 4,
+where similarly to other quantization solutions, all intervals
+have the same maximum relative redundancy and averages
+¯Lk
+def=
+1
+tk+1 − tk
+� tk+1
+tk
+L(σ, ρ∗
+k) dσ.
+(8)
+With sufficiently fine quantization of σ, L(σ, ρ∗
+k) is ap-
+proximately quadratic, and we can use
+ρ∗
+k
+≈
+{ρ : L(tk, ρ) = L(tk+1, ρ)} ,
+(9)
+¯Lk
+≈
+L(tk, ρ∗
+k)
+3
+= L(tk+1, ρ∗
+k)
+3
+.
+(10)
+Neural-based codecs using precise floating-point com-
+putations can be implemented with arrays of pre-computed
+
+Fig. 5.
+Proposed modification of codec in Fig. 3, replacing pdf
+parameter σ with parameter u optimized for quantization.
+Fig. 6. Mapping from non-uniform to uniform quantization.
+code vectors, and the approach shown in the example of
+Fig. 4. However, for practical codecs, which must handle a
+wide range of bit rates, it is necessary to support, for exam-
+ple, σ ∈ [0.1, 1000] with the precision of hyperprior network
+output ˆσ constrained to only 8 bits. This means that {tk}N
+k=0
+must also use only 8 bits, severely constraining the design
+and resulting in significant losses.
+To solve this problem we can exploit the fact that σ is only
+used for rate estimation during training, and while computing
+code vectors. The proposed solution is shown in Fig. 5: hy-
+perprior networks are trained to compute a new pdf parameter
+u ∈ [0, 1] that is converted as σ = T (u). For analysis we de-
+fine u ∈ R, but note that during actual coding the quantized
+networks only need to generate integer k. This approach is
+used in [16], where the conversion is chosen to be in the form
+σ = eαu+β.
+Fig. 6 shows how optimal T (u, N) can be defined, for a
+given N, to obtain constant maximum relative redundancy,
+and the following algorithm can be used for its computation.
+1. Given N and [σmin, σmax], choose initial maximum re-
+dundancy ǫ; initialize t0 = σmin.
+2. For n = 1, 2, . . . , N:
+(a) Set ρ∗
+n−1 = {ρ > tn−1 : L(tn−1, ρ) = ǫ};
+(b) Set tn = {t > ρ∗
+n−1 : L(t, ρ∗
+n−1) = ǫ}.
+3. If tN ≈ σmax then stop.
+4. If tN < σmax then increase ǫ, otherwise decrease ǫ,
+using a method for unidimensional search.
+5. Go to step 2.
+The surprising conclusion from computing T (u, N) for
+several values of N, and interpolating between the sampled
+values, is that all results are practically identical. This means
+that if we compute
+T (u) = lim
+N→∞ T (u, N),
+(11)
+we obtain a single reference function that can be used for all
+network trainings, and the value of N can be chosen and mod-
+ified later, without the need for re-training.
+The rationale for determining T (u) can be seen from the
+lower part of Fig. 6. For increasing values of N the redun-
+dancy curves in all intervals should approximate quadratic
+equations on u, with same second derivatives. This can be
+achieved if we can find T (u) and constant α such that
+∂2
+∂u2 L(T (u), ρ)
+����
+ρ=T (u)
+= α,
+u ∈ [0, 1].
+(12)
+Defining
+ψ(σ)
+def=
+∂2
+∂σ2 L(σ, ρ)
+����
+ρ=σ
+(13)
+it can be shown that, thanks to properties of L(σ, ρ), eq. (12)
+is equivalent to
+d
+du T (u) =
+�
+α
+ψ(T (u)),
+(14)
+and thus function T (u) can be determined by solving this or-
+dinary differential equation (ODE) using boundary conditions
+T (0) = σmin, T (1) = σmax.
+For any sequence of differentiable parameterized proba-
+bilities it can be shown that
+ψ(σ) =
+1
+ln(2)H(σ)
+∞
+�
+n=−∞
+1
+pn(σ)
+�dpn(σ)
+dσ
+�2
+,
+(15)
+and for normal pdf we have
+ψ(σ)
+=
+1
+ln(2)H(σ)
+�
+[φ1(σ)]2
+1 − ω1(σ)+
+(16)
+∞
+�
+n=1
+[φ2n−1(σ) − φ2n+1(σ)]2
+ω2n−1(σ) − ω2n+1(σ)
+�
+,
+where
+ωk(σ)
+def= erfc
+�
+k
+√
+8σ
+�
+,
+φk(σ)
+def= k e−k2/(8σ2)
+√
+2πσ2
+.
+(17)
+4. EXPERIMENTAL RESULTS
+The ODE of eq. (14) was solved using the 4th-order Runge-
+Kutta method [17, §17.1], and boundary conditions σ ∈
+[0.1, 1000]. The solution T (u) is shown in Fig. 7. A quick
+observation may lead to the incorrect conclusion that it can
+be approximated by a function like 104u2−1 (red line).
+Since T (u) is defined by a differential equation, the qual-
+ity of the approximation must be measured by relative errors
+of derivatives.
+Those are shown in Fig. 8, where we can
+
+Fig. 7. The optimal pdf parameter conversion function T (u) and
+two approximations.
+Fig. 8. Derivatives of pdf parameter conversion function T (u) and
+its approximations.
+also see a much better approximation Tπ(u) = 10π(u), where
+polynomial
+π(u) = 2.49284 u3 + 0.93703 u2 + 0.57013 u − 1,
+(18)
+minimizes the maximum relative derivative error.
+Fig. 9 shows how the average relative redundancy ¯L(σ)
+varies with σ when uniform quantization is applied to pdf pa-
+rameter u, and N code vectors are used for entropy coding
+latent variables y, as shown in Fig. 5. Following the condi-
+tion defined by eq. (13), ¯L(σ) does not change when T (u) is
+used for pdf parameter conversion. On the other hand, there
+are relatively small variations when approximation Tπ(u) is
+used instead, and those deviations correspond to errors in ap-
+proximating the derivatives (cf. Fig. 8).
+We also observe that the curves are nearly perfectly paral-
+lel, confirming the predominance of the quadratic term in the
+Taylor series of the relative redundancy. This is also verified
+by the fact that ¯L(σ) quadruples when the N is halved.
+Non-uniform quantizations defined by transformation
+T (u) were used for entropy coding values from a floating-
+point implementation of the scale hyperprior codec [15],
+in the Kodak image dataset [18].
+Redundancy results are
+shown in Table 1, for average bit rates between 0.25 and
+2.5 bits/pixel.
+For values of N larger than about 128 (7 bit represen-
+tation) the redundancy from quantization is 0.01% or below,
+which is within the experimental variation. For smaller values
+of N we can observe that
+Fig. 9. Average relative redundancies from T (u) (dashed) and ap-
+proximation 10π(u) (solid), for different number of quantization in-
+tervals N.
+Table 1. Coding results obtained using scale hyperprior codec ap-
+plied to 24 images of Kodak dataset.
+N
+CV mem.
+Rel. redundancy (%) @ bit rate
+(Kbytes)
+0.25
+0.50
+1.00
+1.50
+2.50
+256
+51.6
+0.00
+0.00
+0.00
+0.00
+0.01
+192
+38.7
+0.00
+0.00
+0.00
+0.01
+0.01
+128
+25.8
+0.04
+0.04
+0.04
+0.03
+0.03
+96
+19.4
+0.03
+0.04
+0.04
+0.04
+0.05
+64
+12.9
+0.13
+0.12
+0.12
+0.12
+0.11
+48
+9.7
+0.24
+0.23
+0.22
+0.22
+0.20
+32
+6.5
+0.38
+0.38
+0.40
+0.41
+0.42
+24
+4.9
+0.85
+0.81
+0.81
+0.82
+0.79
+16
+3.4
+1.79
+1.76
+1.76
+1.78
+1.75
+• The redundancies are practically constant for all bit
+rates, meaning that the proposed coding method achieves
+the design objective of having constant relative redun-
+dancies.
+• The average redundancy levels match those of Fig. 9,
+showing that features predicted by theory match the
+practical implementation.
+The second column of Table 1 contains the total amount
+of memory needed to store the code vectors, when using
+16 bits/element. This shows that the proposed method can
+be directly used to minimize memory. For example, even if
+the network is able to support 8 bit outputs, and redundancy
+below 0.5% is acceptable, the pdf parameter can be further
+quantized to 5 bits, by simply discarding the 3 least significant
+bits, to reduce CV memory from 51.6 to 6.5 Kbytes.
+5. CONCLUSIONS
+It is shown that the effects of quantization on the entropy cod-
+ing can be analyzed by evaluating coding redundancy, and
+this can be used for changing the pdf parameter to be learned
+so that average relative redundancy becomes practically con-
+stant. In consequence, networks do not need to be re-trained
+when the pdf parameter quantization is changed, which we
+show can also be done intentionally to minimize memory for
+code tables.
+
+6. REFERENCES
+[1] S. Ma, X. Zhang, C. Jia, Z. Zhao, S. Wang, and
+S. Wang, “Image and video compression with neural
+networks: a review,” IEEE Trans. Circuits Syst. Video
+Technol., vol. 30, no. 6, pp. 1683–1698, June 2020,
+arXiv:1904.03567v2.
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+arXiv:2101.06341v1.
+[3] Qualcomm AI Research,
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+first-software-based-neural-video-decoder-running-hd-
+format-real-time, June. 2021.
+[4] Qualcomm AI Research, “Demonstration of real time
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+https://www.youtube.com/watch?v=WUnlSHenr08,
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+[5] J. Ascenso, P. Akyazi, F. Pereira, and T. Ebrahimi,
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+Applicat. VI, Apr. 2020, pp. 164–176.
+[6] M. Nagel,
+M. van Baalen,
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+and
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+“Data-free quantization through weight
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+Int. Conf. Comput. Vision, Oct. 2019, pp. 1325–1334,
+arXiv:1906.04721v3.
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+darenko,
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+“Pruning and quantization for deep neural network ac-
+celeration: a survey,” Neurocomputing, vol. 461, pp.
+370–403, Oct. 2021, arXiv:2101.09671.
+[9] G. J. Sullivan, J.-R. Ohm, W.-J. Han, and T. Wie-
+gand, “Overview of the High Efficiency Video Coding
+(HEVC) Standard,” IEEE Trans. Circuits Syst. Video
+Technol., vol. 22, no. 12, pp. 1649–1668, Dec. 2012.
+[10] V. Sze and D. Marpe, “Entropy coding in HEVC,” in
+High Efficiency Video Coding (HEVC): Algorithms and
+Architectures, V. Sze, M. Budagavi, and G. J. Sullivan,
+Eds., chapter 8, pp. 209–274. Springer, 2014.
+[11] B. Bross, Y.-K. Wang, Y. Ye, S. Liu, J. Chen, G. J. Sulli-
+van, and J.-R. Ohm, “Overview of the versatile video
+coding (VVC) standard and its applications,”
+IEEE
+Trans. Circuits Syst. Video Technol., vol. 31, no. 10, pp.
+3736–3764, Oct. 2021.
+[12] P. Elias, “Universal codeword sets and representations
+of the integers,” IEEE Trans. Inf. Theory, vol. 21, no. 2,
+pp. 194–203, Mar. 1975.
+[13] A. Said,
+“Arithmetic coding,”
+in Lossless Compres-
+sion Handbook, K. Sayood, Ed., chapter 5, pp. 101–152.
+Academic Press, San Diego, CA, 2003.
+[14] A. Said,
+“Introduction to arithmetic coding – theory
+and practice,” Technical Report HPL-2004-76, Hewlett
+Packard Laboratories, Palo Alto, CA, USA, Apr.
+2004,
+(http://www.hpl.hp.com/techreports/2004/HPL-
+2004-76.pdf).
+[15] J. Ball´e, D. Minnen, S. Singh, S. J. Hwang, and N. John-
+ston, “Variational image compression with a scale hy-
+perprior,”
+in Sixth Int. Conf. Learning Representa-
+tions, Vancouver, Canada, Apr. 2018, arXiv preprint
+arXiv:1802.01436v2.
+[16] J. Ball´e, N. Johnston, and D. Minnen,
+“Integer
+networks for data compression with latent-variable
+models,”
+in Int. Conf. Learning Representations,
+New Orleans, Louisiana, USA, May 2019,
+URL:
+https://openreview.net/forum?id=S1zz2i0cY7.
+[17] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and
+B. P. Flannery, Numerical Recipes: The Art of Scientific
+Computing, Cambridge University Press, Cambridge,
+UK, third edition, 2007.
+[18] Kodak Image Dataset, URL: http://www.cs.albany.edu/ xy-
+pan/research/snr/Kodak.html.
+
diff --git a/gtFAT4oBgHgl3EQf8x46/content/tmp_files/load_file.txt b/gtFAT4oBgHgl3EQf8x46/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf,len=398
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='08752v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='IV]  20 Jan 2023  OPTIMIZED LEARNED ENTROPY CODING PARAMETERS FOR PRACTICAL  NEURAL-BASED IMAGE AND VIDEO COMPRESSION  Amir Said  Reza Pourreza  Hoang Le  Qualcomm AI Research,∗ San Diego, CA, USA  ABSTRACT  Neural-based image and video codecs are significantly more  power-efficient when weights and activations are quantized  to low-precision integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' While there are general-purpose  techniques for reducing quantization effects, large losses can  occur when specific entropy coding properties are not con-  sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' This work analyzes how entropy coding is affected  by parameter quantizations, and provides a method to min-  imize losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  It is shown that, by using a certain type of  coding parameters to be learned, uniform quantization be-  comes practically optimal, also simplifying the minimization  of code memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' The mathematical properties  of the new representation are presented, and its effectiveness  is demonstrated by coding experiments, showing that good  results can be obtained with precision as low as 4 bits per  network output, and practically no loss with 8 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Index Terms— Learned image and video compression,  neural network quantization, entropy coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' INTRODUCTION  After years of research showing the advantages of neural-  based techniques for video coding [1, 2], and recent demon-  strations of real-time decoding in mobile devices [3, 4], there  is growing interest in their standardization and commercial  deployment [5], and a main practical implementation chal-  lenge is the minimization of decoding power requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Implementing neural network (NN) computations with  low-precision integers significantly reduces power needs,  but requires minimizing the effects of quantizing networks  weights and activations [6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' For neural image and video  codecs, efficacy is strongly defined by the entropy coding  stage and related networks, and can be severely degraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Analyzing quantization effects in NNs can be difficult  because they implement non-linear processes, and learned  variables have unknown properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Techniques based on  re-training under quantization constraints help to reduce the  losses, but have limited efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  In this work it is shown that NN outputs used for entropy  coding have well-defined meaning and properties, which can  be used for predicting quantization effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Furthermore,  those properties can be used for modifying the loss function  ∗Qualcomm AI Research is an initiative of Qualcomm Technologies, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Quantized values from normal distribution with std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' σ,  commonly used in neural-based codecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  used while learning, and it is demonstrated that networks can  be trained to be minimally sensitive to quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Further analysis shows that this optimized representa-  tion is remarkably independent of the quantization, and thus  networks do not need to be retrained even after significant  changes to entropy coding design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Thus, in those  networks the uniform quantization can also be extended be-  yond the hardware requirements, and used to minimize the  memory needed for storing entropy coding tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Experimental results show that the proposed solution can  guarantee good performance (≈ 2% redundancy) with pre-  cision as low as 4 bits, and with relative redundancy reduc-  tion by a factor of 4 for each additional bit, reaching less  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='01% for 8 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' ENTROPY CODING FOR NN-BASED CODECS  Entropy coding in conventional codecs is based on estimat-  ing the probabilities of variables to be coded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' For example,  in video standards like HEVC and VVC [9, 10, 11], adap-  tive binary arithmetic coding contexts are used for estimating  probabilities related to data elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Neural-based codecs use a different principle: a param-  eterized probability distribution (pdf) is chosen for training,  and a learned scheme (table or NN) is used to determine the  parameter value to be used for coding a quantized random  variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 1 shows an example of the commonly used nor-  mal distribution, and unit-step quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  The sequence of probabilities of random variable y ∼  N(0, σ2), quantized to value n ∈ Z, is  pn(σ) = 1  2  �  erfc  �2n − 1  √  8σ  �  − erfc  �2n + 1  √  8σ  ��  ,  (1)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Example of a neural-based codec architecture, including  scale hyperprior networks for entropy coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  with entropy  H(σ)  def= −  ∞  �  n=−∞  pn(σ) log2(pn(σ)) ,  (2)  where the complementary error function is  erfc(x)  def=  2  √π  � ∞  x  e−t2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  (3)  In practical implementations special symbols are used to  code values in underflow and overflow infinite sets, which  have very small but nonzero probabilities, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Within those sets the values are sub-optimally coded with, for  example, Elias universal codes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 1 are the elements of code vector c(σ)  (CV), which contain the probabilities to be used for arithmetic  coding (AC), within a given range R  ci(σ) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  �−R  n=−∞ pn(σ),  i = 0,  pi−R(σ),  i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' , 2R − 1,  �∞  n=R pn(σ),  i = 2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  (4)  To simplify the notation we ignore implementation de-  tails like conversions to integer-valued cumulative sums for  AC [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' It is assumed that R and AC precision are cho-  sen to obtain compression very close to entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  This type of entropy coding is used within a NN-based  codec as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 2, corresponding to the scale hyper-  prior architecture [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' There are other codec configurations,  but our main interest is in the per-element coding stages  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 3, which is used with all factorized priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  The latent y to be coded is quantized with Qy→q, which  is taken into account during floating-point training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' The prob-  lems addressed here are caused by the quantization stage  Qσ→ˆσ, representing the use of low-precision network outputs  during coding, but not during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Since code vectors must be bitwise identical at the en-  coder and decoder [16], given a range U = [σmin, σmax],  we assume they share a monotonic sequence {tk}N  k=0 parti-  tioning U, and that latent y is coded using c(ρ∗  k) whenever  ˆσ ∈ [tk, tk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  To determine the optimal values of ρ∗  k for code vector  computation we measure the average redundancy, defined by  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Per-element entropy coding, including quantization of pdf  parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Example of nonuniform quantization of parameter σ, de-  signed for constant maximum relative redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  the Kullback-Leibler divergence  R(σ, ρ)  def=  ∞  �  n=−∞  pn(σ) log2  �pn(σ)  pn(ρ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  (5)  Since practical image and video codecs are meant to be  efficient in a wide range of bit rates, it is interesting to have  entropy coding designed to minimize relative redundancy  L(σ, ρ)  def= R(σ, ρ)  H(σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  (6)  and to enable good performance for all use cases use  ρ∗  k = arg min  ρ  � tk+1  tk  L(σ, ρ) dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  (7)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' NEW PDF PARAMETERIZATION  From these definitions it is possible to optimize non-uniform  quantization schemes as shown in the example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 4,  where similarly to other quantization solutions, all intervals  have the same maximum relative redundancy and averages  ¯Lk  def=  1  tk+1 − tk  � tk+1  tk  L(σ, ρ∗  k) dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  (8)  With sufficiently fine quantization of σ, L(σ, ρ∗  k) is ap-  proximately quadratic, and we can use  ρ∗  k  ≈  {ρ : L(tk, ρ) = L(tk+1, ρ)} ,  (9)  ¯Lk  ≈  L(tk, ρ∗  k)  3  = L(tk+1, ρ∗  k)  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  (10)  Neural-based codecs using precise floating-point com-  putations can be implemented with arrays of pre-computed  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Proposed modification of codec in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 3, replacing pdf  parameter σ with parameter u optimized for quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Mapping from non-uniform to uniform quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  code vectors, and the approach shown in the example of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' However, for practical codecs, which must handle a  wide range of bit rates, it is necessary to support, for exam-  ple, σ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='1, 1000] with the precision of hyperprior network  output ˆσ constrained to only 8 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' This means that {tk}N  k=0  must also use only 8 bits, severely constraining the design  and resulting in significant losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  To solve this problem we can exploit the fact that σ is only  used for rate estimation during training, and while computing  code vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' The proposed solution is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 5: hy-  perprior networks are trained to compute a new pdf parameter  u ∈ [0, 1] that is converted as σ = T (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' For analysis we de-  fine u ∈ R, but note that during actual coding the quantized  networks only need to generate integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' This approach is  used in [16], where the conversion is chosen to be in the form  σ = eαu+β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 6 shows how optimal T (u, N) can be defined, for a  given N, to obtain constant maximum relative redundancy,  and the following algorithm can be used for its computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Given N and [σmin, σmax], choose initial maximum re-  dundancy ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' initialize t0 = σmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' For n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' , N:  (a) Set ρ∗  n−1 = {ρ > tn−1 : L(tn−1, ρ) = ǫ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  (b) Set tn = {t > ρ∗  n−1 : L(t, ρ∗  n−1) = ǫ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' If tN ≈ σmax then stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' If tN < σmax then increase ǫ, otherwise decrease ǫ,  using a method for unidimensional search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Go to step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  The surprising conclusion from computing T (u, N) for  several values of N, and interpolating between the sampled  values, is that all results are practically identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' This means  that if we compute  T (u) = lim  N→∞ T (u, N),  (11)  we obtain a single reference function that can be used for all  network trainings, and the value of N can be chosen and mod-  ified later, without the need for re-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  The rationale for determining T (u) can be seen from the  lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' For increasing values of N the redun-  dancy curves in all intervals should approximate quadratic  equations on u, with same second derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' This can be  achieved if we can find T (u) and constant α such that  ∂2  ∂u2 L(T (u), ρ)  ����  ρ=T (u)  = α,  u ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  (12)  Defining  ψ(σ)  def=  ∂2  ∂σ2 L(σ, ρ)  ����  ρ=σ  (13)  it can be shown that, thanks to properties of L(σ, ρ), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' (12)  is equivalent to  d  du T (u) =  �  α  ψ(T (u)),  (14)  and thus function T (u) can be determined by solving this or-  dinary differential equation (ODE) using boundary conditions  T (0) = σmin, T (1) = σmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  For any sequence of differentiable parameterized proba-  bilities it can be shown that  ψ(σ) =  1  ln(2)H(σ)  ∞  �  n=−∞  1  pn(σ)  �dpn(σ)  dσ  �2  ,  (15)  and for normal pdf we have  ψ(σ)  =  1  ln(2)H(σ)  �  [φ1(σ)]2  1 − ω1(σ)+  (16)  ∞  �  n=1  [φ2n−1(σ) − φ2n+1(σ)]2  ω2n−1(σ) − ω2n+1(σ)  �  ,  where  ωk(σ)  def= erfc  �  k  √  8σ  �  ,  φk(σ)  def= k e−k2/(8σ2)  √  2πσ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  (17)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' EXPERIMENTAL RESULTS  The ODE of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' (14) was solved using the 4th-order Runge-  Kutta method [17, §17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='1], and boundary conditions σ ∈  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='1, 1000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' The solution T (u) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' A quick  observation may lead to the incorrect conclusion that it can  be approximated by a function like 104u2−1 (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Since T (u) is defined by a differential equation, the qual-  ity of the approximation must be measured by relative errors  of derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Those are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 8, where we can  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' The optimal pdf parameter conversion function T (u) and  two approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Derivatives of pdf parameter conversion function T (u) and  its approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  also see a much better approximation Tπ(u) = 10π(u), where  polynomial  π(u) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='49284 u3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='93703 u2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='57013 u − 1,  (18)  minimizes the maximum relative derivative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 9 shows how the average relative redundancy ¯L(σ)  varies with σ when uniform quantization is applied to pdf pa-  rameter u, and N code vectors are used for entropy coding  latent variables y, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Following the condi-  tion defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' (13), ¯L(σ) does not change when T (u) is  used for pdf parameter conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' On the other hand, there  are relatively small variations when approximation Tπ(u) is  used instead, and those deviations correspond to errors in ap-  proximating the derivatives (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  We also observe that the curves are nearly perfectly paral-  lel, confirming the predominance of the quadratic term in the  Taylor series of the relative redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' This is also verified  by the fact that ¯L(σ) quadruples when the N is halved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Non-uniform quantizations defined by transformation  T (u) were used for entropy coding values from a floating-  point implementation of the scale hyperprior codec [15],  in the Kodak image dataset [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Redundancy results are  shown in Table 1, for average bit rates between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='25 and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='5 bits/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  For values of N larger than about 128 (7 bit represen-  tation) the redundancy from quantization is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='01% or below,  which is within the experimental variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' For smaller values  of N we can observe that  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Average relative redundancies from T (u) (dashed) and ap-  proximation 10π(u) (solid), for different number of quantization in-  tervals N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Coding results obtained using scale hyperprior codec ap-  plied to 24 images of Kodak dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  N  CV mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' redundancy (%) @ bit rate  (Kbytes)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
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+page_content='50  256  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
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+page_content='42  24  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
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+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='79  16  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='79  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='78  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='75  The redundancies are practically constant for all bit  rates, meaning that the proposed coding method achieves  the design objective of having constant relative redun-  dancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  The average redundancy levels match those of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 9,  showing that features predicted by theory match the  practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  The second column of Table 1 contains the total amount  of memory needed to store the code vectors, when using  16 bits/element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' This shows that the proposed method can  be directly used to minimize memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' For example, even if  the network is able to support 8 bit outputs, and redundancy  below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='5% is acceptable, the pdf parameter can be further  quantized to 5 bits, by simply discarding the 3 least significant  bits, to reduce CV memory from 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='6 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='5 Kbytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' CONCLUSIONS  It is shown that the effects of quantization on the entropy cod-  ing can be analyzed by evaluating coding redundancy, and  this can be used for changing the pdf parameter to be learned  so that average relative redundancy becomes practically con-  stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' In consequence, networks do not need to be re-trained  when the pdf parameter quantization is changed, which we  show can also be done intentionally to minimize memory for  code tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' REFERENCES  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Jia, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Wang, and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Wang, “Image and video compression with neural  networks: a review,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
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+page_content=' Video  Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
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+page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' 1683–1698, June 2020,  arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='03567v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Ding, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
+page_content=' Ma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFAT4oBgHgl3EQf8x46/content/2301.08752v1.pdf'}
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+1
+Shape the Future of ITS –
+Optimization and scheduling for a large scale urban transportation system –
+in a fast-changing world
+Yi Zhang
+I. INTRODUCTION
+The rapid growth of the population facilitates the urban
+sprawl, automobile production and leads to heavy traffic con-
+gestions, pollution, noises and traffic fatalities. Therefore, it is
+indispensable to develop an eco-friendly sustainable transport
+with high performance of the existing traffic network. Although
+many transformational products have been developed and used
+on the road, such as autonomous vehicles (AV) and drones,
+and their quality is continually upgraded, 100% replacement of
+traditional cars to AVs requires not only the safety and security
+evaluation but also corresponding physical infrastructures, e.g.,
+non-signalized intersection, charging stations. Also, we still
+have a long way to go in order to completely commercialize
+AVs with level 5 automation. Thus, it is predictable that traffic
+signals will still play an important role for a certain long time
+in this hybrid transportation world, which involves both human-
+driven cars and AVs under different automated levels. At the
+period of transition, a feasible and implementable approach is
+more helpful and practical to reshape the transportation system
+in the near future.
+The concept of the pedestrian/transit-oriented transport,
+which is designed for the human body instead of the car body,
+has been proposed by researchers [2]: “cities with very high
+walking, cycling and transit mode share (i.e., 75% or more)
+typically have high density, mixed use urban centres at or
+above 100-200 people per hectare and are supported by a
+transportation strategy that prioritizes pedestrians first, then
+cyclists followed by transit users.” However, vehicle traffic
+still gets excessively high attention in many cities’ guidelines,
+and walkability is seldom taken into account. Also, walking is
+the access to public transport (PT), thereby PT services could
+accordingly be promoted if the pedestrian walking safety and
+pleasantness can be satisfied. PT system has been studied for
+several decades due to its large ridership and sustainability
+on economic efficiency, environmental protection and social
+equity. Proper bus dispatching and operation can attract more
+passengers, which encourages commuters to change their travel
+mode from private automobiles to public buses to further
+alleviate the traffic congestion and air pollution.
+However, PT system, a typical ride-sharing transport serving
+a variety of access needs and providing an equal social value,
+now faces the challenges from the mobility-on-demand (MOD)
+system, which is famed for its easy transactions and convenient
+access via mobile phones. Therefore, we need to explore and
+create a win-win cooperation model between both PT and AV
+systems at current transitional period. The flexible demand-
+Yi Zhang is affiliated with with the Institute for Infocomm Research (I2R),
+Agency for Science, Technology and Research (A*STAR), Singapore 138632.
+Emails: yzhang120@e.ntu.edu.sg(zhang yi@i2r.a-star.edu.sg).
+driven pattern in AVs could make up the shortages due to fixed
+routes in the PT system. On the other hand, PT system is still
+coping with the high-volume transfer tasks, which may not be
+feasible for AVs due to the safety concern.
+The objective of this essay is to propose a set of coordi-
+nated technological solutions to transform existing transport
+system to a more intelligent interactive system by adopting
+optimization and control methods implementable in the near
+future, thereby improving public services and quality of life for
+residents. In this essay, three different application scenes that
+closely related to people’s daily life are discussed. We firstly
+propose a traffic light scheduling strategy via model predic-
+tive control (MPC) method, with the aim to fairly minimize
+both pedestrians’ and vehicles’ delay. After that, a combined
+dispatching-operation system is proposed to further increase
+the control flexibility, and corresponding implementation solu-
+tion for boarding control is also illustrated. Finally, a possible
+scheme to combine both PT and AV systems is proposed to
+improve existing PT system.
+II. WORKING PACKAGES AND TASKS
+A. Traffic light scheduling for vehicle-pedestrian mixed-flow
+networks
+Most traffic signal controllers differentiate vehicles and
+pedestrians, and focus on their great contributions to vehicle
+flows, this is reasonable when pedestrian volume is low. How-
+ever, in the downtown areas where large number of pedestrians
+interfere with vehicular traffic, optimizing traffic signals only
+for vehicles may create more conflicts between both traffic
+participants and potentially reduce the economic interest, since
+pedestrians in CBD areas are usually potential customers of
+nearby shopping malls. According to SGS Economics and
+Planning [1], optimized pedestrian flow can increase $1.3
+billion a year for Melbourne CBD area. On the other hand,
+rich data can be obtained with the help of the advanced sensor
+equipments, such as V2X, 5G communication, Lidar and so
+on. Powerful machine learning algorithms can be effectively
+utilized to help predict the traffic flows based on the large
+amount of database, which better serve the optimized signal
+controller.
+In view of this, we propose this signal controller with the
+aim to fairly minimize both vehicles’ and pedestrians’ delay
+[4][7]. Fig. 1 illustrates the framework of the proposed real-
+time traffic light scheduling strategy implemented in simula-
+tor software VISSIM. The macroscopic flow model for both
+pedestrians and vehicles are developed, and the impacts of
+the signal on pedestrian crossing capacity is captured in the
+model. Benefitted from the advanced sensors, signal priority
+level could also be incorporated into the model to enable
+arXiv:2301.12087v1  [eess.SY]  28 Jan 2023
+
+2
+higher priority for public buses. The mixed-flow model is
+then solved by adopting the commercial optimization solver
+or evolutionary algorithms. After that, the optimized signal
+phases and duration are sent to the simulator VISSIM, which
+mimics the real urban traffic environment. Meanwhile, the
+traffic information, such as traffic volumes, turning ratios, is
+stored in the back-end database, which is used to re-train the AI
+models by adopting the machine learning algorithms to predict
+the required information. The predicted traffic parameters and
+current real-time information are all sent to the controller side,
+which is solved in a rolling horizon manner, and the whole
+process continues with the evolve of the traffic system to form
+a closed-loop control strategy.
+B. A combined dispatching-operation strategy for public bus
+management incorporated with the boarding control
+The bus operation systems in most today’s studies are
+proposed based on known bus dispatching time or schedule
+headways/frequencies. On the other hand, the bus dispatching
+systems seldom consider operation control for on-road buses.
+It is understandable that separately deal with these two prob-
+lems could reduce the computational complexity significantly
+compared with the complicated combined model. However, the
+increasing application and perfection of telematics (e.g., Auto-
+matic Vehicle Location (AVL), Automatic Passenger Counting
+(APC), etc.) in the bus management system, the collection of
+the real-time information becomes possible, also, the constantly
+upgraded computer with more computing power is keeping
+breaking the records, thus, it is predictable that to solve a
+combined optimization problem is implementable [5]. Decision
+variables, such as bus dispatching time, bus speed between any
+two adjacent stops, bus dwell time at each stop, OD-based
+boarding volume, could all be described into a holistic model
+to enable this combined dispatching-operation bus management
+system [6]. Also, boarding control has been studied in some
+literature and proved its high efficiency in improving the
+bus service quality. However, it still has not been widely
+implemented in the real world.
+Fig. 2 gives an graph illustration of future bus stop, which
+makes the boarding control applicable. Imagine a message via
+mobile apps or screen board located at the bus stop could be
+sent or displayed to passengers when a bus is approaching the
+stop, and only passengers at the front queue will be selected
+to enter the designated boarding area at the bus stop. The
+identification of the front-queue passenger can be realized by
+the camera or Lidar installed at the bus stop. When the bus
+reaches the bus stop, only passengers who are waiting at the
+designated area are allowed to board the bus. This requires
+infrastructural enhancement at the bus stop, not only the area
+re-design but also the advanced sensors.
+C. Autonomous bus fleet management for mobility-on-demand
+system
+To increase the vehicle utilization and reduce the carbon
+dioxide emissions, the on-line carsharing services have been
+carried out in many cities in the past few years. Also, the adop-
+tion of the electric vehicles could potentially reduce emissions
+to promote a sustainable environment, meanwhile, the forth-
+coming commercialization of AVs has constantly reshaping
+the transportation system. All of the above emerging concepts
+and technologies have combined and led to an autonomous
+mobility-on-demand system (AMOD). AMOD system is not a
+new concept and has been proposed since 2014 [3], however,
+the implementation and research studies about AMOD system
+mainly focus on private cars and taxi services, and this may
+bring challenges on the traditional public transport, which
+has been regarded as the symbol of equity and accessibility.
+Therefore, it is strongly necessary to identify and explore the
+synergistic possibilities between AMOD and public transports,
+especially under this transitional period.
+Fig. 3 illustrates one possible case capturing how AVs could
+support the existing public bus lines. The left subfigure de-
+scribes two traditional bus lines served by corresponding buses
+dispatched from the terminal. The orange bar represents the
+demand size at each bus stop. All control variables mentioned
+in Section II-B could be adopted and tuned to facilitate the op-
+eration of the traditional bus service. Meanwhile, we could also
+dispatch AVs to serve bus stops on multiple bus lines but with
+high demands due to its flexible on-demand characteristics.
+AVs are normally commercialized as electric cars with limited
+battery capacity, thus, it is inevitable to develop a smart AV
+routing strategy catering for the demands of the users as well
+as the battery capacity level, as illustrated in Fig. 4. As AVs
+do not follow a fixed bus route and on-board passengers need
+to be sent to their destined stops, thus, on-demand autonomous
+bus fleet dispatching strategy inevitably requires the boarding
+control at each bus stop, and this enables the system to consider
+not only the routing decision but also the volume dynamics,
+which is different with current one passenger pick-up and drop-
+off problem. The optimization involving in routing, subsequent
+bus stop selection and corresponding boarding volume makes
+the problem more challenging but also interesting.
+III. CONCLUSION
+In
+this
+essay,
+technological
+solutions
+from
+a
+pedestrian/transit-oriented perspective have been provided.
+Firstly,
+an
+adaptive
+traffic
+signal
+control
+framework
+is
+presented, with consideration of a macroscopic mixed-flow
+optimization model, VISSIM simulation platform, historical
+database and machine learning-enabled AI prediction model.
+The experiment results in our paper [4] have demonstrated
+that the proposed strategy in a Manhattan-shaped network
+can strike a good balance between pedestrians’ and vehicle
+drivers’ needs. Therefore, we are optimistic to implement this
+strategy in the real world to enhance the existing traffic signals
+in the near future. Next, the bus dispatching and its on-road
+operation, especially the boarding control, are combined
+together to minimize the passenger delay time and the
+operating bus vacancy, which makes our strategy more flexible
+and adaptive to meet the passenger demand. In our previous
+study [6], a multi-bus dispatching and boarding control
+strategy can reduce roughly 50% of remaining passenger
+volumes when compared with the timetable-based fixed
+schedule, which is quite promising and exciting. Moreover,
+to achieve a synergistic objective between PT system and
+
+3
+Fig. 1: Framework of traffic light scheduling for vehicle-pedestrian mixed-flow networks
+Fig. 2: Graphic illustration of future bus stop
+Fig. 3: Application of autonomous buses on supporting tradi-
+tional public bus lines
+Fig. 4: Autonomous bus routing for multi-loading and on-road
+charging
+AV system, a scheme is proposed, where AVs are dispatched
+to pick up passengers at high-demand stops to support the
+PT lines. This scheme shall be further studied in my future
+work to incorporate multiple optimized variables, including
+AV routing, volume dynamics and boarding limits. Overall,
+the essay presents three application scenes in the intelligent
+transportation field. The suggested strategies and guidelines
+can assist in developing an intelligent future urban transport
+system in order to provide citizens a smarter, safer and more
+interactive transportation experience.
+REFERENCES
+[1] SGS Economics and Planning. CBD pedestrian analysis, technical report
+city of melbourn. Technical report, 2014.
+[2] Jennie Moore, Kirstin Mille, Richard Registe, Sarah Campbell, Julian
+Zelazny, and Sven Eberlein.
+International ecocity standards(brochure).
+Technical report, 2017.
+[3] Kevin Spieser, Kyle Treleaven, Rick Zhang, Emilio Frazzoli, Daniel
+Morton, and Marco Pavone. Toward a systematic approach to the design
+and evaluation of automated mobility-on-demand systems: A case study
+in singapore. In Road vehicle automation, pages 229–245. Springer, 2014.
+[4] Yi Zhang, Kaizhou Gao, Yicheng Zhang, and Rong Su.
+Traffic light
+scheduling for pedestrian-vehicle mixed-flow networks. IEEE Transac-
+tions on Intelligent Transportation Systems, 20(4):1468–1483, 2018.
+[5] Yi Zhang, Rong Su, Yicheng Zhang, and Nadeesha Sandamali Gammana
+Guruge. A multi-bus dispatching strategy based on boarding control. IEEE
+Transactions on Intelligent Transportation Systems, 2021.
+[6] Yi Zhang, Rong Su, Yicheng Zhang, and Bohui Wang. Dynamic multi-bus
+dispatching strategy with boarding and holding control for passenger delay
+alleviation and schedule reliability: A combined dispatching-operation
+system. IEEE Transactions on Intelligent Transportation Systems, 2021.
+[7] Yi Zhang, Yicheng Zhang, and Rong Su. Pedestrian-safety-aware traffic
+light control strategy for urban traffic congestion alleviation.
+IEEE
+Transactions on Intelligent Transportation Systems, 22(1):178–193, 2019.
+Yi Zhang (S’17-M’21) Yi Zhang received her Bache-
+lor degree of Engineering from Shandong University,
+China in 2014, and the PhD degree in Electrical and
+Electronic Engineering from Nanyang Technological
+University, Singapore in 2020. She is currently a
+research scientist at the Institute for Infocomm Re-
+search (I2R) in the Agency for Science, Technology
+and Research, Singapore (A*STAR). Her research
+interests focus on intelligent transportation system,
+including urban traffic flow management, model-
+based traffic signal scheduling, lane change predic-
+tion and bus dispatching and operation management.
+
+Mixed-Flow Network-based
+Controller Implementationin
+VissiM SignalActivation via
+OptimizationMode
+Gurobi/EvolutionaryAlgorithms
+CoMInterface
+Optimization ParameterPrediction via
+Database
+Real-Time
+Machine LearningAlgorithms
+Database
+Give Way to
+Intra-day
+Pedestrians
+Database
+00:00-
+03:00
+07:00
+02:59
+05:59
+09:59
+Monthly
+Database
+Weekt
+Week4
+Historical
+Database
+JAN
+FEB
+10Waiting
+Area
+(OD-based)
+Notification Screen
+Area
+(Bus-based)Terminal
+Droppoints
+Pick-up
+Points
+(larger
+points
+denote
+larger
+demands)
+Charging
+pointsShape the Future of ITS –
+Optimization and scheduling for a large scale urban transportation system
+Yi Zhang
+I. INTRODUCTION
+Even if the quality of AVs is continually upgrade, 100% re-
+placement of traditional cars (gasoline cars, hybrid cars, elec-
+tronic cars are not Automation 6) to fully 6-level autonomous
+cars requires not only the safety and security evaluation
+but also corresponding matched physical infrastructures, e.g.,
+non-signalized intersection, thus, it is predictable that traffic
+signal still play an important role in the near future. At the
+period of transition, a feasible and implementable approach
+is proposed in this essay.
+II. OBJECTIVES
+The objective of this essay is to propose a set of coordi-
+nated technological solutions to transform existing transport
+system to a more intelligent interactive system by adopting
+implementable optimization and control methods in the near
+future, thereby improve public services and quality of life
+for residents. In this essay, three different application scenes
+that closely related to people’s daily life are discussed.
+III. WORKING PACKAGES AND TASKS
+A. Traffic light scheduling for vehicle-pedestrian mixed-flow
+networks
+Most traffic signal controllers differentiate vehicles and
+pedestrians, and focus on their great contributions to vehicle
+flows, this is reasonable when pedestrian volume is low.
+However, in the downtown areas where large number of
+pedestrians interfere with vehicular traffic, optimizing traffic
+signals only for vehicles may create more conflicts between
+both traffic participants and potentially reduce the economic
+interest, since pedestrians in CBD areas are usually potential
+customers of nearby shopping malls. According to SGS
+Economics and Planning [], optimized pedestrian flow can
+increase $1.3 billion a year for Melbourne CBD area. On the
+other hand, rich data can be obtained with the help of the
+advanced sensor equipments, such as V2X, 5G communica-
+tion, Lidar and so on. Powerful machine learning algorithms
+can be effectively utilized to help predict the traffic flows
+based on the large amount of database, which better serve
+the optimized signal controller.
+In view of this, we propose this signal controller with the
+aim to fairly minimize both vehicles’ and pedestrians’ delay
+[]. Fig. 1 illustrates the framework of the proposed real-time
+traffic light scheduling strategy implemented in simulator
+Yi Zhang is affiliated with with the Institute for Infocomm Research
+(I2R), Agency for Science, Technology and Research (A*STAR), Singapore
+138632. Emails: zhang yi@i2r.a-star.edu.sg.
+software VISSIM. The macroscopic flow model for both
+pedestrians and vehicles are developed, and the impacts of
+the signal on pedestrian crossing capacity is captured in the
+model. Benefitted from the advanced sensors, signal priority
+level could also be incorporated into the model to enable
+higher priority for public buses. The mixed-flow model is
+then solved by adopting the commercial optimization solver
+or evolutionary algorithms. After that, the optimized signal
+phases and duration are sent to the simulator VISSIM, which
+mimics the real urban traffic environment. Meanwhile, the
+traffic information, such as traffic volumes, turning ratios, is
+stored in the back-end database, which is used to re-train
+the AI models by adopting the machine learning algorithms
+to predict the required information. The predicted traffic
+parameters and current real-time information are all sent
+to the controller side, which is solved in a rolling horizon
+manner, and the whole process continues with the evolve of
+the traffic system to form a closed-loop control strategy.
+B. A combined dispatching-operation strategy for public bus
+management incorporated with the boarding control
+The bus operation systems in most today’s studies are
+proposed based on known bus dispatching time or schedule
+headways/frequencies. On the other hand, the bus dispatch-
+ing systems seldom consider operation control for on-road
+buses. It is understandable that separately deal with these two
+problems could reduce the computational complexity sig-
+nificantly compared with the complicated combined model.
+However, the increasing application and perfection of telem-
+atics (e.g., Automatic Vehicle Location (AVL), Automatic
+Passenger Counting (APC), etc.) in the bus management
+system, the collection of the real-time information becomes
+possible, also, the constantly upgraded computer with more
+computing power is keeping breaking the records, thus, it is
+predictable that to solve a combined optimization problem is
+implementable. Decision variables, such as bus dispatching
+time, bus speed between any two adjacent stops, bus dwell
+time at each stop, OD-based boarding volume, could all
+be described into a holistic model to enable this combined
+dispatching-operation bus management system.
+Also, boarding control has been studied in some literature
+and proved its high efficiency in improving the bus service
+quality. However, it still has not been widely implemented in
+the real world. Fig. 2 gives an graph illustration of future bus
+stop, which makes the boarding control applicable. Imagine
+a message via mobile apps or screen board located at the bus
+stop could be sent or displayed to passengers when a bus is
+approaching the stop, and only passengers at the front queue
+arXiv:2301.12087v1  [eess.SY]  28 Jan 2023
+
+Fig. 1: Framework of traffic light scheduling for vehicle-pedestrian mixed-flow networks
+will be selected to enter the designated boarding area at the
+bus stop. The identification of the front-queue passenger can
+be realized by the camera or Lidar installed at the bus stop.
+When the bus reaches the bus stop, only passengers who are
+waiting at the designated area are allowed to board the bus.
+This requires infrastructural enhancement at the bus stop, not
+only the area re-design but also the advanced sensors.
+Fig. 2: Graphic illustration of future bus stop
+C. Autonomous bus fleet management for mobility-on-
+demand system
+To increase the vehicle utilization and reduce the carbon
+dioxide emissions, the on-line carsharing services have been
+carried out in many cities in the past few years. Also, the
+adoption of the electric vehicles could potentially reduce
+emissions to promote a sustainable environment, meanwhile,
+the forth-coming commercialization of AVs has constantly
+reshaping the transportation system. All of the above emerg-
+ing concepts and technologies have combined and led to an
+autonomous mobility-on-demand system (AMOD). AMOD
+system is not a new concept and has been proposed since
+2014, however, the implementation and research studies
+about AMOD system mainly focus on private cars and
+taxi services, which may bring challenges on the traditional
+public transport, which has been regarded as the symbol of
+equity and accessibility. Therefore, it is strongly necessary
+to identify and explore the synergistic possibilities between
+AMOD and public transports, especially under this transi-
+tional period.
+Fig. 3 illustrates one possible case how AVs could support
+the existing public bus lines. The left subfigure describes
+two traditional bus lines served by corresponding buses
+dispatched from the terminal. The orange bar represents the
+demand size at each bus stop. All control variables mentioned
+in Section III-B could be adopted and tuned to facilitate
+the operation of the traditional bus service. Meanwhile, we
+could also dispatch AVs to serve bus stops on multiple bus
+lines but with high demands due to its flexible on-demand
+characteristics. AVs are normally commercialized as electric
+cars with limited battery capacity, thus, it is inevitable to
+develop a smart AV routing strategy catering for the demands
+of the users as well as the battery capacity level, as illustrated
+in Fig. 4. As AVs do not follow a fixed bus route and on-
+board passengers need to be sent to their destined stops,
+thus, on-demand autonomous bus fleet dispatching strategy
+inevitably requires the boarding control at each bus stop,
+and this enables the system to consider not only the routing
+decision but also the volume dynamics, which is different
+with current one passenger pick-up and drop-off problem.
+The optimization involving in routing, subsequent bus stop
+selection and corresponding boarding volume makes the
+problem more challenging but also interesting.
+Fig. 3: Application of autonomous buses on supporting
+traditional public bus lines
+IV. CONCLUSION
+
+Mixed-Flow Network-based
+Controller Implementationin
+VissiM SignalActivation via
+OptimizationMode
+Gurobi/EvolutionaryAlgorithms
+CoMInterface
+Optimization ParameterPrediction via
+Database
+Real-Time
+Machine LearningAlgorithms
+Database
+Give Way to
+Intra-day
+Pedestrians
+Database
+00:00-
+03:00
+07:00
+02:59
+05:59
+09:59
+Monthly
+Database
+Weekt
+Week4
+Historical
+Database
+JAN
+FEB
+10Waiting
+Area
+(OD-based)
+Notification Screen
+Area
+(Bus-based)Fig. 4: Autonomous bus routing for multi-loading and on-
+road charging
+
+Terminal
+Droppoints
+Pick-up
+Points
+(larger
+points
+denote
+larger
+demands)
+Charging
+pointsShape the Future of ITS –
+Optimization and scheduling for a large scale urban transportation system
+Yi Zhang
+I. INTRODUCTION
+Even if the quality of AVs is continually upgrade, 100% re-
+placement of traditional cars (gasoline cars, hybrid cars, elec-
+tronic cars are not Automation 6) to fully 6-level autonomous
+cars requires not only the safety and security evaluation
+but also corresponding matched physical infrastructures, e.g.,
+non-signalized intersection, thus, it is predictable that traffic
+signal still play an important role in the near future. At the
+period of transition, a feasible and implementable approach
+is proposed in this essay.
+II. OBJECTIVES
+The objective of this essay is to propose a set of coordi-
+nated technological solutions to transform existing transport
+system to a more intelligent interactive system by adopting
+implementable optimization and control methods in the near
+future, thereby improve public services and quality of life
+for residents. In this essay, three different application scenes
+that closely related to people’s daily life are discussed.
+III. WORKING PACKAGES AND TASKS
+A. Traffic light scheduling for vehicle-pedestrian mixed-flow
+networks
+Most traffic signal controllers differentiate vehicles and
+pedestrians, and focus on their great contributions to vehicle
+flows, this is reasonable when pedestrian volume is low.
+However, in the downtown areas where large number of
+pedestrians interfere with vehicular traffic, optimizing traffic
+signals only for vehicles may create more conflicts between
+both traffic participants and potentially reduce the economic
+interest, since pedestrians in CBD areas are usually potential
+customers of nearby shopping malls. According to SGS
+Economics and Planning [], optimized pedestrian flow can
+increase $1.3 billion a year for Melbourne CBD area. On the
+other hand, rich data can be obtained with the help of the
+advanced sensor equipments, such as V2X, 5G communica-
+tion, Lidar and so on. Powerful machine learning algorithms
+can be effectively utilized to help predict the traffic flows
+based on the large amount of database, which better serve
+the optimized signal controller.
+In view of this, we propose this signal controller with the
+aim to fairly minimize both vehicles’ and pedestrians’ delay
+[]. Fig. 1 illustrates the framework of the proposed real-time
+traffic light scheduling strategy implemented in simulator
+Yi Zhang is affiliated with with the Institute for Infocomm Research
+(I2R), Agency for Science, Technology and Research (A*STAR), Singapore
+138632. Emails: zhang yi@i2r.a-star.edu.sg.
+software VISSIM. The macroscopic flow model for both
+pedestrians and vehicles are developed, and the impacts of
+the signal on pedestrian crossing capacity is captured in the
+model. Benefitted from the advanced sensors, signal priority
+level could also be incorporated into the model to enable
+higher priority for public buses. The mixed-flow model is
+then solved by adopting the commercial optimization solver
+or evolutionary algorithms. After that, the optimized signal
+phases and duration are sent to the simulator VISSIM, which
+mimics the real urban traffic environment. Meanwhile, the
+traffic information, such as traffic volumes, turning ratios, is
+stored in the back-end database, which is used to re-train
+the AI models by adopting the machine learning algorithms
+to predict the required information. The predicted traffic
+parameters and current real-time information are all sent
+to the controller side, which is solved in a rolling horizon
+manner, and the whole process continues with the evolve of
+the traffic system to form a closed-loop control strategy.
+B. A combined dispatching-operation strategy for public bus
+management incorporated with the boarding control
+The bus operation systems in most today’s studies are
+proposed based on known bus dispatching time or schedule
+headways/frequencies. On the other hand, the bus dispatch-
+ing systems seldom consider operation control for on-road
+buses. It is understandable that separately deal with these two
+problems could reduce the computational complexity sig-
+nificantly compared with the complicated combined model.
+However, the increasing application and perfection of telem-
+atics (e.g., Automatic Vehicle Location (AVL), Automatic
+Passenger Counting (APC), etc.) in the bus management
+system, the collection of the real-time information becomes
+possible, also, the constantly upgraded computer with more
+computing power is keeping breaking the records, thus, it is
+predictable that to solve a combined optimization problem is
+implementable. Decision variables, such as bus dispatching
+time, bus speed between any two adjacent stops, bus dwell
+time at each stop, OD-based boarding volume, could all
+be described into a holistic model to enable this combined
+dispatching-operation bus management system.
+Also, boarding control has been studied in some literature
+and proved its high efficiency in improving the bus service
+quality. However, it still has not been widely implemented in
+the real world. Fig. 2 gives an graph illustration of future bus
+stop, which makes the boarding control applicable. Imagine
+a message via mobile apps or screen board located at the bus
+stop could be sent or displayed to passengers when a bus is
+approaching the stop, and only passengers at the front queue
+arXiv:2301.12087v1  [eess.SY]  28 Jan 2023
+
+Fig. 1: Framework of traffic light scheduling for vehicle-pedestrian mixed-flow networks
+will be selected to enter the designated boarding area at the
+bus stop. The identification of the front-queue passenger can
+be realized by the camera or Lidar installed at the bus stop.
+When the bus reaches the bus stop, only passengers who are
+waiting at the designated area are allowed to board the bus.
+This requires infrastructural enhancement at the bus stop, not
+only the area re-design but also the advanced sensors.
+Fig. 2: Graphic illustration of future bus stop
+C. Autonomous bus fleet management for mobility-on-
+demand system
+To increase the vehicle utilization and reduce the carbon
+dioxide emissions, the on-line carsharing services have been
+carried out in many cities in the past few years. Also, the
+adoption of the electric vehicles could potentially reduce
+emissions to promote a sustainable environment, meanwhile,
+the forth-coming commercialization of AVs has constantly
+reshaping the transportation system. All of the above emerg-
+ing concepts and technologies have combined and led to an
+autonomous mobility-on-demand system (AMOD). AMOD
+system is not a new concept and has been proposed since
+2014, however, the implementation and research studies
+about AMOD system mainly focus on private cars and
+taxi services, which may bring challenges on the traditional
+public transport, which has been regarded as the symbol of
+equity and accessibility. Therefore, it is strongly necessary
+to identify and explore the synergistic possibilities between
+AMOD and public transports, especially under this transi-
+tional period.
+Fig. 3 illustrates one possible case how AVs could support
+the existing public bus lines. The left subfigure describes
+two traditional bus lines served by corresponding buses
+dispatched from the terminal. The orange bar represents the
+demand size at each bus stop. All control variables mentioned
+in Section III-B could be adopted and tuned to facilitate
+the operation of the traditional bus service. Meanwhile, we
+could also dispatch AVs to serve bus stops on multiple bus
+lines but with high demands due to its flexible on-demand
+characteristics. AVs are normally commercialized as electric
+cars with limited battery capacity, thus, it is inevitable to
+develop a smart AV routing strategy catering for the demands
+of the users as well as the battery capacity level, as illustrated
+in Fig. 4. As AVs do not follow a fixed bus route and on-
+board passengers need to be sent to their destined stops,
+thus, on-demand autonomous bus fleet dispatching strategy
+inevitably requires the boarding control at each bus stop,
+and this enables the system to consider not only the routing
+decision but also the volume dynamics, which is different
+with current one passenger pick-up and drop-off problem.
+The optimization involving in routing, subsequent bus stop
+selection and corresponding boarding volume makes the
+problem more challenging but also interesting.
+Fig. 3: Application of autonomous buses on supporting
+traditional public bus lines
+IV. CONCLUSION
+
+Mixed-Flow Network-based
+Controller Implementationin
+VissiM SignalActivation via
+OptimizationMode
+Gurobi/EvolutionaryAlgorithms
+CoMInterface
+Optimization ParameterPrediction via
+Database
+Real-Time
+Machine LearningAlgorithms
+Database
+Give Way to
+Intra-day
+Pedestrians
+Database
+00:00-
+03:00
+07:00
+02:59
+05:59
+09:59
+Monthly
+Database
+Weekt
+Week4
+Historical
+Database
+JAN
+FEB
+10Waiting
+Area
+(OD-based)
+Notification Screen
+Area
+(Bus-based)Fig. 4: Autonomous bus routing for multi-loading and on-
+road charging
+
+Terminal
+Droppoints
+Pick-up
+Points
+(larger
+points
+denote
+larger
+demands)
+Charging
+points
\ No newline at end of file
diff --git a/lNFLT4oBgHgl3EQfdi98/content/tmp_files/load_file.txt b/lNFLT4oBgHgl3EQfdi98/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/lNFLT4oBgHgl3EQfdi98/content/tmp_files/load_file.txt
@@ -0,0 +1,314 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf,len=313
+page_content='1  Shape the Future of ITS –  Optimization and scheduling for a large scale urban transportation system –  in a fast-changing world  Yi Zhang  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' INTRODUCTION  The rapid growth of the population facilitates the urban  sprawl, automobile production and leads to heavy traffic con-  gestions, pollution, noises and traffic fatalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Therefore, it is  indispensable to develop an eco-friendly sustainable transport  with high performance of the existing traffic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Although  many transformational products have been developed and used  on the road, such as autonomous vehicles (AV) and drones,  and their quality is continually upgraded, 100% replacement of  traditional cars to AVs requires not only the safety and security  evaluation but also corresponding physical infrastructures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=',  non-signalized intersection, charging stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Also, we still  have a long way to go in order to completely commercialize  AVs with level 5 automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Thus, it is predictable that traffic  signals will still play an important role for a certain long time  in this hybrid transportation world, which involves both human-  driven cars and AVs under different automated levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' At the  period of transition, a feasible and implementable approach is  more helpful and practical to reshape the transportation system  in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  The concept of the pedestrian/transit-oriented transport,  which is designed for the human body instead of the car body,  has been proposed by researchers [2]: “cities with very high  walking, cycling and transit mode share (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=', 75% or more)  typically have high density, mixed use urban centres at or  above 100-200 people per hectare and are supported by a  transportation strategy that prioritizes pedestrians first, then  cyclists followed by transit users.” However, vehicle traffic  still gets excessively high attention in many cities’ guidelines,  and walkability is seldom taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Also, walking is  the access to public transport (PT), thereby PT services could  accordingly be promoted if the pedestrian walking safety and  pleasantness can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' PT system has been studied for  several decades due to its large ridership and sustainability  on economic efficiency, environmental protection and social  equity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Proper bus dispatching and operation can attract more  passengers, which encourages commuters to change their travel  mode from private automobiles to public buses to further  alleviate the traffic congestion and air pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  However, PT system, a typical ride-sharing transport serving  a variety of access needs and providing an equal social value,  now faces the challenges from the mobility-on-demand (MOD)  system, which is famed for its easy transactions and convenient  access via mobile phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Therefore, we need to explore and  create a win-win cooperation model between both PT and AV  systems at current transitional period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The flexible demand-  Yi Zhang is affiliated with with the Institute for Infocomm Research (I2R),  Agency for Science, Technology and Research (A*STAR), Singapore 138632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Emails: yzhang120@e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='sg(zhang yi@i2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='a-star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  driven pattern in AVs could make up the shortages due to fixed  routes in the PT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' On the other hand, PT system is still  coping with the high-volume transfer tasks, which may not be  feasible for AVs due to the safety concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  The objective of this essay is to propose a set of coordi-  nated technological solutions to transform existing transport  system to a more intelligent interactive system by adopting  optimization and control methods implementable in the near  future, thereby improving public services and quality of life for  residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' In this essay, three different application scenes that  closely related to people’s daily life are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' We firstly  propose a traffic light scheduling strategy via model predic-  tive control (MPC) method, with the aim to fairly minimize  both pedestrians’ and vehicles’ delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' After that, a combined  dispatching-operation system is proposed to further increase  the control flexibility, and corresponding implementation solu-  tion for boarding control is also illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Finally, a possible  scheme to combine both PT and AV systems is proposed to  improve existing PT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' WORKING PACKAGES AND TASKS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Traffic light scheduling for vehicle-pedestrian mixed-flow  networks  Most traffic signal controllers differentiate vehicles and  pedestrians, and focus on their great contributions to vehicle  flows, this is reasonable when pedestrian volume is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' How-  ever, in the downtown areas where large number of pedestrians  interfere with vehicular traffic, optimizing traffic signals only  for vehicles may create more conflicts between both traffic  participants and potentially reduce the economic interest, since  pedestrians in CBD areas are usually potential customers of  nearby shopping malls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' According to SGS Economics and  Planning [1], optimized pedestrian flow can increase $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='3  billion a year for Melbourne CBD area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' On the other hand,  rich data can be obtained with the help of the advanced sensor  equipments, such as V2X, 5G communication, Lidar and so  on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Powerful machine learning algorithms can be effectively  utilized to help predict the traffic flows based on the large  amount of database, which better serve the optimized signal  controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  In view of this, we propose this signal controller with the  aim to fairly minimize both vehicles’ and pedestrians’ delay  [4][7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 1 illustrates the framework of the proposed real-  time traffic light scheduling strategy implemented in simula-  tor software VISSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The macroscopic flow model for both  pedestrians and vehicles are developed, and the impacts of  the signal on pedestrian crossing capacity is captured in the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Benefitted from the advanced sensors, signal priority  level could also be incorporated into the model to enable  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='12087v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='SY]  28 Jan 2023  2  higher priority for public buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The mixed-flow model is  then solved by adopting the commercial optimization solver  or evolutionary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' After that, the optimized signal  phases and duration are sent to the simulator VISSIM, which  mimics the real urban traffic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Meanwhile, the  traffic information, such as traffic volumes, turning ratios, is  stored in the back-end database, which is used to re-train the AI  models by adopting the machine learning algorithms to predict  the required information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The predicted traffic parameters and  current real-time information are all sent to the controller side,  which is solved in a rolling horizon manner, and the whole  process continues with the evolve of the traffic system to form  a closed-loop control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' A combined dispatching-operation strategy for public bus  management incorporated with the boarding control  The bus operation systems in most today’s studies are  proposed based on known bus dispatching time or schedule  headways/frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' On the other hand, the bus dispatching  systems seldom consider operation control for on-road buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  It is understandable that separately deal with these two prob-  lems could reduce the computational complexity significantly  compared with the complicated combined model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' However, the  increasing application and perfection of telematics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=', Auto-  matic Vehicle Location (AVL), Automatic Passenger Counting  (APC), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=') in the bus management system, the collection of  the real-time information becomes possible, also, the constantly  upgraded computer with more computing power is keeping  breaking the records, thus, it is predictable that to solve a  combined optimization problem is implementable [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Decision  variables, such as bus dispatching time, bus speed between any  two adjacent stops, bus dwell time at each stop, OD-based  boarding volume, could all be described into a holistic model  to enable this combined dispatching-operation bus management  system [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Also, boarding control has been studied in some  literature and proved its high efficiency in improving the  bus service quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' However, it still has not been widely  implemented in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 2 gives an graph illustration of future bus stop, which  makes the boarding control applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Imagine a message via  mobile apps or screen board located at the bus stop could be  sent or displayed to passengers when a bus is approaching the  stop, and only passengers at the front queue will be selected  to enter the designated boarding area at the bus stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The  identification of the front-queue passenger can be realized by  the camera or Lidar installed at the bus stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' When the bus  reaches the bus stop, only passengers who are waiting at the  designated area are allowed to board the bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' This requires  infrastructural enhancement at the bus stop, not only the area  re-design but also the advanced sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Autonomous bus fleet management for mobility-on-demand  system  To increase the vehicle utilization and reduce the carbon  dioxide emissions, the on-line carsharing services have been  carried out in many cities in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Also, the adop-  tion of the electric vehicles could potentially reduce emissions  to promote a sustainable environment, meanwhile, the forth-  coming commercialization of AVs has constantly reshaping  the transportation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' All of the above emerging concepts  and technologies have combined and led to an autonomous  mobility-on-demand system (AMOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' AMOD system is not a  new concept and has been proposed since 2014 [3], however,  the implementation and research studies about AMOD system  mainly focus on private cars and taxi services, and this may  bring challenges on the traditional public transport, which  has been regarded as the symbol of equity and accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Therefore, it is strongly necessary to identify and explore the  synergistic possibilities between AMOD and public transports,  especially under this transitional period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 3 illustrates one possible case capturing how AVs could  support the existing public bus lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The left subfigure de-  scribes two traditional bus lines served by corresponding buses  dispatched from the terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The orange bar represents the  demand size at each bus stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' All control variables mentioned  in Section II-B could be adopted and tuned to facilitate the op-  eration of the traditional bus service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Meanwhile, we could also  dispatch AVs to serve bus stops on multiple bus lines but with  high demands due to its flexible on-demand characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  AVs are normally commercialized as electric cars with limited  battery capacity, thus, it is inevitable to develop a smart AV  routing strategy catering for the demands of the users as well  as the battery capacity level, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' As AVs  do not follow a fixed bus route and on-board passengers need  to be sent to their destined stops, thus, on-demand autonomous  bus fleet dispatching strategy inevitably requires the boarding  control at each bus stop, and this enables the system to consider  not only the routing decision but also the volume dynamics,  which is different with current one passenger pick-up and drop-  off problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The optimization involving in routing, subsequent  bus stop selection and corresponding boarding volume makes  the problem more challenging but also interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' CONCLUSION  In  this  essay,  technological  solutions  from  a  pedestrian/transit-oriented perspective have been provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Firstly,  an  adaptive  traffic  signal  control  framework  is  presented, with consideration of a macroscopic mixed-flow  optimization model, VISSIM simulation platform, historical  database and machine learning-enabled AI prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  The experiment results in our paper [4] have demonstrated  that the proposed strategy in a Manhattan-shaped network  can strike a good balance between pedestrians’ and vehicle  drivers’ needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Therefore, we are optimistic to implement this  strategy in the real world to enhance the existing traffic signals  in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Next, the bus dispatching and its on-road  operation, especially the boarding control, are combined  together to minimize the passenger delay time and the  operating bus vacancy, which makes our strategy more flexible  and adaptive to meet the passenger demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' In our previous  study [6], a multi-bus dispatching and boarding control  strategy can reduce roughly 50% of remaining passenger  volumes when compared with the timetable-based fixed  schedule, which is quite promising and exciting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Moreover,  to achieve a synergistic objective between PT system and  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 1: Framework of traffic light scheduling for vehicle-pedestrian mixed-flow networks  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 2: Graphic illustration of future bus stop  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 3: Application of autonomous buses on supporting tradi-  tional public bus lines  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 4: Autonomous bus routing for multi-loading and on-road  charging  AV system, a scheme is proposed, where AVs are dispatched  to pick up passengers at high-demand stops to support the  PT lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' This scheme shall be further studied in my future  work to incorporate multiple optimized variables, including  AV routing, volume dynamics and boarding limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Overall,  the essay presents three application scenes in the intelligent  transportation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The suggested strategies and guidelines  can assist in developing an intelligent future urban transport  system in order to provide citizens a smarter, safer and more  interactive transportation experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  REFERENCES  [1] SGS Economics and Planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' CBD pedestrian analysis, technical report  city of melbourn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Technical report, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  [2] Jennie Moore, Kirstin Mille, Richard Registe, Sarah Campbell, Julian  Zelazny, and Sven Eberlein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  International ecocity standards(brochure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Technical report, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  [3] Kevin Spieser, Kyle Treleaven, Rick Zhang, Emilio Frazzoli, Daniel  Morton, and Marco Pavone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Toward a systematic approach to the design  and evaluation of automated mobility-on-demand systems: A case study  in singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' In Road vehicle automation, pages 229–245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  [4] Yi Zhang, Kaizhou Gao, Yicheng Zhang, and Rong Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Traffic light  scheduling for pedestrian-vehicle mixed-flow networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' IEEE Transac-  tions on Intelligent Transportation Systems, 20(4):1468–1483, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  [5] Yi Zhang, Rong Su, Yicheng Zhang, and Nadeesha Sandamali Gammana  Guruge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' A multi-bus dispatching strategy based on boarding control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' IEEE  Transactions on Intelligent Transportation Systems, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  [6] Yi Zhang, Rong Su, Yicheng Zhang, and Bohui Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Dynamic multi-bus  dispatching strategy with boarding and holding control for passenger delay  alleviation and schedule reliability: A combined dispatching-operation  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' IEEE Transactions on Intelligent Transportation Systems, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  [7] Yi Zhang, Yicheng Zhang, and Rong Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Pedestrian-safety-aware traffic  light control strategy for urban traffic congestion alleviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  IEEE  Transactions on Intelligent Transportation Systems, 22(1):178–193, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Yi Zhang (S’17-M’21) Yi Zhang received her Bache-  lor degree of Engineering from Shandong University,  China in 2014, and the PhD degree in Electrical and  Electronic Engineering from Nanyang Technological  University, Singapore in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' She is currently a  research scientist at the Institute for Infocomm Re-  search (I2R) in the Agency for Science, Technology  and Research, Singapore (A*STAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Her research  interests focus on intelligent transportation system,  including urban traffic flow management, model-  based traffic signal scheduling, lane change predic-  tion and bus dispatching and operation management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Mixed-Flow Network-based  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Controller Implementationin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='VissiM SignalActivation via  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='OptimizationMode  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Gurobi/EvolutionaryAlgorithms  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
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+page_content='Database  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
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+page_content='Database  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Give Way to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
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+page_content='Pedestrians  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Database  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
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+page_content='Week4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Historical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Database  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='JAN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='FEB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='10Waiting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Area  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='(OD-based)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Notification Screen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Area  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='(Bus-based)Terminal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Droppoints  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Pick-up  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Points  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='(larger  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='points  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='denote  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='larger  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='demands)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Charging  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='pointsShape the Future of ITS –  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Optimization and scheduling for a large scale urban transportation system  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='Yi Zhang  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' INTRODUCTION  Even if the quality of AVs is continually upgrade, 100% re-  placement of traditional cars (gasoline cars, hybrid cars, elec-  tronic cars are not Automation 6) to fully 6-level autonomous  cars requires not only the safety and security evaluation  but also corresponding matched physical infrastructures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=',  non-signalized intersection, thus, it is predictable that traffic  signal still play an important role in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' At the  period of transition, a feasible and implementable approach  is proposed in this essay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' OBJECTIVES  The objective of this essay is to propose a set of coordi-  nated technological solutions to transform existing transport  system to a more intelligent interactive system by adopting  implementable optimization and control methods in the near  future, thereby improve public services and quality of life  for residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' In this essay, three different application scenes  that closely related to people’s daily life are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' WORKING PACKAGES AND TASKS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Traffic light scheduling for vehicle-pedestrian mixed-flow  networks  Most traffic signal controllers differentiate vehicles and  pedestrians, and focus on their great contributions to vehicle  flows, this is reasonable when pedestrian volume is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  However, in the downtown areas where large number of  pedestrians interfere with vehicular traffic, optimizing traffic  signals only for vehicles may create more conflicts between  both traffic participants and potentially reduce the economic  interest, since pedestrians in CBD areas are usually potential  customers of nearby shopping malls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' According to SGS  Economics and Planning [], optimized pedestrian flow can  increase $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='3 billion a year for Melbourne CBD area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' On the  other hand, rich data can be obtained with the help of the  advanced sensor equipments, such as V2X, 5G communica-  tion, Lidar and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Powerful machine learning algorithms  can be effectively utilized to help predict the traffic flows  based on the large amount of database, which better serve  the optimized signal controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  In view of this, we propose this signal controller with the  aim to fairly minimize both vehicles’ and pedestrians’ delay  [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 1 illustrates the framework of the proposed real-time  traffic light scheduling strategy implemented in simulator  Yi Zhang is affiliated with with the Institute for Infocomm Research  (I2R), Agency for Science, Technology and Research (A*STAR), Singapore  138632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Emails: zhang yi@i2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='a-star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  software VISSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The macroscopic flow model for both  pedestrians and vehicles are developed, and the impacts of  the signal on pedestrian crossing capacity is captured in the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Benefitted from the advanced sensors, signal priority  level could also be incorporated into the model to enable  higher priority for public buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The mixed-flow model is  then solved by adopting the commercial optimization solver  or evolutionary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' After that, the optimized signal  phases and duration are sent to the simulator VISSIM, which  mimics the real urban traffic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Meanwhile, the  traffic information, such as traffic volumes, turning ratios, is  stored in the back-end database, which is used to re-train  the AI models by adopting the machine learning algorithms  to predict the required information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The predicted traffic  parameters and current real-time information are all sent  to the controller side, which is solved in a rolling horizon  manner, and the whole process continues with the evolve of  the traffic system to form a closed-loop control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' A combined dispatching-operation strategy for public bus  management incorporated with the boarding control  The bus operation systems in most today’s studies are  proposed based on known bus dispatching time or schedule  headways/frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' On the other hand, the bus dispatch-  ing systems seldom consider operation control for on-road  buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' It is understandable that separately deal with these two  problems could reduce the computational complexity sig-  nificantly compared with the complicated combined model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  However, the increasing application and perfection of telem-  atics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=', Automatic Vehicle Location (AVL), Automatic  Passenger Counting (APC), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=') in the bus management  system, the collection of the real-time information becomes  possible, also, the constantly upgraded computer with more  computing power is keeping breaking the records, thus, it is  predictable that to solve a combined optimization problem is  implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Decision variables, such as bus dispatching  time, bus speed between any two adjacent stops, bus dwell  time at each stop, OD-based boarding volume, could all  be described into a holistic model to enable this combined  dispatching-operation bus management system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Also, boarding control has been studied in some literature  and proved its high efficiency in improving the bus service  quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' However, it still has not been widely implemented in  the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 2 gives an graph illustration of future bus  stop, which makes the boarding control applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Imagine  a message via mobile apps or screen board located at the bus  stop could be sent or displayed to passengers when a bus is  approaching the stop, and only passengers at the front queue  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='12087v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='SY]  28 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 1: Framework of traffic light scheduling for vehicle-pedestrian mixed-flow networks  will be selected to enter the designated boarding area at the  bus stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The identification of the front-queue passenger can  be realized by the camera or Lidar installed at the bus stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  When the bus reaches the bus stop, only passengers who are  waiting at the designated area are allowed to board the bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  This requires infrastructural enhancement at the bus stop, not  only the area re-design but also the advanced sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 2: Graphic illustration of future bus stop  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Autonomous bus fleet management for mobility-on-  demand system  To increase the vehicle utilization and reduce the carbon  dioxide emissions, the on-line carsharing services have been  carried out in many cities in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Also, the  adoption of the electric vehicles could potentially reduce  emissions to promote a sustainable environment, meanwhile,  the forth-coming commercialization of AVs has constantly  reshaping the transportation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' All of the above emerg-  ing concepts and technologies have combined and led to an  autonomous mobility-on-demand system (AMOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' AMOD  system is not a new concept and has been proposed since  2014, however, the implementation and research studies  about AMOD system mainly focus on private cars and  taxi services, which may bring challenges on the traditional  public transport, which has been regarded as the symbol of  equity and accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Therefore, it is strongly necessary  to identify and explore the synergistic possibilities between  AMOD and public transports, especially under this transi-  tional period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 3 illustrates one possible case how AVs could support  the existing public bus lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The left subfigure describes  two traditional bus lines served by corresponding buses  dispatched from the terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The orange bar represents the  demand size at each bus stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' All control variables mentioned  in Section III-B could be adopted and tuned to facilitate  the operation of the traditional bus service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Meanwhile, we  could also dispatch AVs to serve bus stops on multiple bus  lines but with high demands due to its flexible on-demand  characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' AVs are normally commercialized as electric  cars with limited battery capacity, thus, it is inevitable to  develop a smart AV routing strategy catering for the demands  of the users as well as the battery capacity level, as illustrated  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' As AVs do not follow a fixed bus route and on-  board passengers need to be sent to their destined stops,  thus, on-demand autonomous bus fleet dispatching strategy  inevitably requires the boarding control at each bus stop,  and this enables the system to consider not only the routing  decision but also the volume dynamics, which is different  with current one passenger pick-up and drop-off problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  The optimization involving in routing, subsequent bus stop  selection and corresponding boarding volume makes the  problem more challenging but also interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 3: Application of autonomous buses on supporting  traditional public bus lines  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' CONCLUSION  Mixed-Flow Network-based  Controller Implementationin  VissiM SignalActivation via  OptimizationMode  Gurobi/EvolutionaryAlgorithms  CoMInterface  Optimization ParameterPrediction via  Database  Real-Time  Machine LearningAlgorithms  Database  Give Way to  Intra-day  Pedestrians  Database  00:00-  03:00  07:00  02:59  05:59  09:59  Monthly  Database  Weekt  Week4  Historical  Database  JAN  FEB  10Waiting  Area  (OD-based)  Notification Screen  Area  (Bus-based)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 4: Autonomous bus routing for multi-loading and on-  road charging  Terminal  Droppoints  Pick-up  Points  (larger  points  denote  larger  demands)  Charging  pointsShape the Future of ITS –  Optimization and scheduling for a large scale urban transportation system  Yi Zhang  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' INTRODUCTION  Even if the quality of AVs is continually upgrade, 100% re-  placement of traditional cars (gasoline cars, hybrid cars, elec-  tronic cars are not Automation 6) to fully 6-level autonomous  cars requires not only the safety and security evaluation  but also corresponding matched physical infrastructures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=',  non-signalized intersection, thus, it is predictable that traffic  signal still play an important role in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' At the  period of transition, a feasible and implementable approach  is proposed in this essay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' OBJECTIVES  The objective of this essay is to propose a set of coordi-  nated technological solutions to transform existing transport  system to a more intelligent interactive system by adopting  implementable optimization and control methods in the near  future, thereby improve public services and quality of life  for residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' In this essay, three different application scenes  that closely related to people’s daily life are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' WORKING PACKAGES AND TASKS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Traffic light scheduling for vehicle-pedestrian mixed-flow  networks  Most traffic signal controllers differentiate vehicles and  pedestrians, and focus on their great contributions to vehicle  flows, this is reasonable when pedestrian volume is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  However, in the downtown areas where large number of  pedestrians interfere with vehicular traffic, optimizing traffic  signals only for vehicles may create more conflicts between  both traffic participants and potentially reduce the economic  interest, since pedestrians in CBD areas are usually potential  customers of nearby shopping malls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' According to SGS  Economics and Planning [], optimized pedestrian flow can  increase $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='3 billion a year for Melbourne CBD area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' On the  other hand, rich data can be obtained with the help of the  advanced sensor equipments, such as V2X, 5G communica-  tion, Lidar and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Powerful machine learning algorithms  can be effectively utilized to help predict the traffic flows  based on the large amount of database, which better serve  the optimized signal controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  In view of this, we propose this signal controller with the  aim to fairly minimize both vehicles’ and pedestrians’ delay  [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 1 illustrates the framework of the proposed real-time  traffic light scheduling strategy implemented in simulator  Yi Zhang is affiliated with with the Institute for Infocomm Research  (I2R), Agency for Science, Technology and Research (A*STAR), Singapore  138632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Emails: zhang yi@i2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='a-star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  software VISSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The macroscopic flow model for both  pedestrians and vehicles are developed, and the impacts of  the signal on pedestrian crossing capacity is captured in the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Benefitted from the advanced sensors, signal priority  level could also be incorporated into the model to enable  higher priority for public buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The mixed-flow model is  then solved by adopting the commercial optimization solver  or evolutionary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' After that, the optimized signal  phases and duration are sent to the simulator VISSIM, which  mimics the real urban traffic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Meanwhile, the  traffic information, such as traffic volumes, turning ratios, is  stored in the back-end database, which is used to re-train  the AI models by adopting the machine learning algorithms  to predict the required information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The predicted traffic  parameters and current real-time information are all sent  to the controller side, which is solved in a rolling horizon  manner, and the whole process continues with the evolve of  the traffic system to form a closed-loop control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' A combined dispatching-operation strategy for public bus  management incorporated with the boarding control  The bus operation systems in most today’s studies are  proposed based on known bus dispatching time or schedule  headways/frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' On the other hand, the bus dispatch-  ing systems seldom consider operation control for on-road  buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' It is understandable that separately deal with these two  problems could reduce the computational complexity sig-  nificantly compared with the complicated combined model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  However, the increasing application and perfection of telem-  atics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=', Automatic Vehicle Location (AVL), Automatic  Passenger Counting (APC), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=') in the bus management  system, the collection of the real-time information becomes  possible, also, the constantly upgraded computer with more  computing power is keeping breaking the records, thus, it is  predictable that to solve a combined optimization problem is  implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Decision variables, such as bus dispatching  time, bus speed between any two adjacent stops, bus dwell  time at each stop, OD-based boarding volume, could all  be described into a holistic model to enable this combined  dispatching-operation bus management system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Also, boarding control has been studied in some literature  and proved its high efficiency in improving the bus service  quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' However, it still has not been widely implemented in  the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 2 gives an graph illustration of future bus  stop, which makes the boarding control applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Imagine  a message via mobile apps or screen board located at the bus  stop could be sent or displayed to passengers when a bus is  approaching the stop, and only passengers at the front queue  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='12087v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='SY]  28 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 1: Framework of traffic light scheduling for vehicle-pedestrian mixed-flow networks  will be selected to enter the designated boarding area at the  bus stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The identification of the front-queue passenger can  be realized by the camera or Lidar installed at the bus stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  When the bus reaches the bus stop, only passengers who are  waiting at the designated area are allowed to board the bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  This requires infrastructural enhancement at the bus stop, not  only the area re-design but also the advanced sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 2: Graphic illustration of future bus stop  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Autonomous bus fleet management for mobility-on-  demand system  To increase the vehicle utilization and reduce the carbon  dioxide emissions, the on-line carsharing services have been  carried out in many cities in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Also, the  adoption of the electric vehicles could potentially reduce  emissions to promote a sustainable environment, meanwhile,  the forth-coming commercialization of AVs has constantly  reshaping the transportation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' All of the above emerg-  ing concepts and technologies have combined and led to an  autonomous mobility-on-demand system (AMOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' AMOD  system is not a new concept and has been proposed since  2014, however, the implementation and research studies  about AMOD system mainly focus on private cars and  taxi services, which may bring challenges on the traditional  public transport, which has been regarded as the symbol of  equity and accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Therefore, it is strongly necessary  to identify and explore the synergistic possibilities between  AMOD and public transports, especially under this transi-  tional period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 3 illustrates one possible case how AVs could support  the existing public bus lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The left subfigure describes  two traditional bus lines served by corresponding buses  dispatched from the terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' The orange bar represents the  demand size at each bus stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' All control variables mentioned  in Section III-B could be adopted and tuned to facilitate  the operation of the traditional bus service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' Meanwhile, we  could also dispatch AVs to serve bus stops on multiple bus  lines but with high demands due to its flexible on-demand  characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' AVs are normally commercialized as electric  cars with limited battery capacity, thus, it is inevitable to  develop a smart AV routing strategy catering for the demands  of the users as well as the battery capacity level, as illustrated  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' As AVs do not follow a fixed bus route and on-  board passengers need to be sent to their destined stops,  thus, on-demand autonomous bus fleet dispatching strategy  inevitably requires the boarding control at each bus stop,  and this enables the system to consider not only the routing  decision but also the volume dynamics, which is different  with current one passenger pick-up and drop-off problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  The optimization involving in routing, subsequent bus stop  selection and corresponding boarding volume makes the  problem more challenging but also interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 3: Application of autonomous buses on supporting  traditional public bus lines  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' CONCLUSION  Mixed-Flow Network-based  Controller Implementationin  VissiM SignalActivation via  OptimizationMode  Gurobi/EvolutionaryAlgorithms  CoMInterface  Optimization ParameterPrediction via  Database  Real-Time  Machine LearningAlgorithms  Database  Give Way to  Intra-day  Pedestrians  Database  00:00-  03:00  07:00  02:59  05:59  09:59  Monthly  Database  Weekt  Week4  Historical  Database  JAN  FEB  10Waiting  Area  (OD-based)  Notification Screen  Area  (Bus-based)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
+page_content=' 4: Autonomous bus routing for multi-loading and on-  road charging  Terminal  Droppoints  Pick-up  Points  (larger  points  denote  larger  demands)  Charging  points' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFLT4oBgHgl3EQfdi98/content/2301.12087v1.pdf'}
diff --git a/ldE1T4oBgHgl3EQf0wUL/content/tmp_files/2301.03459v1.pdf.txt b/ldE1T4oBgHgl3EQf0wUL/content/tmp_files/2301.03459v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8c44b0ff05c8e2ef21fdddd0b5b38110f4e265c5
--- /dev/null
+++ b/ldE1T4oBgHgl3EQf0wUL/content/tmp_files/2301.03459v1.pdf.txt
@@ -0,0 +1,287 @@
+RaspberryPi to measure EEG, ECG, EMG and EOG with Shield PiEEG  
+ 
+In this paper presented hardware and software for shield PiEEG for reading signals through the families 
+of single-board computers - RaspberryPi, OrangePi, BananaPi, etc. For the most part, the paper provides 
+technical information on how to implement this device. This device is designed to be familiar with 
+neuroscience and is one of the easiest ways to get started with EEG measurements. 
+ 
+Ildar Rakhmatulin, PhD, PiEEG, ildarr2016@gmail.com  
+Source https://github.com/Ildaron/EEGwithRaspberryPI  
+Presentation https://youtu.be/uK8QF2liO5U  
+ 
+Keywords: RaspberryPi and EEG, ECG, EMG and EOG; brain-computer interface; shield for 
+RaspberryPi 
+ 
+1. Introduction 
+The brain-computer interface is a device for reading brain signals in order to identify any correlations that 
+can be used for practical purposes. In 2021 we developed the brain-computer interface - ironbci [1,2,3] 
+but the chip shortage significantly increased the cost of the device, after which we switched to PiEEG 
+shield, which made it possible to reduce the cost of the device and simplify the installation process. The 
+PiEEG device was presented in general terms at the conference [4] and in the publication [10]. In this 
+paper we will focus more on the technical details of the implementation of this device. 
+ 
+2. Safety recommendation 
+The developed device was tested only for Raspberry Pi. During the test, it is forbidden connect the device 
+to the electrical source, this need for safety and to avoid network interference. Cannot be used this device 
+when powered via an electrical network and used it only when working with 5V batteries (capacity– not 
+mote 2000 mAh). General view of the complete assembly of the device in Fig.1. 
+ 
+ 
+Fig.1 General View of assembled device: 1 – hat EEG, 2 – monitor, 3 – RaspberryPI and shield EEG, 4 – 
+power bank 
+ 
+All parts should be connected only with battery – monitor, keyboard, mouse, RaspberryPI and PiEEG 
+device. The device is connected only directly through GPIO40 to the Raspberry Pi. There must not be any 
+connections for any elements with the electrical network.  
+Also observe the following protective measures: 
+- 
+If the device fails (heating of components, smoke etc. ), switch off immediately; 
+- 
+When you have finished using any electrical appliance, be sure to turn it off and unplug it; 
+- 
+Never try to fix a malfunction in an electrical appliance yourself; 
+- 
+Do not touch the components on the board when the device is turned on. 
+ 
+3. Connecting the device 
+The device is designed to allow single-board computers to read the EEG signal via the SPI protocol from 
+ADS1299. The next single computers can be connected (not tested yet) - OrangePi, BananaPi, and the 
+device can be connected via wires to the input of the microcontroller if necessary, Fig.2. 
+
+2
+3
+4 
+Fig.2. Device connection scheme, GPIO40 
+ 
+4. Software  
+Data transfer between Raspberry Pi and ADS1299 using the SPI protocol is written in C at a rate of 250 
+samples per second for each channel. Data processing realized in Python 3. Software files details from 
+https://github.com/Ildaron/EEGwithRaspberryPI/tree/master/GUI :  
+libperiphery.a – static library compiled from https://github.com/vsergeev/c-periphery , 
+real_time.py – script for editing processing signal and data visualization, 
+real_time_massive.c – header call, 
+real_time_massive.h – register setting for data transfer protocol, 
+super_real_time_massive.so – compiled dynamic library. 
+ 
+To start the reading EEG signals is possible to write software or download from the git 
+https://github.com/Ildaron/EEGwithRaspberryPI , then go to the folder - GUI and run the file - 
+real_time.py. An example of reading is shown in the figure in Fig.3. 
+ 
+Fig.3. EEG signals after band pass filter 1-30 Hz  
+ 
+Through the python script, is possible to configure the parameters for band pass filters. Signal processing 
+is implemented using the standard scipy library. Through the C file, once per second we receive a data 
+packet of 2000 data, 250 samples per second for each channel. 
+Currently we used the next setting for registers for ADS1299:                  
+ 
+          write_reg (0x14,0x80);//led 
+ 
+          write_reg (0x05,0x00);//ch1 
+ 
+          write_reg (0x06,0x01);//ch2 
+ 
+          write_reg (0x07,0x01);//ch3 
+ 
+          write_reg (0x08,0x01);//ch4 
+ 
+           write_reg (0x0D,0xFF);//bias N           
+ 
+           write_reg (0x0E,0x00);//bias P 
+           
+ 
+          write_reg (0x09,0x01);//ch5 
+ 
+          write_reg (0x0A,0x01);//ch6 
+ 
+          write_reg (0x0B,0x01);//ch7 
+ 
+          write_reg (0x0C,0x01);//ch8 
+
+PWR
+CS
+Ground
+GPIO14
+UARTO TXD
+GPIO15
+UARTORXD
+GPIO18
+Ground
+GPIO23
+GPIO24
+Ground
+GPIO25
+SPIOCE1N
+IDEEPRO
+PCM.CLK
+GPI08
+SPIOCEO
+GPI07
+Ground
+GPIO12
+Ground
+GPIO16
+GPIO20
+GPI021
+GO
+6
+8
+2
+2
+2
+忆
+2
+2
+3
+3
+3
+5
+9
+3
+5
+9
+7
+1
+5
+2
+3
+3
+EAE
+GPIO2
+GPIO3
+SCL112C
+GPIO4
+aJIM-T
+Ground
+GPIO17
+GPIO27
+GPIO22
+EAE
+GPIO10
+SONOldS
+GPIO9
+GPIO11
+Ground
+IDSD
+GPIO5
+GPIO6
+GPIO13
+GPIO19
+GPIO26
+Ground
+MOSI
+SCLK
+DRDYGND
+MISO
+JMP1
+JMP3
+JMP250
+50
+50
+50
+50
+50
+100 
+          write_reg (0x15,0x20);// mics 
+ 
+          write_reg (0x01,0x96);// reg1 
+ 
+          write_reg (0x02,0xD4);// reg2  
+ 
+          write_reg (0x03,0xE0);// reg3 
+ 
+          send_command (0x10);//sdatc 
+ 
+          send_command (0x08);//start 
+ 
+To change the ADS1299 registers, needs to work with the header file - real_time_massive.h, after any 
+changes, need to recompile the static library, following the next steps for GPIO library for Linux from 
+https://github.com/vsergeev/c-periphery:  
+$ mkdir build 
+$ cd build 
+$ cmake .. 
+$ make 
+That allows receiving the Static Library - libperiphery.a and finally to create a dynamic library for .so - 
+gcc 
+-shared 
+-home/pi/Desktop/new_spi/c-periphery-master/src 
+real_time_massive.c 
+/home/pi/Desktop/new_spi/c-periphery-master/build/libperiphery.a -o super_real_time_massive.so  
+This library can be imported in python script with signal processing (example - real_time.py) via 
+standard library – ctypes. 
+In case of errors incompatibility in a python IDE, it is necessary to check the versions of installed libraries 
+https://github.com/Ildaron/EEGwithRaspberryPI/blob/master/pip.txt 
+The PIEEG device has a diode connected to the ADS1299, which lights up when the PiEEG device is 
+initialized.  If the life bit is not burning, it is necessary to check the power (Fig.4) in the next points: 
+TP1: – 2.5V 
+TP2:  3.3V 
+TP3:  2.5V 
+ 
+Fig.4. Test points in the PiEEG board  
+ 
+If there is noise in the measured EEG signals that goes beyond the limits declared in the device it is 
+necessary to check the voltage in the next points: 
+           VCAP1 = -1.2V 
+           VCAP2 = 0.001V 
+           VCAP3 = 4.2V 
+           VCAP4=-0.125V 
+ 
+5. Conclusion and perspectives 
+This device is positioned as an easy way to start measuring EEG signals. We introduced hardware and 
+software for reading and processing data in real-time in open source format. We demonstrated how to 
+control a robot using a device [5], in the future we will work on the P300 and on other paradigms, then we 
+plan to gather a team of like-minded people and collect the necessary amount of data to control robots 
+through the motor imagery method. Fig.5 shows a block diagram for the possibility areas of use of the 
+device. 
+ 
+
+2.5V
+I-2.5V
+Power led
+3.3V
+PIEEG
+Led of life 
+Fig.5. possibility areas of use of the PiEEG 
+ 
+Additional Information 
+            Machine learning to identify alcoholism [9], 
+            Overview of dry electrodes [6], 
+            Signal processing [7,8] . 
+ 
+References  
+1. Rakhmatulin, I., Parfenov, A., Traylor, Z. et al. (2021). Low-cost brain computer interface for everyday 
+use. ExpBrain Res 239, 3573–3583. https://doi.org/10.1007/s00221-021-06231-49  
+2. Rakhmatulin, I. (2021). ironbci - Open-source Brain-computer interface with the embedded board to 
+monitor the physiological subject's condition and environmental parameters arXiv:2111.03656 
+3. Rakhmatulin, I . (2021). The electronic board to replace the reference voltage on the earlobe for EEG 
+measurement. March 2021Measurement 173 DOI: 10.1016/j.measurement.2020.108673 
+4. Rakhmatulin, I.  et al. (2021).  Raspberry PI Shield - for measure EEG (PIEEG). Conference: 2021 5th 
+International Conference on Electrical, Electronics, Communication, Computer Technologies and 
+Optimization 
+Techniques 
+(ICEECCOT). 
+DOI: 
+10.1109/ICEECCOT52851.2021.9707969 
+5. I Rakhmatulin, S Volkl. (2022). Brain-Computer-Interface controlled robot via RaspberryPi and PiEEG, 
+arXiv preprint arXiv:2202.01936 
+6. Rakhmatulin, I., Gan, Y. Review Dry and Non-Contact EEG Electrodes for 2010-2021 Years. Preprints 
+2021, 2021030555. doi:10.20944/preprints202103.0555.v1 
+7.  Rakhmatulin, I. (2020). Review of EEG Feature Selection by Neural Networks. Int. J. Sci. Bus. 2020, 
+4, 101–112. http://dx.doi.org/10.2139/ssrn.3675950  
+8. Ildar, R. (2021). Progress in neural networks for EEG signal recognition in 2021. arXiv preprint 
+arXiv:2103.15755 
+9. Rakhmatulin, I. (2020). Deep learning and machine learning for EEG signal processing on the example 
+of recognizing the disease of alcoholism. https://doi.org/10.1101/2021.06.02.21258251 
+10. I Rakhmatulin, S Volkl. (2022). PIEEG: Turn a Raspberry Pi into a Brain-Computer-Interface to 
+measure biosignals. arXiv:2201.02228 
+ 
+ 
+
+mechanicalarm
+Controlbybrain
+Read signals from
+computer
+brain
+exoskeleton
+unmannedaerialvehicle
+Processing
+toycars
\ No newline at end of file
diff --git a/ldE1T4oBgHgl3EQf0wUL/content/tmp_files/load_file.txt b/ldE1T4oBgHgl3EQf0wUL/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7b4358ac6aeb8d26b1435b6ce278bbf54d48f508
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+++ b/ldE1T4oBgHgl3EQf0wUL/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf,len=184
+page_content='RaspberryPi to measure EEG, ECG, EMG and EOG with Shield PiEEG  In this paper presented hardware and software for shield PiEEG for reading signals through the families of single-board computers - RaspberryPi, OrangePi, BananaPi, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' For the most part, the paper provides technical information on how to implement this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' This device is designed to be familiar with neuroscience and is one of the easiest ways to get started with EEG measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  Ildar Rakhmatulin, PhD, PiEEG, ildarr2016@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='com Source https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='com/Ildaron/EEGwithRaspberryPI Presentation https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='be/uK8QF2liO5U  Keywords: RaspberryPi and EEG, ECG, EMG and EOG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' brain-computer interface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' shield for RaspberryPi  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Introduction The brain-computer interface is a device for reading brain signals in order to identify any correlations that can be used for practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' In 2021 we developed the brain-computer interface - ironbci [1,2,3] but the chip shortage significantly increased the cost of the device, after which we switched to PiEEG shield, which made it possible to reduce the cost of the device and simplify the installation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' The PiEEG device was presented in general terms at the conference [4] and in the publication [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' In this paper we will focus more on the technical details of the implementation of this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Safety recommendation The developed device was tested only for Raspberry Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' During the test, it is forbidden connect the device to the electrical source, this need for safety and to avoid network interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Cannot be used this device when powered via an electrical network and used it only when working with 5V batteries (capacity– not mote 2000 mAh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' General view of the complete assembly of the device in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='1 General View of assembled device: 1 – hat EEG, 2 – monitor, 3 – RaspberryPI and shield EEG, 4 – power bank  All parts should be connected only with battery – monitor, keyboard, mouse, RaspberryPI and PiEEG device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' The device is connected only directly through GPIO40 to the Raspberry Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' There must not be any connections for any elements with the electrical network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Also observe the following protective measures: - If the device fails (heating of components, smoke etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' ), switch off immediately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' - When you have finished using any electrical appliance, be sure to turn it off and unplug it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' - Never try to fix a malfunction in an electrical appliance yourself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' - Do not touch the components on the board when the device is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Connecting the device The device is designed to allow single-board computers to read the EEG signal via the SPI protocol from ADS1299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' The next single computers can be connected (not tested yet) - OrangePi, BananaPi, and the device can be connected via wires to the input of the microcontroller if necessary, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  2 3 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Device connection scheme, GPIO40  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Software Data transfer between Raspberry Pi and ADS1299 using the SPI protocol is written in C at a rate of 250 samples per second for each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Data processing realized in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Software files details from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='com/Ildaron/EEGwithRaspberryPI/tree/master/GUI : libperiphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='a – static library compiled from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='com/vsergeev/c-periphery , real_time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='py – script for editing processing signal and data visualization, real_time_massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='c – header call, real_time_massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='h – register setting for data transfer protocol, super_real_time_massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='so – compiled dynamic library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  To start the reading EEG signals is possible to write software or download from the git https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='com/Ildaron/EEGwithRaspberryPI , then go to the folder - GUI and run the file - real_time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' An example of reading is shown in the figure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' EEG signals after band pass filter 1-30 Hz  Through the python script, is possible to configure the parameters for band pass filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Signal processing is implemented using the standard scipy library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Through the C file, once per second we receive a data packet of 2000 data, 250 samples per second for each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Currently we used the next setting for registers for ADS1299:  write_reg (0x14,0x80);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//led  write_reg (0x05,0x00);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//ch1  write_reg (0x06,0x01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//ch2  write_reg (0x07,0x01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//ch3  write_reg (0x08,0x01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//ch4  write_reg (0x0D,0xFF);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//bias N  write_reg (0x0E,0x00);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//bias P  write_reg (0x09,0x01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//ch5  write_reg (0x0A,0x01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//ch6  write_reg (0x0B,0x01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//ch7  write_reg (0x0C,0x01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//ch8  PWR  CS  Ground  GPIO14  UARTO TXD  GPIO15  UARTORXD  GPIO18  Ground  GPIO23  GPIO24  Ground  GPIO25  SPIOCE1N  IDEEPRO  PCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='CLK  GPI08  SPIOCEO  GPI07  Ground  GPIO12  Ground  GPIO16  GPIO20  GPI021  GO  6  8  2  2  2  忆  2  2  3  3  3  5  9  3  5  9  7  1  5  2  3  3  EAE  GPIO2  GPIO3  SCL112C  GPIO4  aJIM T  Ground  GPIO17  GPIO27  GPIO22  EAE  GPIO10  SONOldS  GPIO9  GPIO11  Ground  IDSD  GPIO5  GPIO6  GPIO13  GPIO19  GPIO26  Ground  MOSI  SCLK  DRDYGND  MISO  JMP1  JMP3  JMP250  50  50  50  50  50  100  write_reg (0x15,0x20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='// mics  write_reg (0x01,0x96);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='// reg1  write_reg (0x02,0xD4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='// reg2  write_reg (0x03,0xE0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='// reg3  send_command (0x10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//sdatc  send_command (0x08);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='//start  To change the ADS1299 registers, needs to work with the header file - real_time_massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='h, after any changes, need to recompile the static library, following the next steps for GPIO library for Linux from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='com/vsergeev/c-periphery: $ mkdir build $ cd build $ cmake .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='. $ make That allows receiving the Static Library - libperiphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='a and finally to create a dynamic library for .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='so - gcc -shared -home/pi/Desktop/new_spi/c-periphery-master/src real_time_massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='c /home/pi/Desktop/new_spi/c-periphery-master/build/libperiphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='a -o super_real_time_massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='so This library can be imported in python script with signal processing (example - real_time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='py) via standard library – ctypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' In case of errors incompatibility in a python IDE, it is necessary to check the versions of installed libraries https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='com/Ildaron/EEGwithRaspberryPI/blob/master/pip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='txt The PIEEG device has a diode connected to the ADS1299, which lights up when the PiEEG device is initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' If the life bit is not burning, it is necessary to check the power (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='4) in the next points: TP1: – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='5V TP2:  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='3V TP3:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='5V  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Test points in the PiEEG board  If there is noise in the measured EEG signals that goes beyond the limits declared in the device it is necessary to check the voltage in the next points: VCAP1 = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='2V VCAP2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='001V VCAP3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='2V VCAP4=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='125V  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Conclusion and perspectives This device is positioned as an easy way to start measuring EEG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' We introduced hardware and software for reading and processing data in real-time in open source format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' We demonstrated how to control a robot using a device [5], in the future we will work on the P300 and on other paradigms, then we plan to gather a team of like-minded people and collect the necessary amount of data to control robots through the motor imagery method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='5 shows a block diagram for the possibility areas of use of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='5V I-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='5V Power led 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='3V PIEEG Led of life Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' possibility areas of use of the PiEEG  Additional Information Machine learning to identify alcoholism [9], Overview of dry electrodes [6], Signal processing [7,8] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Rakhmatulin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=', Parfenov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=', Traylor, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Low-cost brain computer interface for everyday use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' ExpBrain Res 239, 3573–3583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='1007/s00221-021-06231-49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Rakhmatulin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=" ironbci - Open-source Brain-computer interface with the embedded board to monitor the physiological subject's condition and environmental parameters arXiv:2111." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='03656 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Rakhmatulin, I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' The electronic board to replace the reference voltage on the earlobe for EEG measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' March 2021Measurement 173 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='108673 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Rakhmatulin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Raspberry PI Shield - for measure EEG (PIEEG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Conference: 2021 5th International Conference on Electrical, Electronics, Communication, Computer Technologies and Optimization Techniques (ICEECCOT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='1109/ICEECCOT52851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='9707969 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' I Rakhmatulin, S Volkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Brain-Computer-Interface controlled robot via RaspberryPi and PiEEG, arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='01936 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Rakhmatulin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=', Gan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Review Dry and Non-Contact EEG Electrodes for 2010-2021 Years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Preprints 2021, 2021030555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='20944/preprints202103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='0555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='v1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Rakhmatulin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Review of EEG Feature Selection by Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' 2020, 4, 101–112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='2139/ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='3675950 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Ildar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Progress in neural networks for EEG signal recognition in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='15755 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' Rakhmatulin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='  Deep learning and machine learning for EEG signal processing on the example of recognizing the disease of alcoholism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='1101/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='21258251 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' I Rakhmatulin, S Volkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' PIEEG: Turn a Raspberry Pi into a Brain-Computer-Interface to measure biosignals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content=' arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
+page_content='02228  mechanicalarm  Controlbybrain  Read signals from  computer  brain  exoskeleton  unmannedaerialvehicle  Processing  toycars' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE1T4oBgHgl3EQf0wUL/content/2301.03459v1.pdf'}
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+Unlocking Slot Attention by Changing Optimal Transport Costs
+Yan Zhang * 1 David W. Zhang * 2 Simon Lacoste-Julien 1 3 4 Gertjan J. Burghouts 5 Cees G. M. Snoek 2
+Abstract
+Slot attention is a powerful method for object-
+centric modeling in images and videos. However,
+its set-equivariance limits its ability to handle
+videos with a dynamic number of objects because
+it cannot break ties. To overcome this limitation,
+we first establish a connection between slot atten-
+tion and optimal transport. Based on this new per-
+spective we propose MESH (Minimize Entropy
+of Sinkhorn): a cross-attention module that com-
+bines the tiebreaking properties of unregularized
+optimal transport with the speed of regularized op-
+timal transport. We evaluate slot attention using
+MESH on multiple object-centric learning bench-
+marks and find significant improvements over slot
+attention in every setting.
+1. Introduction
+Suppose we have an image containing two cats and one dog.
+Given a query like [cat, cat, dog], our task is to provide
+instance-specific information for each query element, such
+as their positions in the image. When constructing a neural
+network to solve this problem, cross-attention is a natural
+choice to relate the queries to our image (Vaswani et al.,
+2017; Wei et al., 2020). With such a model, the dog can be
+located perfectly, but our two queries for the cats inevitably
+end up with an undesirable result: the average of the two
+cats’ positions. The problem is that with our model, multiple
+copies of the same query element must have the same result
+(Zhang et al., 2022); it is impossible to receive different
+answers for the same query, even if the context makes it
+obvious what is desired.
+Models that rely on cross-attention, such as slot attention
+(Locatello et al., 2020), can run into this issue when trying
+to extract objects from images and other data modalities.
+This has especially been a problem in the video domain
+due to occlusions and new objects appearing (Kipf et al.,
+*Equal contribution
+1Samsung - SAIT AI Lab, Montreal
+2University of Amsterdam 3Mila, Universit´e de Montreal 4Canada
+CIFAR AI Chair
+5TNO. Correspondence to:
+Yan Zhang
+<yan@cyan.zone>, David W. Zhang <w.d.zhang@uva.nl>.
+Preprint.
+2022). For example, Wu et al. (2022) observe that when
+there are two objects but five “slots” (equivalent to queries
+in the previous example), the three slots that don’t have a
+specific object to model become nearly identical. Once that
+has happened, the “cat problem” from before applies: these
+very similar slots (queries) must receive essentially the same
+information. Consequently, if multiple new objects appear
+in the video, these three slots can—at best—all bind to only
+a single object, which is clearly undesirable.
+This issue is present because cross-attention is set-
+equivariant, a property traditionally considered desirable.
+However, Zhang et al. (2022) show that set-equivariance
+is too restrictive when applied to multisets: sets with re-
+peated elements allowed, which are prevalent in deep learn-
+ing (background in Section 2). This manifests itself in two
+related problems:
+1. Soft assignments. The model tends to mix several
+inputs into each slot (query) rather than making a hard
+decision of one slot corresponding to exactly one input.
+This leads to difficulties when the information from
+each input must be kept distinct.
+2. Lack of tiebreaking. Similar slots are processed sim-
+ilarly, so they will likely contain similar information.
+Multiple similar slots prefer to capture an average of
+the relevant inputs rather than each slot capturing a
+different input, which leads to the problem described
+earlier where two cats cannot be localized individually.
+To avoid these issues, a property called exclusive multiset-
+equivariance is necessary (Zhang et al., 2022). So far,
+only models from the Deep Set Prediction Networks family
+(Zhang et al., 2019; 2022) are known to have this property.
+However, they lack the object-centric inductive bias useful
+for object-centric learning tasks where the set-equivariant
+slot attention (Locatello et al., 2020) shines. Fortunately,
+introducing even a single exclusively multiset-equivariant
+module in a model is enough to give the entire model this
+property. In this paper, we develop a module that enhances
+cross-attention in order to make the object-centric slot at-
+tention exclusively multiset-equivariant, thereby addressing
+the problems of soft assignments and tiebreaking.
+In particular, we will look towards the field of optimal trans-
+port for inspiration, which is a natural fit for attention mod-
+arXiv:2301.13197v1  [cs.LG]  30 Jan 2023
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+els (Sander et al., 2022). This is because optimal transport
+concerns itself with computing the “best” assignment from
+one set to another given some pairwise costs (Villani, 2009),
+while cross-attention learns exactly such costs. This opti-
+mal transport perspective is especially useful because some
+optimal transport algorithms are able to break ties, which
+makes them relevant for multiset-equivariance. Unfortu-
+nately, these algorithms tend to be slow and difficult to
+parallelize. On the other hand, solutions to the entropy-
+regularized optimal transport problem (Cuturi, 2013) are
+fast but are unable to break ties. We aim to combine the best
+of both worlds.
+Contributions.
+1. We establish that slot attention already uses an approx-
+imation of regularized optimal transport (Section 3).
+We use this to motivate variants of slot attention where
+either the approximation, or both the approximation
+and regularization are removed—with the latter being
+exclusively multiset-equivariant. However, this comes
+with a significant speed penalty and a lack of gradients,
+which can hinder learning.
+2. To avoid these issues, we introduce the MESH idea:
+minimize the entropy of Sinkhorn (Section 4).
+It
+combines the benefits of our two proposed slot atten-
+tion variants: speed, gradients, and exclusive multiset-
+equivariance. We also show why this is more effective
+for reducing entropy and maintaining useful gradients
+than Sinkhorn alone.
+3. We evaluate our method in slot attention (SA-MESH1)
+on two object detection and two unsupervised object
+discovery tasks (Section 5). We find that our optimal
+transport-based variants generally outperform slot at-
+tention. Crucially, SA-MESH almost always has the
+best results—often by a significant margin.
+2. Background
+Multisets are generalizations of sets by allowing repetitions
+of elements. In deep learning, sets and multisets are repre-
+sented as Rn×c matrices with n being the number of ele-
+ments and c the feature dimension per element. The unique-
+ness property of sets is rarely enforced in deep learning, so
+most models should be thought of as operating on multisets
+rather than sets (Zhang et al., 2022). These models must
+then be careful to not rely on the arbitrary order of the n
+elements. To guarantee this, they should satisfy certain
+equivariances.
+1Samesh is an Indian name meaning “lord of equality”, fitting
+for our method addressing a problem with equal elements.
+2.1. Permutation equivariances
+The standard definition of permutation-equivariant (set-
+equivariant) functions f states that a permutation of the
+input X should result in the same permutation of the out-
+put (Zaheer et al., 2017). With Π as the space of n × n
+permutation matrices:
+∀X ∈ Rn×c, ∀P ∈ Π :
+f(P X) = P f(X).
+(1)
+However, this means that a set-equivariant function must
+always produce the same result when there are equal inputs
+(Zhang et al., 2022): f([a, a]) = [c, d] is not possible for
+c ̸= d. Zhang et al. (2022) therefore introduce a more appro-
+priate equivariance for multisets, multiset-equivariance:
+∀X ∈ Rn×c, ∀P1 ∈ Π, ∃P2 ∈ Π :
+f(P1X) = P2f(X) ∧ P1X = P2X.
+(2)
+It states that when there are interchangeable elements in
+X (so P1X = P2X for P1 ̸= P2), then there are multi-
+ple permutations of the output that are valid for achieving
+equivariance. This relaxation of set-equivariance makes the
+tiebreaking in f([a, a]) = [c, d] possible.
+While this property specifies what happens with equal el-
+ements, the continuity of most machine learning models
+suggests that the primary benefit in practice is with similar
+elements (Zhang et al., 2022): similar elements no longer
+have to result in similar outputs. All set-equivariant mod-
+els are also multiset-equivariant, which means that only
+models that are exclusively multiset-equivariant (multiset-
+equivariant, but not set-equivariant) are capable of tiebreak-
+ing. Unfortunately, most operations in the multiset learning
+literature are set-equivariant and thus unable to break ties.
+2.2. Slot attention
+Slot attention (SA) (Locatello et al., 2020) can be used to
+abstract the contents of an image into a multiset of “slots”.
+Each slot can be thought of as a “container” that combines
+related information from the input into a vector. The model
+learns how to route information from the input into these
+slots—in object-centric learning, these slots often learn to
+represent individual objects. It does this by alternating two
+steps: 1. cross-attention between the multisets of input fea-
+tures and slot features to compute updates for each slot, and
+2. utilizing a GRU (Cho et al., 2014) to apply the computed
+updates to the corresponding slots.
+The input features are represented by a matrix X ∈ Rn×c,
+and the slots are randomly initialized as a matrix Z(0) ∈
+Rm×d, with m being the number of slots and d the num-
+ber of dimensions per slot. Cross-attention in slot attention
+utilizes the standard key-query-value mechanism to com-
+pute updates for each slot. The computed updates are then
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+applied to the corresponding slots using a GRU update.
+Q(l) = Z(l)WQ,
+K = XWK,
+V = XWV
+(3)
+A(l) = normalize(softmax(Q(l)K⊤))
+(4)
+Z(l+1) = GRU(Z(l), A(l)⊤V ).
+(5)
+In Locatello et al. (2020), softmax forces each sum over the
+m slots to be 1 while normalize forces each sum over
+the n input elements to be 1. All operations used in slot
+attention are set-equivariant with respect to the slots, which
+makes the model set-equivariant (Locatello et al., 2020). In
+this paper, we introduce a module to make slot attention
+exclusively multiset-equivariant, which allows it to break
+ties between similar slots.
+3. Connecting slot attention to optimal
+transport
+Optimal transport (Villani, 2009) identifies the most cost-
+effective method for redistributing mass between two distri-
+butions. This process typically involves sampling from both
+distributions, calculating the pairwise distances between the
+samples, and applying an optimal transport algorithm to
+determine the transport map that minimizes the total cost.
+In the context of cross-attention, we are comparing two mul-
+tisets containing the input and slot features rather than two
+distributions, but the same principles of computing pairwise
+distances and finding an optimal transport map still apply.
+By making this connection between slot attention and opti-
+mal transport more explicit, we can leverage the powerful
+algorithms of optimal transport to enhance the performance
+of slot attention.
+Entropy-regularized optimal transport.
+A critical com-
+ponent when applying cross-attention in slot attention is
+the normalization of the attention matrix. Equation 4 first
+exponentiates all the entries, then normalizes one dimension
+of the attention matrix to sum to 1, then the other dimension
+to sum to 1. This sequence of operations is also known
+as applying the Sinkhorn algorithm for a single step. The
+Sinkhorn algorithm solves the entropy-regularized optimal
+transport problem (Cuturi, 2013) by repeatedly alternating
+these two normalizations, which results in a doubly stochas-
+tic matrix at convergence. We can therefore think of Equa-
+tion 4 as approximating this entropy-regularized optimal
+transport (by using only one Sinkhorn iteration) to deter-
+mine how the information from the input should be associ-
+ated with the slots. This connection between transformers
+(which use a slightly different normalization) and optimal
+transport has also been made by Sander et al. (2022).
+A simple extension is thus to consider slot attention using
+more than one Sinkhorn iteration, which we will refer to as
+SA-SH. In this variant, instead of using a similarity score, a
+distance function d (e.g. L2 norm2) is used to compute the
+attention matrix as follows:
+Cij = d(Qi, Kj)
+(6)
+A = sinkhorn(C).
+(7)
+Although the proposed extension properly computes the opti-
+mal transport map, it remains limited by its set-equivariance
+and is unable to perform tiebreaking. One additional detail
+is that the Sinkhorn algorithm must handle non-square ma-
+trices since the number of slots is usually much smaller than
+the number of inputs; na¨ıvely applying the algorithm on
+such matrices does not converge. We describe the details of
+how to handle this in Appendix A as they are not important
+to the following discussion.
+Unregularized optimal transport.
+By using the full
+Sinkhorn algorithm, SA-SH replaces the usual attention
+matrix with the transport map of the regularized optimal
+transport problem. This raises the question whether other
+optimal transport algorithms can be used in the context of
+slot attention too. A benefit of optimal transport without reg-
+ularization is that it can be exclusively multiset-equivariant:
+many algorithms naturally include tiebreaking, which leads
+to low entropy solutions. By replacing entropy-regularized
+optimal transport with unregularized optimal transport, we
+can make slot attention exclusively multiset-equivariant to
+avoid the issues we pointed out in Section 1.
+We thus propose the SA-EMD (Earth Mover’s Distance)
+variant, wherein we use the EMD algorithm (Bonneel et al.,
+2011) that is part of the POT package (Flamary et al., 2021).
+The EMD algorithm provides a sparse solution to the un-
+regularized optimal transport problem and has the ability to
+break ties. However, using the EMD algorithm also presents
+some challenges. The gradients of unregularized optimal
+transport problems are piecewise constant, which prevents
+learning of the cost matrix C. Thus, we need to estimate
+gradients, for which many different techniques exist (Gould
+et al., 2016; Fung et al., 2022; Bai et al., 2019; Poganˇci´c
+et al., 2020). In practice, we observe that the gradient of the
+Sinkhorn operator is also a good descent direction for EMD.
+We find that we obtain the best empirical results through:
+A = emd(C) + sinkhorn(C).
+(8)
+Another issue is that the EMD algorithm is relatively slow.
+The standard solvers use a network simplex algorithm (a
+variant of the simplex algorithm for graphs), which is dif-
+ficult to parallelize efficiently on GPUs. Can we get the
+benefits of unregularized optimal transport with its exclu-
+sive multiset-equivariance, while still being fast and differ-
+entiable like entropy-regularized optimal transport?
+2Sander et al. (2022) point out that for Sinkhorn, this is equiv-
+alent to the usual (negative) dot product used in regular cross-
+attention.
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+4. MESH: minimizing the entropy of
+Sinkhorn
+Previously, we focused on different ways of turning costs
+C into a transport map A. Now, we turn our attention to
+changing the costs themselves, followed by using the com-
+putational efficiency of the Sinkhorn algorithm to compute
+the transport map for these modified costs. A key differ-
+ence between unregularized optimal transport and entropy-
+regularized optimal transport is the entropy in the resulting
+transport map. The idea is to change the costs in such a way
+that even after entropy-regularized optimal transport, the
+entropy remains low. This objective allows the tiebreaking
+necessary for exclusive multiset-equivariance. To imple-
+ment this idea, we aim to find a new cost C′ that minimizes
+the entropy H(P) = − �
+i,j Pij log Pij.
+MESH(C) = arg min
+C′∈V(C)
+H(sinkhorn(C′))
+(9)
+A = sinkhorn(MESH(C)).
+(10)
+V(C) denotes a neighborhood around C, which is implic-
+itly defined by how we implement the arg min. We refer
+to this variant of slot attention as SA-MESH (Minimize
+Entropy of Sinkhorn). Equation 9 changes the cost so that
+the resulting transport map has low entropy (preferring 0s
+and 1s) despite the entropy regularization. The final trans-
+port map is calculated from this new cost matrix efficiently
+using the Sinkhorn algorithm. In summary, the costs are
+changed to make the transport map look more like the output
+of unregularized optimal transport.
+To solve this optimization problem, we propose to use gra-
+dient descent for a small, fixed number of steps starting
+from the original cost matrix C. The last point ensures
+that the new cost matrix remains close to C so that the new
+optimal transport problem will be similar to the original
+one. Differentiating through this can be done with standard
+automatic differentiation. While the process described so
+far does not feature any tiebreaking explicitly (and would in
+fact struggle to minimize entropy successfully when exactly
+equal rows or columns are present in C), we can simply
+add a small amount of noise at the start of the optimization:
+C′(0) = C + ϵ, ϵij∼N(0, 10−6). Another important detail
+is to normalize the gradient to have a fixed norm: the small
+amount of noise is amplified (to break ties) only when slots
+are similar. This is done without having to resort to large
+learning rates, which would impact the stability of optimiza-
+tion. We thus propose to repeat the following for T steps
+with a learning rate λ:
+C′(t+1) = C′(t) − λ ∇C′(t)A′(t)
+��∇C′(t)A′(t)��
+(11)
+with A′(t) = H(sinkhorn(C′(t))).
+(12)
+Note that exact minimization is not needed, nor necessarily
+desirable. A critical aspect is that C′(T ) is still related to
+the initial C. For example, if C is square, then C′ = 1023I
+would always “successfully” minimize Equation 9, but also
+solve a problem unrelated to C and provide no signal to
+learn from. This is why we initialize C′(0) as C with a
+small amount of noise. Note that if this noise is too high (e.g.
+ϵij∼N(0, 1)), the solution can again lose correspondence
+with the actual cost matrix C that we are trying to compute
+the transport map for. We develop a more complicated
+version that explicitly enforces C and C′ to be similar
+(Appendix B) but find that it is not necessary in practice as
+long as the noise in the initialization is reasonably low.
+4.1. Properties
+Equivariance.
+SA-MESH is now exclusively multiset-
+equivariant: it is multiset-equivariant because all the in-
+dividual operations are multiset-equivariant, but it is not
+set-equivariant because equal slots will no longer receive
+the same transport plans due to the tiebreaking from the
+noise and subsequent optimization. This gives our method
+more representational power than standard SA and SA-SH
+because it is no longer restricted by set-equivariance.
+A useful side-effect is that random initialization of slots
+is no longer necessary. Since ties can be broken by this
+entropy minimization, it is no problem to initialize all slots
+to be the same vector. In standard slot attention, the amount
+of noise in Z(0) needs to be just right: too low, and the
+set-equivariant model has difficulties breaking ties between
+these similar slots (Wu et al., 2022); too high, and the model
+can become unreliable from the noisiness (Kipf et al., 2022).
+Furthermore, randomly initializing the slots does not prevent
+them from collapsing to the same values in a later iteration—
+such as when there are fewer objects to model than slots.
+In contrast, a tiny amount of noise in C′ (as long as it
+is above machine precision after sinkhorn) is sufficient for
+tiebreaking and avoiding collapse, and any other effect of the
+noise can be optimized away through the gradient descent.
+Speed.
+SA-MESH offers a significant improvement in
+computation time when compared to SA-EMD. This is be-
+cause it only requires the evaluation of the Sinkhorn al-
+gorithm for a small number of optimization steps (in our
+experiments, we found little improvement above four steps).
+While it may not be as fast as SA-SH, its exclusive multiset-
+equivariance can speed up learning by making the model
+more powerful. In many cases, the computation required
+for SA-MESH is outweighed by other components of the
+model such as the image processing part that creates the in-
+put multiset for SA. Since each optimization step in MESH
+needs to solve a similar optimal transport problem, we can
+reuse the result from the previous iterations to improve the
+efficiency, as we describe in Appendix C.
+Gradients.
+Another benefit over SA-EMD is that gradient
+computation is simple since we can fully rely on automatic
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+10−2
+10−1
+100
+101
+Scaling factor on input
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized entropy of transport map
+MESH λ = 1
+MESH λ = 2
+MESH λ = 3
+SH τ = 0.3
+SH τ = 0.1
+SH τ = 0.03
+10−2
+10−1
+100
+101
+Scaling factor on input
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized gradient norm w.r.t. C
+MESH λ = 1
+MESH λ = 2
+MESH λ = 3
+SH τ = 0.3
+SH τ = 0.1
+SH τ = 0.03
+Figure 1. MESH reduces entropy while maintaining reasonable gradients across a large range of attention values, in particular when
+attention is uncertain (low scaling factors). Meanwhile, Sinkhorn (SH) provides nontrivial gradients in a much smaller range with entropy
+reduction being ineffective when the scaling is too low. Entropy of transport map (left) and corresponding gradient norm (right) when
+varying the scaling factor on the input. The entropy is normalized to have 1 as maximum entropy. The gradient norm of each method is
+normalized to have a maximum of 1, we show the unnormalized gradient norms in Appendix D.
+differentiation instead of having to manually estimate gra-
+dients for the black-box EMD solver in SA-EMD. We find
+experimentally that we do not even need to differentiate the
+gradient updates in Equation 11 themselves; the gradients of
+C′(T ) can simply be passed along to C in a straight-through
+manner (Bengio et al., 2013) without reduction in perfor-
+mance. SA-MESH also has benefits in terms of the “quality”
+of gradients over SA-SH, which we explain in the following.
+4.2. Comparison to changing temperature
+An alternative for breaking ties is to add noise to the cost
+matrix (like SA-MESH), but then simply reduce the temper-
+ature of the Sinkhorn algorithm, which corresponds to re-
+ducing the amount of entropy regularization (Cuturi, 2013).
+For sufficiently low temperatures, this should also be able to
+map equal inputs to different slots. How does this much sim-
+pler approach compare to MESH, which minimizes entropy
+by gradient descent?
+Sinkhorn with low temperatures can be thought of as analo-
+gous to softmax with low temperatures. As the temperature
+decreases, the behavior of softmax becomes more similar
+to an argmax, but the gradients become ill-behaved as a re-
+sult. Similarly, with Sinkhorn, low temperatures may result
+in gradients that make it difficult or impossible to learn a
+good cost matrix. We now make this notion more concrete
+and show that MESH can reduce entropy while maintaining
+“good” gradients for a much larger range of inputs.
+In Figure 1, we have the following set-up. Starting from a
+10 × 10 identity matrix, we scale it by a varying factor to
+obtain the cost matrix, then apply either Sinkhorn (for dif-
+ferent temperatures τ) or MESH (for different learning rates
+λ). We then measure the entropy of the resulting transport
+map (left), as well as the norm of the gradient of this entropy
+with respect to the cost matrix (right). Gradient norms close
+to zero slow down learning because the gradients provide
+little to no information on how to learn the cost matrix.
+When the scaling factor is large (e.g. >10, attention is con-
+fident), the behaviors are similar (low entropy and small
+gradients). When the scaling factor is small (e.g. <0.1, at-
+tention is not confident), MESH is still always able to reduce
+entropy while maintaining reasonable gradients. In contrast,
+Sinkhorn only has nontrivial gradients in a relatively small
+range; outside of this range, learning is difficult because the
+gradient norms become close to 0. Thus, a trade-off has to
+be made for Sinkhorn, which is not necessary for MESH:
+either τ is high and low scaling factors result in virtually
+zero gradients (with no reduction to the entropy), or τ is low
+and higher scaling factors result in virtually zero gradients
+instead. The MESH learning rate λ can be used to control
+the shape of how entropy is reduced, while the Sinkhorn
+temperature τ only changes the location and does not in-
+crease the range of nontrivial gradients. In summary, MESH
+effectively reduces entropy while maintaining well-behaved
+gradients for a large range of inputs.
+5. Experiments
+We now experimentally evaluate SA with our optimal trans-
+port variants, with a particular focus on comparing SA
+to SA-MESH. We open-source all of our code https:
+//github.com/davzha/MESH and provide extra ex-
+perimental details in Appendix E.
+5.1. Random objects detection
+First, we evaluate the effect of the different SA variants in
+a simplified object detection setting. The aim is to assess a
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+Table 1. Random objects detection, measured in RMSE divided
+by standard deviation σ of random objects (lower is better). An
+always-predict-zeros baseline has a normalized RMSE of 1. Av-
+erage of 5 random seeds, except ∗ denotes exclusion of one bad
+outlier. Standard deviation between seeds is negligible (< 0.01).
+Model
+σ = 1.0
+σ = 0.1
+σ = 0.01
+SA
+0.44∗
+0.53
+0.65
+SA-SH
+0.27
+0.32
+0.39∗
+SA-EMD
+0.23
+0.52
+1.00
+SA-MESH
+0.24
+0.27
+0.31
+model’s ability to detect and distinguish similar objects in
+a controlled setting. Given a multiset containing k random
+32d vectors sampled from N(0, σ2I) and h zero vectors,
+the goal is to copy only the k random vectors into the k
+slots. In the context of object detection on images, this can
+be thought of as detecting k different objects, each of which
+occupies exactly one position in a feature map.
+In this setting, we want the object information to be pre-
+served as accurately as possible. To vary the difficulty of
+this task, we change the standard deviation σ of the random
+vectors to be copied and measure the error relative to this
+σ. A lower standard deviation corresponds to a harder task
+because the elements become more similar to each other
+and to the background zero vectors.
+Results.
+Table 1 shows our results for k = 5 objects and
+h = 100 background elements for varying difficulties σ.
+First, we see that SA-SH and SA-MESH can detect the ob-
+jects on the hardest setting of σ = 0.01 more accurately
+than SA on σ = 1. This demonstrates the general value
+of our proposed optimal transport perspective in the con-
+text of attention. SA-MESH with its lower entropy outper-
+forms SA-SH: the SA-MESH performance for a specific
+σ is roughly equivalent to SA-SH at a σ ten times higher.
+This shows the benefits of reducing entropies in the trans-
+port map through MESH, which helps objects stay distinct
+from each other. Meanwhile, SA-EMD (also with low en-
+tropy transport maps) performs similarly to SA-MESH on
+σ = 1, but degenerates to the always-predict-zeros baseline
+on σ = 0.01. We attribute this to the inherently imprecise
+gradient estimation leading to learning problems.
+5.2. CLEVR property prediction
+Next, we test SA and our proposed variants on a more realis-
+tic object detection task. CLEVR (Johnson et al., 2017) is a
+synthetic dataset containing images with up to ten objects in
+a 3d scene. Each object is sampled with varying sizes, mate-
+rials, shapes, and colors. The task is to predict the multiset
+of objects with their properties and 3d position. Following
+Zhang et al. (2019), we evaluate using average precision
+(AP) at different distance thresholds for the 3d coordinates
+of the predicted objects.
+Results.
+Table 2 shows that SA-MESH achieves the best
+SA results (and state-of-the-art results on some metrics)
+while only increasing run time by a small amount. These
+results are followed by SA-EMD, which has slightly worse
+results (likely due to the inherently imprecise gradient esti-
+mation) but takes over three times longer to train. Again, we
+see that all three optimal transport-based methods greatly
+outperform the baseline SA, which validates the benefits of
+the optimal transport perspective in attention.
+Note that we only provide the results for iDSPN (which is
+also exclusively multiset-equivariant) for context; this is not
+supposed to be a direct comparison due to the significant
+difference in approach. In general, slot attention through the
+use of attention has the benefit of not needing to compress
+the input into a single vector (global scene representation)
+like iDSPN. The resulting object-centric inductive bias and
+the relative simplicity have allowed for wider adoption and
+success of SA over iDSPN (Kipf et al., 2022; Hu et al.,
+2020; Li et al., 2021; Sajjadi et al., 2022), which makes our
+improvements to SA meaningful, even if in this case some
+metrics are worse than iDSPN.
+5.3. Unsupervised object discovery on images
+In this task, the objective is to discover objects without the
+supervision of what the objects are. We follow Locatello
+et al. (2020) and set up an image reconstruction task with
+Table 2. CLEVR property prediction, average precision (AP) in % (mean ± standard deviation) over 5 random seeds, higher is better.
+SA-MESH improves over all other SA variants at only a small computational cost. Note that the exclusively multiset-equivariant iDSPN
+is not object-centric, so they are not directly comparable. SA (original) results are copied from Locatello et al. (2020), iDSPN results from
+Zhang et al. (2022). Models with † use the improvement by Chang et al. (2022), see Appendix F for our results without †.
+Model
+AP∞
+AP1
+AP0.5
+AP0.25
+AP0.125
+AP0.0625
+Time
+iDSPN (Zhang et al., 2022)
+98.8 ±0.5
+98.5 ±0.6
+98.2 ±0.6
+95.8 ±0.7
+76.9 ±2.5
+32.3 ±3.9
+—
+SA (original) (Locatello et al., 2020)
+94.3±1.1
+86.7±1.4
+56.0±3.6
+10.8±1.7
+0.9±0.2
+—
+—
+SA†
+94.3±0.4
+85.7±1.6
+77.2±1.5
+53.1±2.7
+16.7±1.8
+4.0±0.7
+2.2 h
+SA-SH†
+98.9±0.2
+97.7±0.5
+95.2±0.9
+83.3±0.8
+38.5±2.0
+10.0±1.4
+2.3 h
+SA-EMD†
+99.3±0.3
+98.1±0.4
+95.9±0.8
+85.8±1.1
+42.0±2.0
+11.4±1.3
+9.3 h
+SA-MESH†
+99.4±0.1
+99.2±0.2
+98.9±0.2
+91.1±1.1
+47.6±0.8
+12.5±0.4
+2.4 h
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+Table 3. Object discovery on images results in Multi-dSprites in %
+(mean ± standard deviation) over 5 random seeds, higher is better.
+SA-MESH outperforms the other models and has lower variance.
+Model
+FG-ARI
+FG-mIoU
+SA (Locatello et al., 2020)
+91.3±0.3
+—
+SA
+92.2±0.5
+24.3±5.4
+SA-SH
+87.2±1.8
+84.0±3.1
+SA-MESH
+95.6±0.2
+86.2±2.2
+Table 4. Object discovery on images results in ClevrTex in %
+(mean ± standard deviation) over 5 random seeds, higher is better.
+SA-MESH outperforms the other models and has lower variance.
+Model
+FG-ARI
+FG-mIoU
+SA
+52.8±14.9
+26.3±14.9
+SA-SH
+70.8±5.8
+35.3±4.4
+SA-MESH
+79.0±2.6
+43.2±4.0
+slots as the latent bottleneck using SA. These slots are in-
+dividually decoded into object-specific images, each com-
+prising the RGB color channels and an alpha mask. These
+object-specific images are then combined to form the fi-
+nal reconstructed image. To evaluate the performance, we
+compare the per-slot alpha masks to the actual object seg-
+mentation masks. The goal is thus for image reconstruction
+with a multiset bottleneck to lead to a decomposition of
+the scene into individual objects, with each object being
+modeled by a distinct slot.
+We evaluate on the Multi-dSprites dataset, which is the
+only benchmark presented by Locatello et al. (2020) that
+still presented a challenge (possibly due to the presence
+of highly overlapping objects). Additionally, we test on
+ClevrTex (Karazija et al., 2021), a synthetic dataset similar
+to CLEVR that introduces the added challenge of different
+textures. In line with prior work (Kipf et al., 2022), we
+evaluate the Foreground Adjusted Rand Index (FG-ARI)
+and Foreground mean Intersection over Union (FG-mIoU).
+To ensure FG-mIoU is permutation-insensitive, we use the
+Hungarian algorithm to find the best matching between the
+masks. See Section E.3 for more details. Note that we no
+longer test SA-EMD because the larger input size compared
+to Section 5.2 makes its training time infeasible.
+Results.
+Table 3 and Table 4 show that on both Multi-
+dSprites and ClevrTex, SA-MESH achieves significantly
+higher FG-ARI and FG-mIoU compared to all baselines.
+SA-SH and SA-MESH improve especially in mIoU, which
+Karazija et al. (2021) argue is a better metric than ARI to
+evaluate the accuracy of object masks.
+Analysis.
+In Figure 2, we observe that minimizing the
+entropy leads to much sparser attention maps for SA-MESH
+Figure 2. Attention maps (first row) and alpha masks (second row)
+for each of the six slots in SA-MESH on Multi-dSprites. The
+attention focuses on only a small region inside each object, but the
+model still reconstructs the full objects. See Appendix H for more
+examples.
+in comparison to SA: SA-MESH only attends to a small
+part within each object, rather than the whole object as is
+usually the case for SA. Keep in mind that attention maps
+are only used to route the required information to each slot,
+and it is the slots themselves that represent the individual
+objects. Attending to a small part of each object is sufficient
+because the receptive field of the CNN image encoder lets
+it move information from the edge of an object into the
+center. The accuracy of the final alpha masks of SA-MESH
+shows that this happens successfully. In Appendix H, we
+see that SA-MESH attends less to the background than SA
+on Multi-dSprites.
+Similar to Locatello et al. (2020), we observe that some SA
+runs fail to separate the individual objects into different slots
+despite having a low reconstruction error. In those cases,
+the attention maps divide the image into separate regions,
+independent of the image content. SA-MESH ensures that
+the attention maps are sparse, which helps to avoid these
+kinds of failure modes. Appendix H shows this difference
+between SA-MESH and SA on ClevrTex: SA-MESH dis-
+covers objects successfully in most cases, and usually, only
+the background is split into spatial regions.
+5.4. Unsupervised object discovery on video
+As we mentioned in Section 1, Wu et al. (2022) observed
+issues with SA when applied to videos where multiple ob-
+jects can enter the scene. To evaluate our method in this
+scenario, we build two variants of the CLEVRER video
+dataset (Yi et al., 2019) where the number of visible ob-
+jects varies over time. We only use two frames from each
+video: the first frame, and either the 16th frame (short time
+difference, CLEVRER-S) or the 128th frame (long time dif-
+ference, CLEVRER-L). We do this to evaluate SA without
+the dynamics prediction component that Wu et al. (2022)
+are concerned with. In CLEVRER-S, the total number of
+objects increases by two or more in 9.5% of the videos,
+while in CLEVRER-L, this occurs for 68.8% of the videos.
+We evaluate FG-ARI and FG-mIoU on the two frames indi-
+vidually, which is then averaged. We also compute a tempo-
+ral consistency (TC) metric as the fraction of objects that are
+correctly captured by the same slot (details in Section E.4).
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+Table 5. Video object discovery results on CLEVRER-S and CLEVRER-L in % (mean ± standard deviation) over 5 random seeds.
+SA-MESH outperforms all other models in terms of quality of predicted masks (ARI, mIoU) and achieves high temporal consistency
+(TC). See Appendix H for example masks.
+CLEVRER-S
+CLEVRER-L
+Model
+FG-ARI
+FG-mIoU
+TC
+FG-ARI
+FG-mIoU
+TC
+SA
+78.1±14.0
+16.8±9.8
+26.3±22.5
+69.6±14.9
+12.2±6.6
+12.8±8.7
+SA fixed noise
+71.0±34.0
+17.1±11.4
+42.8±19.3
+79.4±5.9
+11.9±6.2
+18.5±13.0
+SA learned noise
+80.2±14.1
+13.2±4.9
+21.3±13.5
+84.7±5.7
+10.2±1.2
+19.4±7.7
+SA-SH
+89.3±2.3
+10.0±2.8
+89.7±2.4
+82.9±1.5
+7.0±0.2
+26.8±0.5
+SA-MESH
+93.8±1.0
+44.1±7.1
+80.2±10.1
+92.9±2.2
+54.4±8.9
+55.4±5.7
+Results.
+Table 5 shows that SA-MESH outperforms the
+other models by a significant margin. The only exception is
+on CLEVRER-S for temporal consistency—we can close
+this gap simply by reducing the MESH learning rate λ (five
+run average: 90.0% FG-ARI, 22.7% FG-mIoU, 95.1% TC).
+The proposal by Wu et al. (2022) of adding noise to slots
+to prevent them from collapsing helps SA on CLEVRER-L,
+but to a lesser extent than SA-MESH, which uses exclusive
+multiset-equivariance to prevent collapse. This is evidenced
+by the much better FG-mIoU of SA-MESH.
+6. Related work
+Object-centric learning.
+Object-centric learning aims to
+model visual input data in terms of multiple “objects” rather
+than a global representation or grid of feature vectors. This
+decomposition can be considered an abstraction of the input;
+reasoning over a small number of objects is intuitively more
+efficient than over a feature map. Scenarios with multiple
+independent objects are ubiquitous in natural data, so it is
+desirable to model them well. Karazija et al. (2021) classify
+object-centric learning methods into three categories: pixel-
+space approaches which group related pixels together (Greff
+et al., 2019; Pervez et al., 2022), glimpse approaches which
+sequentially extract patches from the input (Crawford &
+Pineau, 2019; Lin et al., 2020; Jiang & Ahn, 2020), and
+sprite approaches which learn a dictionary of object appear-
+ances (Monnier et al., 2021; Smirnov et al., 2021).
+We choose to apply MESH on specifically the pixel-space
+based slot attention because of its simplicity in approach
+(cross-attention with GRU updates), the lack of assumptions
+on what an object is (which makes it a general technique),
+and its set-equivariance. Since Zhang et al. (2022) show
+a specific limitation with set-equivariance, there is a clear
+path towards improvement, namely making it exclusively
+multiset-equivariant. We accomplish this in this paper, with
+strong results backing up the benefits of our approach.
+Optimal transport.
+Another approximation of optimal
+transport can be obtained through the Sliced Wasserstein
+Distance (Bonneel et al., 2015). It performs tiebreaking
+through the use of numerical sorting and is thus exclusively
+multiset-equivariant, but it lacks precise 1-to-1 associations
+between inputs and slots. This can especially be a problem
+with varying input sizes. We tried approaches based on this
+method as a replacement for standard cross-attention but did
+not obtain any competitive results.
+In a similar direction to Sinkhorn, which performs entropy-
+regularized optimal transport, Blondel et al. (2018) study
+L2-regularized optimal transport problems. While their
+solver is faster than unregularized optimal transport and
+obtains lower entropy solutions than Sinkhorn, similarly to
+Sinkhorn the convexity of the problem means that it cannot
+break ties effectively on its own, even with noise. In SA-
+MESH, we can replace Sinkhorn with this method, but we
+found that it was too slow comparatively.
+7. Discussion
+We introduced several variants of slot attention that can
+break ties between slots which enables better modeling of
+objects. In particular, MESH is a promising method that
+enhances cross-attention and, as a result, grants slot atten-
+tion the property of exclusive multiset-equivariance, while
+maintaining its learnability and efficiency.
+While minimizing entropy in MESH has a nice symmetry
+with entropy-regularized optimal transport, it is not clear
+whether a sparse attention map is always desirable. We try to
+learn a neural network on the transport map in Appendix G,
+but find no improvements over simply using the entropy;
+the derivative of the learned objective ends up with a similar
+shape to that of the entropy, which suggests that entropy is
+indeed a reasonable choice to minimize for now.
+An interesting direction for future research is to explore
+the use of MESH in other contexts than slot attention. For
+example, Sinkhorn is used by Pe˜na et al. (2022) for merging
+the weights of two neural networks together. MESH could
+be used as an alternative in this and other contexts to replace
+Sinkhorn. In general, we believe that optimal transport
+will continue to play an important role in deep learning,
+with MESH being a way of bringing tiebreaking into the
+picture without paying the usual speed penalty associated
+with optimal transport.
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+Acknowledgements
+The work of DWZ is part of the research programme
+Perspectief EDL with project number P16-25 project
+3, which is financed by the Dutch Research Coun-
+cil (NWO) domain Applied and Engineering Sciences
+(TTW). This research was partially supported by Calcul
+Qu´ebec (www.calculquebec.ca), Compute Canada
+(www.computecanada.ca) and the Canada CIFAR AI
+Chair Program. Simon Lacoste-Julien is a CIFAR Associate
+Fellow in the Learning Machines & Brains program.
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+2022.
+A. Marginals in Sinkhorn and EMD
+As we mention in the main text, we need to account for the
+(typical) case of the number of inputs n and the number of
+slots m differing, i.e. with a cost matrix C ∈ Rm×n. The
+problem is that it is impossible to make every row and every
+column of the transport map sum to 1 when the number of
+rows and columns is different. Fortunately, there is standard
+practice for how to deal with this case in optimal transport
+(Peyr´e & Cuturi, 2019; Cuturi, 2013; Bonneel et al., 2011).
+We can define non-negative marginals a ∈ Rm and b ∈ Rn
+that specify the row and column sums of the transport map
+respectively. If �
+i ai = �
+j bj, then convergence is as
+normal.
+In our case, we learn both a = m · softmax(ha(Z)) and
+b = m·softmax(hb(X)) with neural networks ha : Rd →
+R and hb : Rc → R that are shared across the m slots or n
+input elements respectively. These allow the model to put
+focus on important input elements (e.g. the inputs corre-
+sponding to objects) and ignore unimportant input elements
+(e.g. the inputs corresponding to the background), as well
+as put focus on the relevant number of slots. Since both
+softmaxes sum to one, we have �
+i ai = �
+j bj = m so
+there is no problem with convergence.
+For the Sinkhorn algorithm, it now repeatedly alternates
+normalizing all the rows to sum to a, then all the columns
+to sum to b. For the EMD solver that we use (Bonneel
+et al., 2011), these marginals are standard parameters in
+the algorithm. In the main text, we omit these marginals
+whenever we refer to sinkhorn or emd for simplicity of
+notation.
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+B. Enforcing C′ to be similar to C
+The following discussion is not critical to understanding
+the main text, since we find empirically that with the right
+initialization (e.g. C′ = C + ϵ with ϵij ∼ N(0, 10−6)),
+learning is not a problem. We only find that this variant is
+necessary with an initialization such as ϵ ∼ N(0, I). As
+we mention in the main text, the amount of noise is not
+important as long as it remains above machine precision
+after applying sinkhorn, which can be easily checked a-
+priori.
+In order to enforce C′ to be related to C more explicitly,
+we define the following objective instead:
+MESH(C) = arg min
+C′
+[H(sinkhorn(C′))
++α||sinkhorn(C′)S − sinkhorn(C)||2]
+(13)
+The second term relates C′ to C directly with a regulariza-
+tion factor α. S is a similarity matrix, which we will define
+shortly. The idea behind it is to allow costs to be freely
+changed among similar slots, but disallow this for dissim-
+ilar slots. The aim of ||sinkhorn(C′)S − sinkhorn(C)||2
+is thus to make sure that sinkhorn(C′) looks the same as
+sinkhorn(C) after allowing weight in the transport map to
+be moved around among similar slots.
+Example
+Consider the case where we have three slots:
+Z = [x, x, y] and three inputs X = [α, β, γ]. Let us
+assume for this example that the cost matrix prefers asso-
+ciating γ with y and both α and β with x. Computing
+unregularized optimal transport solutions would therefore
+give us either of two solutions:
+T1 =
+�
+�
+1
+0
+0
+0
+1
+0
+0
+0
+1
+�
+�
+or
+T2 =
+�
+�
+0
+1
+0
+1
+0
+0
+0
+0
+1
+�
+�
+(14)
+However, the Sinkhorn algorithm is unable to break the
+tie between the two x slots, so even with a temperature
+approaching 0, we obtain the following result:
+sinkhorn(C) =
+�
+�
+0.5
+0.5
+0
+0.5
+0.5
+0
+0
+0
+1
+�
+�
+(15)
+Suppose we have a similarity matrix ˜S ∈ Rm×m (m is the
+number of slots) that measures pairwise similarities ranging
+from 0 to 1:
+˜S =
+�
+�
+1
+1
+0
+1
+1
+0
+0
+0
+1
+�
+�
+(16)
+The first two slots are similar amongst themselves but dis-
+similar to the y slot. If we normalize each column of ˜S to
+sum to 1 to obtain S, then we see the following:
+�
+�
+1
+0
+0
+0
+1
+0
+0
+0
+1
+�
+�
+�
+��
+�
+T1
+�
+�
+0.5
+0.5
+0
+0.5
+0.5
+0
+0
+0
+1
+�
+�
+�
+��
+�
+S
+=
+�
+�
+0.5
+0.5
+0
+0.5
+0.5
+0
+0
+0
+1
+�
+�
+�
+��
+�
+sinkhorn(C)
+(17)
+�
+�
+0
+1
+0
+1
+0
+0
+0
+0
+1
+�
+�
+�
+��
+�
+T2
+�
+�
+0.5
+0.5
+0
+0.5
+0.5
+0
+0
+0
+1
+�
+�
+�
+��
+�
+S
+=
+�
+�
+0.5
+0.5
+0
+0.5
+0.5
+0
+0
+0
+1
+�
+�
+�
+��
+�
+sinkhorn(C)
+(18)
+This means that sinkhorn(C′) = T1 and sinkhorn(C′) =
+T2 are both valid solutions for the minimization of
+||sinkhorn(C′)S − sinkhorn(C)||2. Note that any other
+permutation matrix for T (i.e. one where there is not a 1 in
+the bottom right corner) would not be a valid solution. This
+restricts Equation 9 to only consider transport maps that are
+convex combinations of T1 and T2 for this example, with
+the entropy minimization preferring T1 and T2 specifically.
+The small amount of noise in the C′ initialization arbitrarily
+makes it prefer one of the two.
+Definition
+We define the similarity matrix
+˜Sij
+=
+g(Zi, Zj), where g : Rc × Rc → R is a small neural
+network that takes pairs of slots as input and produces a
+similarity score as output. We then normalize each column
+of ˜S to sum to 1 by applying softmax on each column.
+S = softmax( ˜S)
+(19)
+If we set up a training task for the example described above,
+we observe that S is learned to be virtually the same as the
+S we use in the example.
+C. Sinkhorn algorithm implementation
+Ideally, we want to compute the Sinkhorn algorithm for as
+few steps as possible since it is used in every MESH step,
+which means that we also have to differentiate through the
+Sinkhorn algorithm in every MESH step. However, we also
+need to run it for a sufficient number of steps for (good
+enough) convergence. Fortunately, because we repeatedly
+run the Sinkhorn algorithm on similar inputs over different
+MESH steps, we can optimize its implementation. The
+idea is that the gradient descent for minimizing the entropy
+makes small changes (especially near MESH convergence),
+which allows us to reuse computation between different
+MESH steps.
+If the entropy minimization has converged, then the
+Sinkhorn output does not change. This means that if we keep
+track of the operations we performed in the previous SA-
+MESH iteration, we can simply reapply them. Let us take a
+look at these operations: the Sinkhorn algorithm (multiplica-
+tively) rescales rows and columns repeatedly. Multiplication
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+0
+5
+10
+15
+MESH iteration
+10−8
+10−7
+10−6
+10−5
+Error compared to full convergence
+Baseline
+Reuse u and v
+Figure 3. Mean absolute error gap to the fully converged Sinkhorn
+for the two different Sinkhorn implementations for varying num-
+bers of MESH iterations. Both implementations always use 5
+Sinkhorn iterations and thus have comparable computational costs.
+Reusing u and b is clearly more effective.
+is commutative, so we can collect all the row normalizations
+together into a single row normalizer u and all the column
+normalizations into a single column normalizer v. Applying
+these two normalizers on the (exponentiated) cost matrix
+gives us exactly the same result as if we had manipulated
+the matrix with the normalizations directly. The next time
+we run the Sinkhorn algorithm with a slightly changed cost
+matrix, we can bootstrap the algorithm with the previously-
+found u and v. We can then run a few more iterations to
+account for the changes in the cost matrix, giving us a new
+u and v.
+In Figure 3, we ablate whether there are benefits to reusing
+u and v in SA-MESH. In particular, we compute the ideal
+solution by running the Sinkhorn algorithm until conver-
+gence at every ME iteration. Then we examine whether
+reusing u and v helps close the gap to the ideal solution
+when we limit the number of SH iterations. At 1 ME iter-
+ation there are no u and v for bootstrapping available, so
+both exhibit the same gap. At more than 1 ME iteration we
+observe that reusing u and v helps in narrowing the gap
+to the ideal solution, or equivalently, can achieve the same
+approximation with fewer iterations.
+We do not claim that this technique of collecting Sinkhorn
+operations into u and v is novel, as there are several imple-
+mentations that use this trick for performing the Sinkhorn
+algorithm. Usually, it is a minor implementation detail
+since the approaches of manipulating the matrix directly
+and collecting normalizations into u and v are mathemati-
+cally equivalent. In our case however, this formulation leads
+to a concrete benefit due to our setup where we run the
+Sinkhorn algorithm on similar inputs, which allows us to
+reuse u and v in a beneficial way.
+D. Gradients of tempered SH and MESH,
+without normalizing their scale
+In Figure 4, we show the same gradient norms as in Figure 1.
+However, rather than normalizing each model to have a
+maximum of 1 (which is more useful for visualizing them all
+at once), we maintain their native scaling. This shows that
+changing the SH temperature has a major effect on the scale
+of the gradients while changing the MESH learning rate
+only has minor effects on the gradients (but still significant
+effects on entropy reduction). In other words, the amount of
+entropy minimization can be changed without major impacts
+on the hyperparameters of other parts of the network.
+E. Experimental details
+E.1. Random object detection
+We generate a dataset of 64,000 data points to train on, each
+being a multiset with five 32-dimensional objects sampled
+from N(0, σ2I) and 100 zero vectors. We directly apply
+slot attention on this: the dimensionality of the slot attention
+weights are all 32. Since we know that there are always
+five objects, we set the number of slots to five. The loss is
+computed by computing a mean squared error between all
+pairs of predicted and ground-truth objects, then using the
+Hungarian algorithm find the matching with the lowest loss.
+We find that using implicit differentiation of slots (Chang
+et al., 2022) is significantly more stable on this dataset, so
+we use it for all models. We train all models for 20 epochs
+with a batch size of 64 (1,000 steps each epoch).
+E.2. Object detection on CLEVR
+We largely follow same training setup as DSPN and iDSPN
+(Zhang et al., 2019; 2022) and adapt the slot attention imple-
+mentation to it. Matching Zhang et al. (2022) and Locatello
+et al. (2020), we resize the input images to 128x128. To
+compute the loss, we use the Hungarian algorithm to com-
+pute the least-cost matching between predicted objects and
+ground-truth objects.
+E.3. Object discovery
+We compute the temporal consistency (TC) by first matching
+the predicted objects in each frame to their corresponding
+ground-truth objects. Then, we calculate the proportion
+of objects that have the same slot in both frames out of
+those that appear in both frames. To find the optimal assign-
+ment between the predicted and ground-truth objects, we
+first compute all pairwise IoUs between the predicted and
+ground-truth masks. We then invert these IoUs by applying
+1 − IoU and use the Hungarian matching algorithm to find
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+10−2
+10−1
+100
+101
+Scaling factor on input
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Gradient norm w.r.t. C
+MESH λ = 1
+MESH λ = 2
+MESH λ = 3
+SH τ = 0.3
+SH τ = 0.1
+SH τ = 0.03
+10−2
+10−1
+100
+101
+Scaling factor on input
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+Gradient norm w.r.t. C
+MESH λ = 1
+MESH λ = 2
+MESH λ = 3
+SH τ = 0.3
+SH τ = 0.1
+SH τ = 0.03
+Figure 4. Same gradient norm plot as Figure 1 but without normalizing each model to have a maximum of 1. The only difference between
+left and right is the scale of the y-axis. This shows that the gradient norm for tempered SH varies drastically with temperature, while the
+gradient norm of MESH remains more similar across learning rates. This makes tuning the learning rate hyperparameter easier as it can
+be considered more independently from other hyperparameters of the model.
+the best match—the one with the highest mIoU.
+We compute the mIoU in a similar manner as the TC metric,
+by finding the matching between the ground-truth objects
+and the predicted objects that results in the highest mIoU.
+Multi-dSprites.
+We closely follow the experimental setup
+described by Locatello et al. (2020). Specifically, we use the
+same image encoder, decoder, and hyperparameters where
+applicable. Locatello et al. (2020) used 500k training steps,
+while all of our runs were trained for 530 epochs, which
+results in slightly fewer than 500k training steps.
+ClevrTex.
+We pre-process the images by applying the
+same center crop as suggested by Karazija et al. (2021) and
+resize the images to 64×64 resolution instead of 128×128
+resolution. This allows us to use the same neural network
+architecture as we did for the Multi-dSprites dataset.
+Since the dataset has more complicated visuals we increase
+the model size by increasing the channel sizes. In particular,
+we double the number of channels in the image encoder and
+decoder to 64, and we double the dimensions of the slots to
+128 (with the MLP in slot attention having an intermediate
+dimension of 256). We again train all models for 530 epochs
+which correspond to around 330k gradient update steps in
+this case. The maximum number of objects in an image is
+10, so we set the number of slots to 11.
+E.4. CLEVRER
+We construct the datasets from the 20k videos in the
+CLEVRER dataset, by picking two frames at specific
+timesteps from each video. In the CLEVRER-S dataset,
+we use the first and 16th video frames. In the second dataset,
+we use the first and 128th (last) frames. The proportion of
+examples where new objects appear increases with the time
+gap between the two frames, and similarly the amount of
+displacement for objects that are in both frames increases
+too. We show examples from the two dataset variants in
+Figure 6.
+For the distance function d that computes the cost matrix of
+the optimal transport problem in SA-MESH, we empirically
+find that the cosine distance works better in this case than
+the l2 distance. We suspect that since the l2 distance allows
+the slots to be pushed arbitrarily far apart that learning might
+slow down in the later stages of training.
+We extend the model which we used in the Multi-dSprites
+experiment to video data. Our setup is similar to Kipf et al.
+(2022), but we do not use a predictor model (except for
+the SA learned noise baseline) to update the slots when
+transitioning from one video frame to the next. In particular,
+the model first applies the image encoder to all video frames
+independently to compute the input feature maps. Next,
+the SA (or our proposed variants) is applied to the features
+of one video frame at a time, and every time the slots are
+initialized from the slots of the previous frame. Finally, each
+image is decoded independently. The learned noise baseline
+uses a 2-layer MLP with LayerNorm to predict the mean
+and variance of a Gaussian, from which the initial slots are
+sampled for the next frame, following the stochastic SAVi
+setup by Wu et al. (2022). We use 8 slots.
+F. CLEVR object prediction results
+In Table 6 we show our results for CLEVR object prediction
+without implicit differentiation of slots (Chang et al., 2022).
+All results are slightly lower than the results reported in
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+Table 6. Results on CLEVR object property multiset prediction, average precision (AP) in % (mean ± standard deviation) over 5 random
+seeds, higher is better. All SA results are based on our re-implementation. SA (original) results copied from Locatello et al. (2020),
+iDSPN results from Zhang et al. (2022).
+Model
+AP∞
+AP1
+AP0.5
+AP0.25
+AP0.125
+AP0.0625
+Time
+iDSPN (Zhang et al., 2022)
+98.8 ±0.5
+98.5 ±0.6
+98.2 ±0.6
+95.8 ±0.7
+76.9 ±2.5
+32.3 ±3.9
+—
+SA (original) (Locatello et al., 2020)
+94.3±1.1
+86.7±1.4
+56.0±3.6
+10.8±1.7
+0.9±0.2
+—
+—
+SA
+89.1±1.2
+85.7±1.0
+73.3±1.2
+35.4±1.5
+9.0±0.8
+2.0±0.3
+2.4 h
+SA-SH
+95.6±1.0
+94.0±1.1
+84.5±1.7
+41.3±3.0
+10.4±0.7
+2.5±0.4
+2.5 h
+SA-EMD
+99.2±0.2
+98.7±0.4
+97.0±0.8
+82.4±1.2
+34.0±2.2
+8.3±0.9
+9.7 h
+SA-MESH
+99.2±0.3
+99.1±0.3
+98.8±0.5
+88.3±0.8
+40.8±1.0
+10.6±0.3
+2.5 h
+Table 2, but the overall message remains exactly the same.
+The only major difference is that SA-SH performs much
+worse compared to SA-SH†.
+G. Alternative MESH objective
+In Section 4 we choose the entropy as the inner objective
+function because the goal was to reverse the effect of the
+entropy-regularized optimal transport version. Alternatively,
+it is possible to learn a neural network with scalar inputs
+and outputs in place of the entropy function. For the neural
+network, we choose a simple 2-layer MLP with ReLU acti-
+vations and 32 hidden dimensions. We plot the derivative of
+a learned objective function in Figure 5b. Its shape is similar
+to the derivative of the entropy Figure 5a, but learning it
+incurs additional compute compared to simply using the
+entropy function H.
+Justified by this analysis we can directly use the entropy
+function as the MESH objective for improved efficiency.
+Empirically we observe that we do not even need to back-
+propagate through the gradient descent optimization proce-
+dure of MESH and it suffices to treat MESH as the identity
+function during backprop. One perspective that might ex-
+plain this: since the negative derivative of H is a monotonic
+increasing function, changes to the input will also affect the
+output in the same direction.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−1
+0
+1
+2
+3
+4
+5
+6
+(a) H
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.2
+0.3
+0.4
+0.5
+(b) MLP
+Figure 5. Derivative of the objective function in MESH
+H. Object discovery example results
+In the following, we show examples of SA and SA-MESH
+performing object discovery on the various datasets that
+we use. We always show the original image on the left,
+followed by either the attention maps for each slot or the
+final alpha masks for each slot. The attention map or the
+alpha masks are multiplied with the original image to make
+it easier to tell how precise their locations are.
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+(a) CLEVRER-S
+(b) CLEVRER-L
+Figure 6. Examples from the two datasets derived from CLEVRER. Significant changes like multiple new objects appearing occur less
+frequently in CLEVRER-S. Objects can be significantly displaced from one frame to the other in CLEVRER-L.
+
+Frame 1
+Frame 128Frame 1
+Frame 16Unlocking Slot Attention by Changing Optimal Transport Costs
+(a) SA
+(b) SA-MESH
+Figure 7. Attention maps for all three slot attention iterations for five different examples from the validation split of Multi-dSprites. Note
+how the shade of SA is generally darker indicating higher attention values even in background areas. In the third example, we can see how
+SA models the pink and blue ellipses using one slot while splitting the purple heart over two slots. In contrast, SA-MESH is able to route
+the three objects into three different slots in the second slot attention iteration.
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+(a) SA
+(b) SA-MESH
+Figure 8. Predicted alpha masks from the validation split of Multi-dSprites. In general, the masks for SA-MESH are sharper than for SA.
+For example, in the last row, the brown heart is only recognizable as a blown blob for SA, but is a distinct heart shape for SA-MESH. In
+the third example, SA splits the purple heart into two slots, while SA-MESH models it with one as desired.
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+(a) SA
+(b) SA-MESH
+Figure 9. Attention maps (top) and masks (bottom) for four different examples from the validation split of ClevrTex. SA commonly learns
+that each slot should attend to a spatial region as opposed to a specific object. SA-MESH on the other hand is better able to localize
+individual objects, though the background is often still handled with a region-specific approach.
+
+Unlocking Slot Attention by Changing Optimal Transport Costs
+CLEVRER-S
+CLEVRER-L
+SA
+SA fixed noise
+SA learned noise
+SH
+SA-MESH
+Figure 10. Example alpha masks for CLEVRER-S and CLEVRER-L. Possibly due to the difficulty of handling multiple new objects,
+SA and its noise variants choose a region-based decomposition on CLEVRER-L instead of an object-based decomposition like on
+CLEVRER-S. SA-SH and SA-MESH learn an object-based decomposition for both datasets.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf,len=1272
+page_content='Unlocking Slot Attention by Changing Optimal Transport Costs  Yan Zhang * 1 David W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Zhang * 2 Simon Lacoste-Julien 1 3 4 Gertjan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Burghouts 5 Cees G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Snoek 2  Abstract  Slot attention is a powerful method for object-  centric modeling in images and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' However,  its set-equivariance limits its ability to handle  videos with a dynamic number of objects because  it cannot break ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' To overcome this limitation,  we first establish a connection between slot atten-  tion and optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Based on this new per-  spective we propose MESH (Minimize Entropy  of Sinkhorn): a cross-attention module that com-  bines the tiebreaking properties of unregularized  optimal transport with the speed of regularized op-  timal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We evaluate slot attention using  MESH on multiple object-centric learning bench-  marks and find significant improvements over slot  attention in every setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Introduction  Suppose we have an image containing two cats and one dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Given a query like [cat, cat, dog], our task is to provide  instance-specific information for each query element, such  as their positions in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' When constructing a neural  network to solve this problem, cross-attention is a natural  choice to relate the queries to our image (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=',  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' With such a model, the dog can be  located perfectly, but our two queries for the cats inevitably  end up with an undesirable result: the average of the two  cats’ positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The problem is that with our model, multiple  copies of the same query element must have the same result  (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' it is impossible to receive different  answers for the same query, even if the context makes it  obvious what is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Models that rely on cross-attention, such as slot attention  (Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2020), can run into this issue when trying  to extract objects from images and other data modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  This has especially been a problem in the video domain  due to occlusions and new objects appearing (Kipf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=',  Equal contribution  1Samsung - SAIT AI Lab, Montreal  2University of Amsterdam 3Mila, Universit´e de Montreal 4Canada  CIFAR AI Chair  5TNO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Correspondence to:  Yan Zhang  <yan@cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='zone>, David W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Zhang <w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='zhang@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='nl>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' For example, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022) observe that when  there are two objects but five “slots” (equivalent to queries  in the previous example), the three slots that don’t have a  specific object to model become nearly identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Once that  has happened, the “cat problem” from before applies: these  very similar slots (queries) must receive essentially the same  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Consequently, if multiple new objects appear  in the video, these three slots can—at best—all bind to only  a single object, which is clearly undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  This issue is present because cross-attention is set-  equivariant, a property traditionally considered desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  However, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022) show that set-equivariance  is too restrictive when applied to multisets: sets with re-  peated elements allowed, which are prevalent in deep learn-  ing (background in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This manifests itself in two  related problems:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Soft assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The model tends to mix several  inputs into each slot (query) rather than making a hard  decision of one slot corresponding to exactly one input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  This leads to difficulties when the information from  each input must be kept distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Lack of tiebreaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Similar slots are processed sim-  ilarly, so they will likely contain similar information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Multiple similar slots prefer to capture an average of  the relevant inputs rather than each slot capturing a  different input, which leads to the problem described  earlier where two cats cannot be localized individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  To avoid these issues, a property called exclusive multiset-  equivariance is necessary (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' So far,  only models from the Deep Set Prediction Networks family  (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' 2022) are known to have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  However, they lack the object-centric inductive bias useful  for object-centric learning tasks where the set-equivariant  slot attention (Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2020) shines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Fortunately,  introducing even a single exclusively multiset-equivariant  module in a model is enough to give the entire model this  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In this paper, we develop a module that enhances  cross-attention in order to make the object-centric slot at-  tention exclusively multiset-equivariant, thereby addressing  the problems of soft assignments and tiebreaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In particular, we will look towards the field of optimal trans-  port for inspiration, which is a natural fit for attention mod-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='13197v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='LG]  30 Jan 2023  Unlocking Slot Attention by Changing Optimal Transport Costs  els (Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This is because optimal transport  concerns itself with computing the “best” assignment from  one set to another given some pairwise costs (Villani, 2009),  while cross-attention learns exactly such costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This opti-  mal transport perspective is especially useful because some  optimal transport algorithms are able to break ties, which  makes them relevant for multiset-equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Unfortu-  nately, these algorithms tend to be slow and difficult to  parallelize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' On the other hand, solutions to the entropy-  regularized optimal transport problem (Cuturi, 2013) are  fast but are unable to break ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We aim to combine the best  of both worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We establish that slot attention already uses an approx-  imation of regularized optimal transport (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We use this to motivate variants of slot attention where  either the approximation, or both the approximation  and regularization are removed—with the latter being  exclusively multiset-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' However, this comes  with a significant speed penalty and a lack of gradients,  which can hinder learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' To avoid these issues, we introduce the MESH idea:  minimize the entropy of Sinkhorn (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  It  combines the benefits of our two proposed slot atten-  tion variants: speed, gradients, and exclusive multiset-  equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We also show why this is more effective  for reducing entropy and maintaining useful gradients  than Sinkhorn alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We evaluate our method in slot attention (SA-MESH1)  on two object detection and two unsupervised object  discovery tasks (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We find that our optimal  transport-based variants generally outperform slot at-  tention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Crucially, SA-MESH almost always has the  best results—often by a significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Background  Multisets are generalizations of sets by allowing repetitions  of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In deep learning, sets and multisets are repre-  sented as Rn×c matrices with n being the number of ele-  ments and c the feature dimension per element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The unique-  ness property of sets is rarely enforced in deep learning, so  most models should be thought of as operating on multisets  rather than sets (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' These models must  then be careful to not rely on the arbitrary order of the n  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' To guarantee this, they should satisfy certain  equivariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  1Samesh is an Indian name meaning “lord of equality”, fitting  for our method addressing a problem with equal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Permutation equivariances  The standard definition of permutation-equivariant (set-  equivariant) functions f states that a permutation of the  input X should result in the same permutation of the out-  put (Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' With Π as the space of n × n  permutation matrices:  ∀X ∈ Rn×c, ∀P ∈ Π :  f(P X) = P f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  (1)  However, this means that a set-equivariant function must  always produce the same result when there are equal inputs  (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022): f([a, a]) = [c, d] is not possible for  c ̸= d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022) therefore introduce a more appro-  priate equivariance for multisets, multiset-equivariance:  ∀X ∈ Rn×c, ∀P1 ∈ Π, ∃P2 ∈ Π :  f(P1X) = P2f(X) ∧ P1X = P2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  (2)  It states that when there are interchangeable elements in  X (so P1X = P2X for P1 ̸= P2), then there are multi-  ple permutations of the output that are valid for achieving  equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This relaxation of set-equivariance makes the  tiebreaking in f([a, a]) = [c, d] possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  While this property specifies what happens with equal el-  ements, the continuity of most machine learning models  suggests that the primary benefit in practice is with similar  elements (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022): similar elements no longer  have to result in similar outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' All set-equivariant mod-  els are also multiset-equivariant, which means that only  models that are exclusively multiset-equivariant (multiset-  equivariant, but not set-equivariant) are capable of tiebreak-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Unfortunately, most operations in the multiset learning  literature are set-equivariant and thus unable to break ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Slot attention  Slot attention (SA) (Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2020) can be used to  abstract the contents of an image into a multiset of “slots”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Each slot can be thought of as a “container” that combines  related information from the input into a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The model  learns how to route information from the input into these  slots—in object-centric learning, these slots often learn to  represent individual objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' It does this by alternating two  steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' cross-attention between the multisets of input fea-  tures and slot features to compute updates for each slot, and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' utilizing a GRU (Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2014) to apply the computed  updates to the corresponding slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  The input features are represented by a matrix X ∈ Rn×c,  and the slots are randomly initialized as a matrix Z(0) ∈  Rm×d, with m being the number of slots and d the num-  ber of dimensions per slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Cross-attention in slot attention  utilizes the standard key-query-value mechanism to com-  pute updates for each slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The computed updates are then  Unlocking Slot Attention by Changing Optimal Transport Costs  applied to the corresponding slots using a GRU update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Q(l) = Z(l)WQ,  K = XWK,  V = XWV  (3)  A(l) = normalize(softmax(Q(l)K⊤))  (4)  Z(l+1) = GRU(Z(l), A(l)⊤V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  (5)  In Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2020), softmax forces each sum over the  m slots to be 1 while normalize forces each sum over  the n input elements to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' All operations used in slot  attention are set-equivariant with respect to the slots, which  makes the model set-equivariant (Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In  this paper, we introduce a module to make slot attention  exclusively multiset-equivariant, which allows it to break  ties between similar slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Connecting slot attention to optimal  transport  Optimal transport (Villani, 2009) identifies the most cost-  effective method for redistributing mass between two distri-  butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This process typically involves sampling from both  distributions, calculating the pairwise distances between the  samples, and applying an optimal transport algorithm to  determine the transport map that minimizes the total cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In the context of cross-attention, we are comparing two mul-  tisets containing the input and slot features rather than two  distributions, but the same principles of computing pairwise  distances and finding an optimal transport map still apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  By making this connection between slot attention and opti-  mal transport more explicit, we can leverage the powerful  algorithms of optimal transport to enhance the performance  of slot attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Entropy-regularized optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  A critical com-  ponent when applying cross-attention in slot attention is  the normalization of the attention matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Equation 4 first  exponentiates all the entries, then normalizes one dimension  of the attention matrix to sum to 1, then the other dimension  to sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This sequence of operations is also known  as applying the Sinkhorn algorithm for a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The  Sinkhorn algorithm solves the entropy-regularized optimal  transport problem (Cuturi, 2013) by repeatedly alternating  these two normalizations, which results in a doubly stochas-  tic matrix at convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We can therefore think of Equa-  tion 4 as approximating this entropy-regularized optimal  transport (by using only one Sinkhorn iteration) to deter-  mine how the information from the input should be associ-  ated with the slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This connection between transformers  (which use a slightly different normalization) and optimal  transport has also been made by Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  A simple extension is thus to consider slot attention using  more than one Sinkhorn iteration, which we will refer to as  SA-SH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In this variant, instead of using a similarity score, a  distance function d (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' L2 norm2) is used to compute the  attention matrix as follows:  Cij = d(Qi, Kj)  (6)  A = sinkhorn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  (7)  Although the proposed extension properly computes the opti-  mal transport map, it remains limited by its set-equivariance  and is unable to perform tiebreaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' One additional detail  is that the Sinkhorn algorithm must handle non-square ma-  trices since the number of slots is usually much smaller than  the number of inputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' na¨ıvely applying the algorithm on  such matrices does not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We describe the details of  how to handle this in Appendix A as they are not important  to the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Unregularized optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  By using the full  Sinkhorn algorithm, SA-SH replaces the usual attention  matrix with the transport map of the regularized optimal  transport problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This raises the question whether other  optimal transport algorithms can be used in the context of  slot attention too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' A benefit of optimal transport without reg-  ularization is that it can be exclusively multiset-equivariant:  many algorithms naturally include tiebreaking, which leads  to low entropy solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' By replacing entropy-regularized  optimal transport with unregularized optimal transport, we  can make slot attention exclusively multiset-equivariant to  avoid the issues we pointed out in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We thus propose the SA-EMD (Earth Mover’s Distance)  variant, wherein we use the EMD algorithm (Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=',  2011) that is part of the POT package (Flamary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  The EMD algorithm provides a sparse solution to the un-  regularized optimal transport problem and has the ability to  break ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' However, using the EMD algorithm also presents  some challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The gradients of unregularized optimal  transport problems are piecewise constant, which prevents  learning of the cost matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Thus, we need to estimate  gradients, for which many different techniques exist (Gould  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Fung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Poganˇci´c  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In practice, we observe that the gradient of the  Sinkhorn operator is also a good descent direction for EMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We find that we obtain the best empirical results through:  A = emd(C) + sinkhorn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  (8)  Another issue is that the EMD algorithm is relatively slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  The standard solvers use a network simplex algorithm (a  variant of the simplex algorithm for graphs), which is dif-  ficult to parallelize efficiently on GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Can we get the  benefits of unregularized optimal transport with its exclu-  sive multiset-equivariance, while still being fast and differ-  entiable like entropy-regularized optimal transport?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  2Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022) point out that for Sinkhorn, this is equiv-  alent to the usual (negative) dot product used in regular cross-  attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Unlocking Slot Attention by Changing Optimal Transport Costs  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' MESH: minimizing the entropy of  Sinkhorn  Previously, we focused on different ways of turning costs  C into a transport map A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Now, we turn our attention to  changing the costs themselves, followed by using the com-  putational efficiency of the Sinkhorn algorithm to compute  the transport map for these modified costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' A key differ-  ence between unregularized optimal transport and entropy-  regularized optimal transport is the entropy in the resulting  transport map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The idea is to change the costs in such a way  that even after entropy-regularized optimal transport, the  entropy remains low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This objective allows the tiebreaking  necessary for exclusive multiset-equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' To imple-  ment this idea, we aim to find a new cost C′ that minimizes  the entropy H(P) = − �  i,j Pij log Pij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  MESH(C) = arg min  C′∈V(C)  H(sinkhorn(C′))  (9)  A = sinkhorn(MESH(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  (10)  V(C) denotes a neighborhood around C, which is implic-  itly defined by how we implement the arg min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We refer  to this variant of slot attention as SA-MESH (Minimize  Entropy of Sinkhorn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Equation 9 changes the cost so that  the resulting transport map has low entropy (preferring 0s  and 1s) despite the entropy regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The final trans-  port map is calculated from this new cost matrix efficiently  using the Sinkhorn algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In summary, the costs are  changed to make the transport map look more like the output  of unregularized optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  To solve this optimization problem, we propose to use gra-  dient descent for a small, fixed number of steps starting  from the original cost matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The last point ensures  that the new cost matrix remains close to C so that the new  optimal transport problem will be similar to the original  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Differentiating through this can be done with standard  automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' While the process described so  far does not feature any tiebreaking explicitly (and would in  fact struggle to minimize entropy successfully when exactly  equal rows or columns are present in C), we can simply  add a small amount of noise at the start of the optimization:  C′(0) = C + ϵ, ϵij∼N(0, 10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Another important detail  is to normalize the gradient to have a fixed norm: the small  amount of noise is amplified (to break ties) only when slots  are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This is done without having to resort to large  learning rates, which would impact the stability of optimiza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We thus propose to repeat the following for T steps  with a learning rate λ:  C′(t+1) = C′(t) − λ ∇C′(t)A′(t)  ��∇C′(t)A′(t)��  (11)  with A′(t) = H(sinkhorn(C′(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  (12)  Note that exact minimization is not needed, nor necessarily  desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' A critical aspect is that C′(T ) is still related to  the initial C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' For example, if C is square, then C′ = 1023I  would always “successfully” minimize Equation 9, but also  solve a problem unrelated to C and provide no signal to  learn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This is why we initialize C′(0) as C with a  small amount of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Note that if this noise is too high (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  ϵij∼N(0, 1)), the solution can again lose correspondence  with the actual cost matrix C that we are trying to compute  the transport map for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We develop a more complicated  version that explicitly enforces C and C′ to be similar  (Appendix B) but find that it is not necessary in practice as  long as the noise in the initialization is reasonably low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Properties  Equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  SA-MESH is now exclusively multiset-  equivariant: it is multiset-equivariant because all the in-  dividual operations are multiset-equivariant, but it is not  set-equivariant because equal slots will no longer receive  the same transport plans due to the tiebreaking from the  noise and subsequent optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This gives our method  more representational power than standard SA and SA-SH  because it is no longer restricted by set-equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  A useful side-effect is that random initialization of slots  is no longer necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Since ties can be broken by this  entropy minimization, it is no problem to initialize all slots  to be the same vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In standard slot attention, the amount  of noise in Z(0) needs to be just right: too low, and the  set-equivariant model has difficulties breaking ties between  these similar slots (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' too high, and the model  can become unreliable from the noisiness (Kipf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Furthermore, randomly initializing the slots does not prevent  them from collapsing to the same values in a later iteration—  such as when there are fewer objects to model than slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In contrast, a tiny amount of noise in C′ (as long as it  is above machine precision after sinkhorn) is sufficient for  tiebreaking and avoiding collapse, and any other effect of the  noise can be optimized away through the gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  SA-MESH offers a significant improvement in  computation time when compared to SA-EMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This is be-  cause it only requires the evaluation of the Sinkhorn al-  gorithm for a small number of optimization steps (in our  experiments, we found little improvement above four steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  While it may not be as fast as SA-SH, its exclusive multiset-  equivariance can speed up learning by making the model  more powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In many cases, the computation required  for SA-MESH is outweighed by other components of the  model such as the image processing part that creates the in-  put multiset for SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Since each optimization step in MESH  needs to solve a similar optimal transport problem, we can  reuse the result from the previous iterations to improve the  efficiency, as we describe in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Another benefit over SA-EMD is that gradient  computation is simple since we can fully rely on automatic  Unlocking Slot Attention by Changing Optimal Transport Costs  10−2  10−1  100  101  Scaling factor on input  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  Normalized entropy of transport map  MESH λ = 1  MESH λ = 2  MESH λ = 3  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='03  10−2  10−1  100  101  Scaling factor on input  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  Normalized gradient norm w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' C  MESH λ = 1  MESH λ = 2  MESH λ = 3  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='03  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' MESH reduces entropy while maintaining reasonable gradients across a large range of attention values, in particular when  attention is uncertain (low scaling factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Meanwhile, Sinkhorn (SH) provides nontrivial gradients in a much smaller range with entropy  reduction being ineffective when the scaling is too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Entropy of transport map (left) and corresponding gradient norm (right) when  varying the scaling factor on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The entropy is normalized to have 1 as maximum entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The gradient norm of each method is  normalized to have a maximum of 1, we show the unnormalized gradient norms in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  differentiation instead of having to manually estimate gra-  dients for the black-box EMD solver in SA-EMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We find  experimentally that we do not even need to differentiate the  gradient updates in Equation 11 themselves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' the gradients of  C′(T ) can simply be passed along to C in a straight-through  manner (Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2013) without reduction in perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' SA-MESH also has benefits in terms of the “quality”  of gradients over SA-SH, which we explain in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Comparison to changing temperature  An alternative for breaking ties is to add noise to the cost  matrix (like SA-MESH), but then simply reduce the temper-  ature of the Sinkhorn algorithm, which corresponds to re-  ducing the amount of entropy regularization (Cuturi, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  For sufficiently low temperatures, this should also be able to  map equal inputs to different slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' How does this much sim-  pler approach compare to MESH, which minimizes entropy  by gradient descent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Sinkhorn with low temperatures can be thought of as analo-  gous to softmax with low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' As the temperature  decreases, the behavior of softmax becomes more similar  to an argmax, but the gradients become ill-behaved as a re-  sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Similarly, with Sinkhorn, low temperatures may result  in gradients that make it difficult or impossible to learn a  good cost matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We now make this notion more concrete  and show that MESH can reduce entropy while maintaining  “good” gradients for a much larger range of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In Figure 1, we have the following set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Starting from a  10 × 10 identity matrix, we scale it by a varying factor to  obtain the cost matrix, then apply either Sinkhorn (for dif-  ferent temperatures τ) or MESH (for different learning rates  λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We then measure the entropy of the resulting transport  map (left), as well as the norm of the gradient of this entropy  with respect to the cost matrix (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Gradient norms close  to zero slow down learning because the gradients provide  little to no information on how to learn the cost matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  When the scaling factor is large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' >10, attention is con-  fident), the behaviors are similar (low entropy and small  gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' When the scaling factor is small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1, at-  tention is not confident), MESH is still always able to reduce  entropy while maintaining reasonable gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In contrast,  Sinkhorn only has nontrivial gradients in a relatively small  range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' outside of this range, learning is difficult because the  gradient norms become close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Thus, a trade-off has to  be made for Sinkhorn, which is not necessary for MESH:  either τ is high and low scaling factors result in virtually  zero gradients (with no reduction to the entropy), or τ is low  and higher scaling factors result in virtually zero gradients  instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The MESH learning rate λ can be used to control  the shape of how entropy is reduced, while the Sinkhorn  temperature τ only changes the location and does not in-  crease the range of nontrivial gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In summary, MESH  effectively reduces entropy while maintaining well-behaved  gradients for a large range of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Experiments  We now experimentally evaluate SA with our optimal trans-  port variants, with a particular focus on comparing SA  to SA-MESH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We open-source all of our code https:  //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='com/davzha/MESH and provide extra ex-  perimental details in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Random objects detection  First, we evaluate the effect of the different SA variants in  a simplified object detection setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The aim is to assess a  Unlocking Slot Attention by Changing Optimal Transport Costs  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Random objects detection, measured in RMSE divided  by standard deviation σ of random objects (lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' An  always-predict-zeros baseline has a normalized RMSE of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Av-  erage of 5 random seeds, except ∗ denotes exclusion of one bad  outlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Standard deviation between seeds is negligible (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Model  σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='01  SA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='44∗  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='65  SA-SH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='39∗  SA-EMD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='00  SA-MESH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='31  model’s ability to detect and distinguish similar objects in  a controlled setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Given a multiset containing k random  32d vectors sampled from N(0, σ2I) and h zero vectors,  the goal is to copy only the k random vectors into the k  slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In the context of object detection on images, this can  be thought of as detecting k different objects, each of which  occupies exactly one position in a feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In this setting, we want the object information to be pre-  served as accurately as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' To vary the difficulty of  this task, we change the standard deviation σ of the random  vectors to be copied and measure the error relative to this  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' A lower standard deviation corresponds to a harder task  because the elements become more similar to each other  and to the background zero vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Table 1 shows our results for k = 5 objects and  h = 100 background elements for varying difficulties σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  First, we see that SA-SH and SA-MESH can detect the ob-  jects on the hardest setting of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='01 more accurately  than SA on σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This demonstrates the general value  of our proposed optimal transport perspective in the con-  text of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' SA-MESH with its lower entropy outper-  forms SA-SH: the SA-MESH performance for a specific  σ is roughly equivalent to SA-SH at a σ ten times higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  This shows the benefits of reducing entropies in the trans-  port map through MESH, which helps objects stay distinct  from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Meanwhile, SA-EMD (also with low en-  tropy transport maps) performs similarly to SA-MESH on  σ = 1, but degenerates to the always-predict-zeros baseline  on σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We attribute this to the inherently imprecise  gradient estimation leading to learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' CLEVR property prediction  Next, we test SA and our proposed variants on a more realis-  tic object detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' CLEVR (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2017) is a  synthetic dataset containing images with up to ten objects in  a 3d scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Each object is sampled with varying sizes, mate-  rials, shapes, and colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The task is to predict the multiset  of objects with their properties and 3d position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Following  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2019), we evaluate using average precision  (AP) at different distance thresholds for the 3d coordinates  of the predicted objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Table 2 shows that SA-MESH achieves the best  SA results (and state-of-the-art results on some metrics)  while only increasing run time by a small amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' These  results are followed by SA-EMD, which has slightly worse  results (likely due to the inherently imprecise gradient esti-  mation) but takes over three times longer to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Again, we  see that all three optimal transport-based methods greatly  outperform the baseline SA, which validates the benefits of  the optimal transport perspective in attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Note that we only provide the results for iDSPN (which is  also exclusively multiset-equivariant) for context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' this is not  supposed to be a direct comparison due to the significant  difference in approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In general, slot attention through the  use of attention has the benefit of not needing to compress  the input into a single vector (global scene representation)  like iDSPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The resulting object-centric inductive bias and  the relative simplicity have allowed for wider adoption and  success of SA over iDSPN (Kipf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Sajjadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022), which makes our  improvements to SA meaningful, even if in this case some  metrics are worse than iDSPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Unsupervised object discovery on images  In this task, the objective is to discover objects without the  supervision of what the objects are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We follow Locatello  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2020) and set up an image reconstruction task with  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' CLEVR property prediction, average precision (AP) in % (mean ± standard deviation) over 5 random seeds, higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  SA-MESH improves over all other SA variants at only a small computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Note that the exclusively multiset-equivariant iDSPN  is not object-centric, so they are not directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' SA (original) results are copied from Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2020), iDSPN results from  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Models with † use the improvement by Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022), see Appendix F for our results without †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Model  AP∞  AP1  AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='25  AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='125  AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0625  Time  iDSPN (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022)  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9  —  SA (original) (Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2020)  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  —  —  SA†  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2 h  SA-SH†  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3 h  SA-EMD†  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3 h  SA-MESH†  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4 h  Unlocking Slot Attention by Changing Optimal Transport Costs  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Object discovery on images results in Multi-dSprites in %  (mean ± standard deviation) over 5 random seeds, higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  SA-MESH outperforms the other models and has lower variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Model  FG-ARI  FG-mIoU  SA (Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2020)  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3  —  SA  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  SA-SH  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  SA-MESH  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Object discovery on images results in ClevrTex in %  (mean ± standard deviation) over 5 random seeds, higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  SA-MESH outperforms the other models and has lower variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Model  FG-ARI  FG-mIoU  SA  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9  SA-SH  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  SA-MESH  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  slots as the latent bottleneck using SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' These slots are in-  dividually decoded into object-specific images, each com-  prising the RGB color channels and an alpha mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' These  object-specific images are then combined to form the fi-  nal reconstructed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' To evaluate the performance, we  compare the per-slot alpha masks to the actual object seg-  mentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The goal is thus for image reconstruction  with a multiset bottleneck to lead to a decomposition of  the scene into individual objects, with each object being  modeled by a distinct slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We evaluate on the Multi-dSprites dataset, which is the  only benchmark presented by Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2020) that  still presented a challenge (possibly due to the presence  of highly overlapping objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Additionally, we test on  ClevrTex (Karazija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2021), a synthetic dataset similar  to CLEVR that introduces the added challenge of different  textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In line with prior work (Kipf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022), we  evaluate the Foreground Adjusted Rand Index (FG-ARI)  and Foreground mean Intersection over Union (FG-mIoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  To ensure FG-mIoU is permutation-insensitive, we use the  Hungarian algorithm to find the best matching between the  masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' See Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Note that we no  longer test SA-EMD because the larger input size compared  to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2 makes its training time infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Table 3 and Table 4 show that on both Multi-  dSprites and ClevrTex, SA-MESH achieves significantly  higher FG-ARI and FG-mIoU compared to all baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  SA-SH and SA-MESH improve especially in mIoU, which  Karazija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2021) argue is a better metric than ARI to  evaluate the accuracy of object masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In Figure 2, we observe that minimizing the  entropy leads to much sparser attention maps for SA-MESH  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Attention maps (first row) and alpha masks (second row)  for each of the six slots in SA-MESH on Multi-dSprites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The  attention focuses on only a small region inside each object, but the  model still reconstructs the full objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' See Appendix H for more  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  in comparison to SA: SA-MESH only attends to a small  part within each object, rather than the whole object as is  usually the case for SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Keep in mind that attention maps  are only used to route the required information to each slot,  and it is the slots themselves that represent the individual  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Attending to a small part of each object is sufficient  because the receptive field of the CNN image encoder lets  it move information from the edge of an object into the  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The accuracy of the final alpha masks of SA-MESH  shows that this happens successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In Appendix H, we  see that SA-MESH attends less to the background than SA  on Multi-dSprites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Similar to Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2020), we observe that some SA  runs fail to separate the individual objects into different slots  despite having a low reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In those cases,  the attention maps divide the image into separate regions,  independent of the image content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' SA-MESH ensures that  the attention maps are sparse, which helps to avoid these  kinds of failure modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Appendix H shows this difference  between SA-MESH and SA on ClevrTex: SA-MESH dis-  covers objects successfully in most cases, and usually, only  the background is split into spatial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Unsupervised object discovery on video  As we mentioned in Section 1, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022) observed  issues with SA when applied to videos where multiple ob-  jects can enter the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' To evaluate our method in this  scenario, we build two variants of the CLEVRER video  dataset (Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2019) where the number of visible ob-  jects varies over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We only use two frames from each  video: the first frame, and either the 16th frame (short time  difference, CLEVRER-S) or the 128th frame (long time dif-  ference, CLEVRER-L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We do this to evaluate SA without  the dynamics prediction component that Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022)  are concerned with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In CLEVRER-S, the total number of  objects increases by two or more in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5% of the videos,  while in CLEVRER-L, this occurs for 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8% of the videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We evaluate FG-ARI and FG-mIoU on the two frames indi-  vidually, which is then averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We also compute a tempo-  ral consistency (TC) metric as the fraction of objects that are  correctly captured by the same slot (details in Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Unlocking Slot Attention by Changing Optimal Transport Costs  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Video object discovery results on CLEVRER-S and CLEVRER-L in % (mean ± standard deviation) over 5 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  SA-MESH outperforms all other models in terms of quality of predicted masks (ARI, mIoU) and achieves high temporal consistency  (TC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' See Appendix H for example masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  CLEVRER-S  CLEVRER-L  Model  FG-ARI  FG-mIoU  TC  FG-ARI  FG-mIoU  TC  SA  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  SA fixed noise  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0±34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  SA learned noise  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  SA-SH  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  SA-MESH  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Table 5 shows that SA-MESH outperforms the  other models by a significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The only exception is  on CLEVRER-S for temporal consistency—we can close  this gap simply by reducing the MESH learning rate λ (five  run average: 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0% FG-ARI, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7% FG-mIoU, 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1% TC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  The proposal by Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022) of adding noise to slots  to prevent them from collapsing helps SA on CLEVRER-L,  but to a lesser extent than SA-MESH, which uses exclusive  multiset-equivariance to prevent collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This is evidenced  by the much better FG-mIoU of SA-MESH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Related work  Object-centric learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Object-centric learning aims to  model visual input data in terms of multiple “objects” rather  than a global representation or grid of feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This  decomposition can be considered an abstraction of the input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  reasoning over a small number of objects is intuitively more  efficient than over a feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Scenarios with multiple  independent objects are ubiquitous in natural data, so it is  desirable to model them well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Karazija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2021) classify  object-centric learning methods into three categories: pixel-  space approaches which group related pixels together (Greff  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Pervez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022), glimpse approaches which  sequentially extract patches from the input (Crawford &  Pineau, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Jiang & Ahn, 2020), and  sprite approaches which learn a dictionary of object appear-  ances (Monnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Smirnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We choose to apply MESH on specifically the pixel-space  based slot attention because of its simplicity in approach  (cross-attention with GRU updates), the lack of assumptions  on what an object is (which makes it a general technique),  and its set-equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Since Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022) show  a specific limitation with set-equivariance, there is a clear  path towards improvement, namely making it exclusively  multiset-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We accomplish this in this paper, with  strong results backing up the benefits of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Another approximation of optimal  transport can be obtained through the Sliced Wasserstein  Distance (Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' It performs tiebreaking  through the use of numerical sorting and is thus exclusively  multiset-equivariant, but it lacks precise 1-to-1 associations  between inputs and slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This can especially be a problem  with varying input sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We tried approaches based on this  method as a replacement for standard cross-attention but did  not obtain any competitive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In a similar direction to Sinkhorn, which performs entropy-  regularized optimal transport, Blondel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2018) study  L2-regularized optimal transport problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' While their  solver is faster than unregularized optimal transport and  obtains lower entropy solutions than Sinkhorn, similarly to  Sinkhorn the convexity of the problem means that it cannot  break ties effectively on its own, even with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In SA-  MESH, we can replace Sinkhorn with this method, but we  found that it was too slow comparatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Discussion  We introduced several variants of slot attention that can  break ties between slots which enables better modeling of  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In particular, MESH is a promising method that  enhances cross-attention and, as a result, grants slot atten-  tion the property of exclusive multiset-equivariance, while  maintaining its learnability and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  While minimizing entropy in MESH has a nice symmetry  with entropy-regularized optimal transport, it is not clear  whether a sparse attention map is always desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We try to  learn a neural network on the transport map in Appendix G,  but find no improvements over simply using the entropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  the derivative of the learned objective ends up with a similar  shape to that of the entropy, which suggests that entropy is  indeed a reasonable choice to minimize for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  An interesting direction for future research is to explore  the use of MESH in other contexts than slot attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' For  example, Sinkhorn is used by Pe˜na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022) for merging  the weights of two neural networks together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' MESH could  be used as an alternative in this and other contexts to replace  Sinkhorn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In general, we believe that optimal transport  will continue to play an important role in deep learning,  with MESH being a way of bringing tiebreaking into the  picture without paying the usual speed penalty associated  with optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Unlocking Slot Attention by Changing Optimal Transport Costs  Acknowledgements  The work of DWZ is part of the research programme  Perspectief EDL with project number P16-25 project  3, which is financed by the Dutch Research Coun-  cil (NWO) domain Applied and Engineering Sciences  (TTW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This research was partially supported by Calcul  Qu´ebec (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='calculquebec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='ca), Compute Canada  (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='computecanada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='ca) and the Canada CIFAR AI  Chair Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Simon Lacoste-Julien is a CIFAR Associate  Fellow in the Learning Machines & Brains program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=' Differentiation of blackbox combinatorial  solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In International Conference on Learning Repre-  sentations (ICLR), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Sajjadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=', Mahendran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', van  Steenkiste, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', Paveti´c, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', Luˇci´c, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=', and Kipf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Object scene representation transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In Advances in Neural Information Processing Systems  (NeurIPS), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Sander, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=', and Peyr´e, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Sink-  formers: Transformers with doubly stochastic attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In International Conference on Artificial Intelligence and  Statistics (AISTATS), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Smirnov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=', Guizilini, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=',  and Solomon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=' Advances in Neural Information Processing  Systems (NeurIPS), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Vaswani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=', Jones,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=' Atten-  tion is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In Advances in Neural Information  Processing Systems (NeurIPS), 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Villani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content='  Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=' Multi-  modality cross attention network for image and sentence  matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In Conference on Computer Vision and Pattern  Recognition (CVPR), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=' Slot-  former: Unsupervised visual dynamics simulation with  object-centric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='05861,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Yi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content=' Clevrer: Collision events for video  representation and reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In International Conference  on Learning Representations (ICLR), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Zaheer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', Kottur, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', Ravanbakhsh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', Poczos, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=',  Salakhutdinov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', and Smola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Deep sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In Advances in Neural Information Processing Systems  (NeurIPS), 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', Hare, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', and Pr¨ugel-Bennett, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Deep set pre-  diction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In Advances in Neural Information  Processing Systems (NeurIPS), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', Lacoste-Julien, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', Burghouts,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', and Snoek, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Multiset-equivariant set pre-  diction with approximate implicit differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In Inter-  national Conference on Learning Representations (ICLR),  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Marginals in Sinkhorn and EMD  As we mention in the main text, we need to account for the  (typical) case of the number of inputs n and the number of  slots m differing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' with a cost matrix C ∈ Rm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The  problem is that it is impossible to make every row and every  column of the transport map sum to 1 when the number of  rows and columns is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Fortunately, there is standard  practice for how to deal with this case in optimal transport  (Peyr´e & Cuturi, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Cuturi, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We can define non-negative marginals a ∈ Rm and b ∈ Rn  that specify the row and column sums of the transport map  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' If �  i ai = �  j bj, then convergence is as  normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In our case, we learn both a = m · softmax(ha(Z)) and  b = m·softmax(hb(X)) with neural networks ha : Rd →  R and hb : Rc → R that are shared across the m slots or n  input elements respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' These allow the model to put  focus on important input elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' the inputs corre-  sponding to objects) and ignore unimportant input elements  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' the inputs corresponding to the background), as well  as put focus on the relevant number of slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Since both  softmaxes sum to one, we have �  i ai = �  j bj = m so  there is no problem with convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  For the Sinkhorn algorithm, it now repeatedly alternates  normalizing all the rows to sum to a, then all the columns  to sum to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' For the EMD solver that we use (Bonneel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2011), these marginals are standard parameters in  the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In the main text, we omit these marginals  whenever we refer to sinkhorn or emd for simplicity of  notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Unlocking Slot Attention by Changing Optimal Transport Costs  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Enforcing C′ to be similar to C  The following discussion is not critical to understanding  the main text, since we find empirically that with the right  initialization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' C′ = C + ϵ with ϵij ∼ N(0, 10−6)),  learning is not a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We only find that this variant is  necessary with an initialization such as ϵ ∼ N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' As  we mention in the main text, the amount of noise is not  important as long as it remains above machine precision  after applying sinkhorn, which can be easily checked a-  priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In order to enforce C′ to be related to C more explicitly,  we define the following objective instead:  MESH(C) = arg min  C′  [H(sinkhorn(C′))  +α||sinkhorn(C′)S − sinkhorn(C)||2]  (13)  The second term relates C′ to C directly with a regulariza-  tion factor α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' S is a similarity matrix, which we will define  shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The idea behind it is to allow costs to be freely  changed among similar slots, but disallow this for dissim-  ilar slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The aim of ||sinkhorn(C′)S − sinkhorn(C)||2  is thus to make sure that sinkhorn(C′) looks the same as  sinkhorn(C) after allowing weight in the transport map to  be moved around among similar slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Example  Consider the case where we have three slots:  Z = [x, x, y] and three inputs X = [α, β, γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Let us  assume for this example that the cost matrix prefers asso-  ciating γ with y and both α and β with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Computing  unregularized optimal transport solutions would therefore  give us either of two solutions:  T1 =  �  �  1  0  0  0  1  0  0  0  1  �  �  or  T2 =  �  �  0  1  0  1  0  0  0  0  1  �  �  (14)  However, the Sinkhorn algorithm is unable to break the  tie between the two x slots, so even with a temperature  approaching 0, we obtain the following result:  sinkhorn(C) =  �  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0  0  0  1  �  �  (15)  Suppose we have a similarity matrix ˜S ∈ Rm×m (m is the  number of slots) that measures pairwise similarities ranging  from 0 to 1:  ˜S =  �  �  1  1  0  1  1  0  0  0  1  �  �  (16)  The first two slots are similar amongst themselves but dis-  similar to the y slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' If we normalize each column of ˜S to  sum to 1 to obtain S, then we see the following:  �  �  1  0  0  0  1  0  0  0  1  �  �  �  ��  �  T1  �  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0  0  0  1  �  �  �  ��  �  S  =  �  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0  0  0  1  �  �  �  ��  �  sinkhorn(C)  (17)  �  �  0  1  0  1  0  0  0  0  1  �  �  �  ��  �  T2  �  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0  0  0  1  �  �  �  ��  �  S  =  �  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0  0  0  1  �  �  �  ��  �  sinkhorn(C)  (18)  This means that sinkhorn(C′) = T1 and sinkhorn(C′) =  T2 are both valid solutions for the minimization of  ||sinkhorn(C′)S − sinkhorn(C)||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Note that any other  permutation matrix for T (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' one where there is not a 1 in  the bottom right corner) would not be a valid solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This  restricts Equation 9 to only consider transport maps that are  convex combinations of T1 and T2 for this example, with  the entropy minimization preferring T1 and T2 specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  The small amount of noise in the C′ initialization arbitrarily  makes it prefer one of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Definition  We define the similarity matrix  ˜Sij  =  g(Zi, Zj), where g : Rc × Rc → R is a small neural  network that takes pairs of slots as input and produces a  similarity score as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We then normalize each column  of ˜S to sum to 1 by applying softmax on each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  S = softmax( ˜S)  (19)  If we set up a training task for the example described above,  we observe that S is learned to be virtually the same as the  S we use in the example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Sinkhorn algorithm implementation  Ideally, we want to compute the Sinkhorn algorithm for as  few steps as possible since it is used in every MESH step,  which means that we also have to differentiate through the  Sinkhorn algorithm in every MESH step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' However, we also  need to run it for a sufficient number of steps for (good  enough) convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Fortunately, because we repeatedly  run the Sinkhorn algorithm on similar inputs over different  MESH steps, we can optimize its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The  idea is that the gradient descent for minimizing the entropy  makes small changes (especially near MESH convergence),  which allows us to reuse computation between different  MESH steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  If the entropy minimization has converged, then the  Sinkhorn output does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This means that if we keep  track of the operations we performed in the previous SA-  MESH iteration, we can simply reapply them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Let us take a  look at these operations: the Sinkhorn algorithm (multiplica-  tively) rescales rows and columns repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Multiplication  Unlocking Slot Attention by Changing Optimal Transport Costs  0  5  10  15  MESH iteration  10−8  10−7  10−6  10−5  Error compared to full convergence  Baseline  Reuse u and v  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Mean absolute error gap to the fully converged Sinkhorn  for the two different Sinkhorn implementations for varying num-  bers of MESH iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Both implementations always use 5  Sinkhorn iterations and thus have comparable computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Reusing u and b is clearly more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  is commutative, so we can collect all the row normalizations  together into a single row normalizer u and all the column  normalizations into a single column normalizer v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Applying  these two normalizers on the (exponentiated) cost matrix  gives us exactly the same result as if we had manipulated  the matrix with the normalizations directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The next time  we run the Sinkhorn algorithm with a slightly changed cost  matrix, we can bootstrap the algorithm with the previously-  found u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We can then run a few more iterations to  account for the changes in the cost matrix, giving us a new  u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  In Figure 3, we ablate whether there are benefits to reusing  u and v in SA-MESH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In particular, we compute the ideal  solution by running the Sinkhorn algorithm until conver-  gence at every ME iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Then we examine whether  reusing u and v helps close the gap to the ideal solution  when we limit the number of SH iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' At 1 ME iter-  ation there are no u and v for bootstrapping available, so  both exhibit the same gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' At more than 1 ME iteration we  observe that reusing u and v helps in narrowing the gap  to the ideal solution, or equivalently, can achieve the same  approximation with fewer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We do not claim that this technique of collecting Sinkhorn  operations into u and v is novel, as there are several imple-  mentations that use this trick for performing the Sinkhorn  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Usually, it is a minor implementation detail  since the approaches of manipulating the matrix directly  and collecting normalizations into u and v are mathemati-  cally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In our case however, this formulation leads  to a concrete benefit due to our setup where we run the  Sinkhorn algorithm on similar inputs, which allows us to  reuse u and v in a beneficial way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Gradients of tempered SH and MESH,  without normalizing their scale  In Figure 4, we show the same gradient norms as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  However, rather than normalizing each model to have a  maximum of 1 (which is more useful for visualizing them all  at once), we maintain their native scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This shows that  changing the SH temperature has a major effect on the scale  of the gradients while changing the MESH learning rate  only has minor effects on the gradients (but still significant  effects on entropy reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In other words, the amount of  entropy minimization can be changed without major impacts  on the hyperparameters of other parts of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Experimental details  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Random object detection  We generate a dataset of 64,000 data points to train on, each  being a multiset with five 32-dimensional objects sampled  from N(0, σ2I) and 100 zero vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We directly apply  slot attention on this: the dimensionality of the slot attention  weights are all 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Since we know that there are always  five objects, we set the number of slots to five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The loss is  computed by computing a mean squared error between all  pairs of predicted and ground-truth objects, then using the  Hungarian algorithm find the matching with the lowest loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We find that using implicit differentiation of slots (Chang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022) is significantly more stable on this dataset, so  we use it for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We train all models for 20 epochs  with a batch size of 64 (1,000 steps each epoch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Object detection on CLEVR  We largely follow same training setup as DSPN and iDSPN  (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' 2022) and adapt the slot attention imple-  mentation to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Matching Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022) and Locatello  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2020), we resize the input images to 128x128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' To  compute the loss, we use the Hungarian algorithm to com-  pute the least-cost matching between predicted objects and  ground-truth objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Object discovery  We compute the temporal consistency (TC) by first matching  the predicted objects in each frame to their corresponding  ground-truth objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Then, we calculate the proportion  of objects that have the same slot in both frames out of  those that appear in both frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' To find the optimal assign-  ment between the predicted and ground-truth objects, we  first compute all pairwise IoUs between the predicted and  ground-truth masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We then invert these IoUs by applying  1 − IoU and use the Hungarian matching algorithm to find  Unlocking Slot Attention by Changing Optimal Transport Costs  10−2  10−1  100  101  Scaling factor on input  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  Gradient norm w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' C  MESH λ = 1  MESH λ = 2  MESH λ = 3  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='03  10−2  10−1  100  101  Scaling factor on input  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='030  Gradient norm w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' C  MESH λ = 1  MESH λ = 2  MESH λ = 3  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  SH τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='03  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Same gradient norm plot as Figure 1 but without normalizing each model to have a maximum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The only difference between  left and right is the scale of the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This shows that the gradient norm for tempered SH varies drastically with temperature, while the  gradient norm of MESH remains more similar across learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This makes tuning the learning rate hyperparameter easier as it can  be considered more independently from other hyperparameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  the best match—the one with the highest mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We compute the mIoU in a similar manner as the TC metric,  by finding the matching between the ground-truth objects  and the predicted objects that results in the highest mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Multi-dSprites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We closely follow the experimental setup  described by Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Specifically, we use the  same image encoder, decoder, and hyperparameters where  applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2020) used 500k training steps,  while all of our runs were trained for 530 epochs, which  results in slightly fewer than 500k training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  ClevrTex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We pre-process the images by applying the  same center crop as suggested by Karazija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2021) and  resize the images to 64×64 resolution instead of 128×128  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' This allows us to use the same neural network  architecture as we did for the Multi-dSprites dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Since the dataset has more complicated visuals we increase  the model size by increasing the channel sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In particular,  we double the number of channels in the image encoder and  decoder to 64, and we double the dimensions of the slots to  128 (with the MLP in slot attention having an intermediate  dimension of 256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We again train all models for 530 epochs  which correspond to around 330k gradient update steps in  this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The maximum number of objects in an image is  10, so we set the number of slots to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' CLEVRER  We construct the datasets from the 20k videos in the  CLEVRER dataset, by picking two frames at specific  timesteps from each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In the CLEVRER-S dataset,  we use the first and 16th video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In the second dataset,  we use the first and 128th (last) frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The proportion of  examples where new objects appear increases with the time  gap between the two frames, and similarly the amount of  displacement for objects that are in both frames increases  too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We show examples from the two dataset variants in  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  For the distance function d that computes the cost matrix of  the optimal transport problem in SA-MESH, we empirically  find that the cosine distance works better in this case than  the l2 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We suspect that since the l2 distance allows  the slots to be pushed arbitrarily far apart that learning might  slow down in the later stages of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  We extend the model which we used in the Multi-dSprites  experiment to video data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Our setup is similar to Kipf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  (2022), but we do not use a predictor model (except for  the SA learned noise baseline) to update the slots when  transitioning from one video frame to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In particular,  the model first applies the image encoder to all video frames  independently to compute the input feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Next,  the SA (or our proposed variants) is applied to the features  of one video frame at a time, and every time the slots are  initialized from the slots of the previous frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Finally, each  image is decoded independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The learned noise baseline  uses a 2-layer MLP with LayerNorm to predict the mean  and variance of a Gaussian, from which the initial slots are  sampled for the next frame, following the stochastic SAVi  setup by Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We use 8 slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' CLEVR object prediction results  In Table 6 we show our results for CLEVR object prediction  without implicit differentiation of slots (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  All results are slightly lower than the results reported in  Unlocking Slot Attention by Changing Optimal Transport Costs  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Results on CLEVR object property multiset prediction, average precision (AP) in % (mean ± standard deviation) over 5 random  seeds, higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' All SA results are based on our re-implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' SA (original) results copied from Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2020),  iDSPN results from Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Model  AP∞  AP1  AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='25  AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='125  AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0625  Time  iDSPN (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2022)  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='2 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='9  —  SA (original) (Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=', 2020)  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='1  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='4  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='6  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='5 h  Table 2, but the overall message remains exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  The only major difference is that SA-SH performs much  worse compared to SA-SH†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Alternative MESH objective  In Section 4 we choose the entropy as the inner objective  function because the goal was to reverse the effect of the  entropy-regularized optimal transport version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Alternatively,  it is possible to learn a neural network with scalar inputs  and outputs in place of the entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' For the neural  network, we choose a simple 2-layer MLP with ReLU acti-  vations and 32 hidden dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We plot the derivative of  a learned objective function in Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Its shape is similar  to the derivative of the entropy Figure 5a, but learning it  incurs additional compute compared to simply using the  entropy function H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Justified by this analysis we can directly use the entropy  function as the MESH objective for improved efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Empirically we observe that we do not even need to back-  propagate through the gradient descent optimization proce-  dure of MESH and it suffices to treat MESH as the identity  function during backprop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' One perspective that might ex-  plain this: since the negative derivative of H is a monotonic  increasing function, changes to the input will also affect the  output in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content='0  −1  0  1  2  3  4  5  6  (a) H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+page_content='5  (b) MLP  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Derivative of the objective function in MESH  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Object discovery example results  In the following, we show examples of SA and SA-MESH  performing object discovery on the various datasets that  we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' We always show the original image on the left,  followed by either the attention maps for each slot or the  final alpha masks for each slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' The attention map or the  alpha masks are multiplied with the original image to make  it easier to tell how precise their locations are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Unlocking Slot Attention by Changing Optimal Transport Costs  (a) CLEVRER-S  (b) CLEVRER-L  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Examples from the two datasets derived from CLEVRER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Significant changes like multiple new objects appearing occur less  frequently in CLEVRER-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Objects can be significantly displaced from one frame to the other in CLEVRER-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Frame 1  Frame 128Frame 1  Frame 16Unlocking Slot Attention by Changing Optimal Transport Costs  (a) SA  (b) SA-MESH  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Attention maps for all three slot attention iterations for five different examples from the validation split of Multi-dSprites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Note  how the shade of SA is generally darker indicating higher attention values even in background areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In the third example, we can see how  SA models the pink and blue ellipses using one slot while splitting the purple heart over two slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In contrast, SA-MESH is able to route  the three objects into three different slots in the second slot attention iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Unlocking Slot Attention by Changing Optimal Transport Costs  (a) SA  (b) SA-MESH  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Predicted alpha masks from the validation split of Multi-dSprites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In general, the masks for SA-MESH are sharper than for SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  For example, in the last row, the brown heart is only recognizable as a blown blob for SA, but is a distinct heart shape for SA-MESH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' In  the third example, SA splits the purple heart into two slots, while SA-MESH models it with one as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Unlocking Slot Attention by Changing Optimal Transport Costs  (a) SA  (b) SA-MESH  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Attention maps (top) and masks (bottom) for four different examples from the validation split of ClevrTex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' SA commonly learns  that each slot should attend to a spatial region as opposed to a specific object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' SA-MESH on the other hand is better able to localize  individual objects, though the background is often still handled with a region-specific approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content='  Unlocking Slot Attention by Changing Optimal Transport Costs  CLEVRER-S  CLEVRER-L  SA  SA fixed noise  SA learned noise  SH  SA-MESH  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Example alpha masks for CLEVRER-S and CLEVRER-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' Possibly due to the difficulty of handling multiple new objects,  SA and its noise variants choose a region-based decomposition on CLEVRER-L instead of an object-based decomposition like on  CLEVRER-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
+page_content=' SA-SH and SA-MESH learn an object-based decomposition for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQf5jU_/content/2301.13197v1.pdf'}
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+Transformer Based Geocoding
+Yuval Solaz∗
+yuval.solaz@gmail.com
+Vitaly Shalumov∗
+vitaly.shalumov@gmail.com
+January 4, 2023
+Abstract
+In this paper, we formulate the problem of predicting a geolocation from free text as
+a sequence-to-sequence problem. Using this formulation, we obtain a geocoding model
+by training a T5 encoder-decoder transformer model using free text as an input and
+geolocation as an output. The geocoding model was trained on geo-tagged wikidump
+data with adaptive cell partitioning for the geolocation representation. All of the code
+including Rest-based application, dataset and model checkpoints used in this work are
+publicly available.
+1
+Introduction
+Social media such as Twitter and Wikipedia contains considerable amount of location-related
+text data. In this paper, we develop a model that learns to predict spatial probabilities from
+free text. Given a query sentence, the model outputs a discrete probability distribution over
+the surface earth, by assigning each geographical cell a likelihood that the input text relates
+to the location inside said cell. The resulting model is capable of localizing a large variety
+of sentences. Viewing the task as a hierarchical classification problem allows the model to
+express its uncertainty in the location associated with the text. The resulting model can be
+used for resolving ambiguity of the location references in the text. This capability is central
+to the success of finding exact location from free text. For example, Paris can refer to more
+than one possible location. In a context such as: The International Olympic Committee
+confirmed the city chosen to host the Olympic Games in 2024. The Games will be held in
+Paris, geocoding models like the one proposed in this paper can help in the resolution of
+the correct location.
+This work introduces the following contributions:
+• Synthesizing a dataset for supervised learning, including adaptive cell partitioning.
+• Formulating the geocoding problem as a sequence-to-sequence problem.
+• Training an end-to-end geocoding model using said formulation.
+• Publicly releasing the curated dataset, a Rest-based application and the T5 geocoding
+model.
+∗Equal Contribution
+1
+arXiv:2301.01170v1  [cs.CL]  2 Jan 2023
+
+2
+Related works
+The geolocation prediction from free text was extensively addressed in the literature. The
+authors of [1] presented a multi-level geocoding model that learns to associate texts to
+geographic locations. The downstream task was formulated as a multi-level classification
+problem based on multi-level S2 [2] cells as the output space from a multi-headed model.
+In [3], the surface of the earth was subdivided into thousands of multi-scale geographic
+cells. The authors trained a deep neural network using millions of geo-tagged images. We
+adopt the adaptive partition approach introduced in this paper. The authors of [4] created
+language models of locations using coordinates extracted from geo-tagged Twitter data.
+The locations were modeled at varying levels of granularity, from zip code to the country
+level. In [5], an end-to-end probabilistic model for geocoding text data was presented. In
+addition, the authors collected a novel data set for evaluating the performance of geocoding
+systems. The model-based solution, called ELECTRo-map was compared to the open source
+system available at the time of publication for geocoding texts for event data. An algorithm
+for estimating a distribution over geographic locations from a single image using a purely
+data-driven scene matching approach was provided in [6].
+The aforementioned approaches to geocoding can be viewed as sequential tasks of token
+classification (which is assumed to be solved perfectly) and geocoding. Unfortunately, this
+approach suffers from several drawbacks.
+First, it cannot handle correctly inputs that
+contain several descriptions of locations (for example: ”We live in country X city Y”).
+Second, it lacks the context of free text (for example: ”Country south of France”).
+3
+Text Geolocation with Transformer
+We pose the task of text geolocation as a sequence-to-sequence problem. The model trans-
+lates input text into hierarchy set of geographic cells which represent probability distribution
+over the surface of the earth. The model output is a sequence encoding of hierarchical cell
+representation of the surface of the earth.
+3.1
+Adaptive Cell Partitioning
+We use Google’s open source S2 Geometry library [2] to partition the surface of the earth into
+non-overlapping cells that define the target of our model. The S2 library defines hierarchical
+partitioning of the surface of a sphere by projecting the surfaces of an enclosing cube on
+it. The six sides of the cube are subdivided hierarchically by six quad-trees. A node in a
+quad-tree defines a region on the sphere called an S2 cell. Fig. 1 illustrates the S2 cells in
+several resolutions. Tab. 1 shows the resolution and number of cells for each S2 Geometry
+level.
+There are several reasons for choosing this subdivision scheme over a simple subdivision
+of latitude/longitude coordinates. Firstly, the latitude/longitude regions get elongated near
+the poles while S2 cells keep a close-to-quadratic shape, and secondly, S2 cells are mostly
+uniform in size (the ratio between the largest and smallest S2 cell is 2.08). A naive approach
+to define a tiling of the earth would be to use all S2 cells at a certain fixed depth in the
+hierarchy, resulting in a set of roughly equally sized cells. However, this would produce a
+very imbalanced class distribution since the geographical distribution of wikidata items [7]
+which was adopted in this paper, has strong peaks in densely populated areas see Fig. 2.
+2
+
+Figure 1: S2 Geometry hierarchical partitioning of the earth.
+Table 1: S2 Geometry levels.
+Level
+Average area
+Number of cells
+00
+85M km2
+6
+01
+21M km2
+24
+02
+5M km2
+96
+03
+1.3M km2
+384
+04
+330K km2
+1536
+05
+83K km2
+6K
+06
+20K km2
+24K
+07
+5K km2
+98K
+08
+1297 km2
+393K
+09
+324 km2
+1573
+10
+81 km2
+6M
+..
+..
+..
+29
+2.95 cm2
+1729 ∗ 1015
+30
+0.74 cm2
+7 ∗ 1018
+We therefore perform adaptive subdivision, based on the dataset item location: starting
+at the roots, we recursively descend each quad-tree and subdivide cells until no cell contains
+more than a certain fixed number (max cell samples parameter) of data points. By using
+this approach, sparsely populated areas are covered by larger cells and densely populated
+areas are covered by finer cells. This adaptive tiling has several advantages over a uniform
+one: (i) training classes are more balanced and (ii) it makes effective use of the parameter
+space because more model capacity is spent on densely populated areas. Fig. 3 demonstrates
+the S2 partitioning for our dataset.
+3
+
+Poland
+Germany
+Ukraine
+Kazakhstan
+Frances
+Mongolia
+Spain
+Italy
+NO!
+Turkey
+China
+Japan
+Tantic
+Afghanistan
+SouthKon
+cean
+Aira
+Iran
+Algeria
+Libya
+Egypt
+Pakistan
+Saudi
+Arabia.
+India
+Mali
+Niger
+Sudan
+Chad
+Venezuela
+Nigeria
+Ethiopi
+Colombia
+DRCongo
+Kenya
+Tanzania
+Indonesia
+Papua New
+Brazil
+Guinea
+Peru
+Angola
+Bolivia
+Namibia
+Mad
+agascar
+indian
+South
+Botswana
+Ocean
+Chile
+Atiantic
+AUS
+alta
+Ocean
+SouthAfrica
+Argentina
+New
+ZealandFigure 2: Data points distribution.
+Figure 3: S2 Geometry adaptive partitioning of our dataset.
+3.2
+Dataset
+This section describes the dataset used for training the model and for the evaluation ex-
+periments. The dataset was constructed from the wikidata archive [8]. The archive was
+filtered as to select all records with location label (approximately 8M records). See Tab. 2
+4
+
+80 -
+60
+40-
+to
+-20 -
+-40 -
+-60
+150
+100
+-50
+50
+100
+150for samples of wikidata records.
+Table 2: Wikipedia Geo Data samples.
+Id
+Latitude
+Longitude
+Text
+01
+53.96
+-1.08
+historic county of England
+02
+51.0
+10.0
+country in Central Europe
+03
+35.88
+14.5
+sovereign state in Southern Europe
+04
+57.30
+-6.36
+whisky distillery in Highland, Scotland, UK
+05
+47.39
+0.69
+city and commune in Indre-et-Loire, Centre-Val de Loire, France
+06
+-33.0
+-71.0
+sovereign state in South America
+3.3
+Data Labeling
+Using the adaptive cells, we label each data sample with the cell id containing the sample
+location. In order to keep the hierarchical nature of the cells in the label we use the following
+cell id encoding: the first digit represents the cell cube face with a digit between 0 to 5.
+The next digits represent for each level the corresponding node in the quad tree with a digit
+between 0 to 3. See Tab. 3 for a cell encoding example. Note that the label is a sequence
+of digits with a variant length between 1 and 1+max level.
+Table 3: Cell encoding.
+Cell description
+Cell representation
+Face cell 2
+2
+Subcell 2 of face cell 1
+12
+Subcell 1 of subcell 3 of face 4
+431
+4
+Model
+4.1
+Train
+The model presented in this paper is based on the T5-base (220M) pre-trained sequence-
+to-sequence model [9]. The model was fine-tuned on the wikidata dataset with the text
+records as input and the location cell encoding as the output target sequence. We chose to
+utilize the standard cross-entropy loss in the fine-tuning process. Evaluation of level-based
+weighting in the loss function is left for future work. We trained for 2M steps over 5 epochs
+with batch size of 12. We used the AdamW optimizer with learning rate 1.5e-5 and linear
+decay. We train the model on 80% of the wiki-data dataset and use the other 20% for in-
+domain evaluation. A diagram of our text-to-location framework with a few input/output
+examples is shown in Fig. 4.
+5
+
+Figure 4: A diagram of our text-to-location framework.
+Figure 5: Prediction results for the text ”Haifa”. The predicted S2 cell in blue and its
+ancestor cells in white.
+4.2
+Evaluation
+4.2.1
+Inference
+The inference of our resulting model is as follows: given a sentence we predict the output
+sequence. Naturally it wold be 1 digit between 0 to 5 representing the predicted cube face,
+followed by up to 9 digits between 0-3 representing the predicted s2cell on each level. This
+sequence can easily converted to a s2 cell, which represents the probability distribution over
+the earth for the input text location. We used a beam search with size of 10. See Fig. 5 for
+inference example.
+4.2.2
+Evaluation Metric
+The most straightforward classification metric is the ”accuracy measure” where only predic-
+tions with a full match between the model’s output and the true label counts as a successful
+prediction. We call this metric ”Flat Accuracy” (as oppose to ”Hierarchical Accuracy”).
+6
+
+"historic county of
+England"
+41023
+'sovereignstate in
+Southern Europe'
+1221
+T5Geo-Location
+"sovereign state in
+123331
+South America"CYPRUS
+ARRAQQF
+MAGUSA
+Halfa
+Famagusta
+Hamah
+(Ammochostos,Gazimagjusa)
+SIACypray
+Haifa
+Salamiyah
+AlMayadin
+DHEKELIA (EASTERN)
+TARTUS
+S
+RIA
+LARNAKA
+Tartus
+UNITED KINGDOM
+Hims
+nassol
+Tall Kalakh
+AbuKamal
+Tripoli
+DAYR AZZAWR
+LIBAN
+NORD
+AlQaryatayn
+EL BEQAA
+MON
+LIBAN
+AnNabk
+Beirut
+LEBANON
+Yabrud
+HIMS
+BEYROUTH
+Zahlah
+Sidon
+AL ANBAR
+IMtHenpon
+Damascus
+2814m
+DLMASHQ
+Habbush.
+LIBAN SU
+Nabatiyee Tahta
+Tyre
+NABATIYE
+QUNAYTIRAH
+DAR'A
+RIF DIMASHQ
+ArRutbah
+Haifa (98%)
+Zefat
+AS SUWAYDA'
+Haifa
+IAZAFON
+Nawa
+.AsSuwayda'
+Insid.Dar'a
+OFPALESTINE
+HEFA
+Hadera
+IRBID
+A
+Nablus
+rAjlen
+R
+TEL
+(Shechem)
+AJLLIN
+Y
+R
+AVIM
+E
+ARAH
+S
+TelAvivYafo
+.AzZara
+Assalt
+BANK
+ALMAFRAQ
+Amman
+HAMERKAZ
+TallAsu
+D
+1022mBALOA
+Ashdod
+Jerusalem (Yerushalayim,Al Quds)
+YERUSHALAYINThis metric fails to capture the inherent hierarchical nature of the label. Another measure
+is the ”Mean distance error”. Mean distance error averages the distances between the pre-
+dicted location (center of the predicted S2 cell in our case) and true location of the target
+text. It too fails to capture the hierarchical nature of the label.
+For this reason we prefer a hierarchical classification metric described in Hierarchy-
+Specific Variations on the Regular Classification Metrics [10]. Those are variations of the
+well-known precision, recall and f-score metrics, specifically adapted to fit hierarchical clas-
+sification:
+hierarchical precision (hP):
+hP =
+�
+i |Pi ∩ Ti|
+�
+i |Pi|
+,
+(1)
+hierarchical recall (hR):
+hR =
+�
+i |Pi ∩ Ti|
+�
+i |Ti|
+,
+(2)
+and hierarchical f-measure (hF):
+hF = 2 ∗ hP ∗ hR
+hP + hR
+(3)
+where Pi is the set consisting of the most specific class predicted for each test example i, and
+all of its ancestor classes. Ti is the set consisting of the true most specific class of test example
+i, and all its ancestor classes. Each summation is computed over all of the test set examples.
+5
+Results
+In this section, the performance of the developed geocoding model is presented and analyzed.
+To demonstrate the performance of the model, let us first present several inference examples
+of the model. The examples of the predicted samples are given in Tab. 4.
+Table 4: Inference examples - true and predicted labels.
+Text
+Predicted Label
+True Label
+townland in Drummaan, County Clare, Ireland
+21002321
+21002321
+lake in Eksj¨o Municipality, Sweden
+20302303
+20302303
+ancient monument in Denmark (2976)
+20331122
+20331122
+school in Cheshire West and Chester, UK
+210033112
+210033113
+mountain in Iran
+1333313
+133302
+railway stop in Harburg, Germany
+20331203
+20331022
+It is difficult to compare two sequences due to the fact that two adjacent cells can have
+very different sequences in terms of string comparison.
+This results from the fact that
+two adjacent cells can originate from different parent cells. To visualize the results, a web
+demo application powered by the fune-tuned model was constructed. This demo application
+can facilitate an intuitive comparison between the predicted and the true label. Inference
+examples using this application are given in Fig. 6, Fig. 7, and Fig. 8.
+7
+
+Figure 6: Prediction results for the text ”Pacific ocean”.
+Figure 7: Prediction results for the text ”South”. We can assume large amount of South
+African samples biased the prediction.
+Evaluation results on the wikidata test split using the metrics described in section 4.2.2
+are given in Tab. 5.
+Table 5: Evaluation results.
+Evaluation metric
+Results
+Flat accuracy
+0.51547
+Hierarchy accuracy
+0.791
+8
+
+Pacific ocean
+Q
+Dhaka
+ROFICOF
+HongKong
+Kokata
+AOS
+Haiphong
+Cacutta)NeyPui Ta
+ISLANDS
+BHAILAND
+South
+Bay Raynay
+China
+Bengi
+Sea
+Manila
+GUA
+CAMB
+VIETNAM
+PHILIPPINES
+Ho Chi
+Pacific ocean (7%)
+Minh City
+SRILANK!
+Kotte
+KualaLumpur
+BRUNEL
+EOU
+MALAYSIA
+Celebes Sea
+ngapore
+Samarinda
+Mamuju
+INDONESIA
+PAPUA
+Surabaya
+NEW
+Dili
+GUINEA
+SOLOMONISLANDS
+TIMOR
+Honara
+EASTTIMOR
+Port Moresb
+Timar
+Arafurd
+SeaOPIA
+South
+CAMEROON
+Kampala
+Wos
+KENYA
+South
+DEMOCRATIC
+GABON
+REPUBLICOF
+Kigali
+Nairobi
+BURUNDI
+ANDPRINCIPE
+THE CONGO
+Braz
+TANZANIA
+Somali
+Kinshasa
+Mbuji-Mayi
+Dodoma
+Basin
+Dares
+Salaam
+Luanda
+ANGOLA
+ZAMBIA
+Lilongwe
+Angola
+MALAWI
+Lusaka
+Basin
+Harare
+SAINTHELEN,
+ZIMBABWE
+NAMIBIA
+Windhoek
+BOTSWANA
+MADAGASCAR
+TROFICCF CAFRI
+(Tshwane)
+Maputo
+CRN
+PortLou
+Johannesburg
+SWAZILAND
+Bloenfontein
+Natal
+LESOTHDurban
+Basin
+SOUTH AFRICA
+Mad
+asir
+agascar
+Plat
+CapeTown
+Cape
+Basin
+South (14%)
+Agulhas
+Plateau
+Agulhan
+BasiFigure 8: Prediction results for the text ”Paris”.
+6
+Conclusions
+In this paper we formulated the problem of geocoding as a sequence-to-sequence problem
+and trained a transformer model for geocoding. To leverage the capabilities of a language
+model, a pre-trained T5 model was used for fine-tuning. For the sequence representation
+of the geolocation label, an adaptive cell partitioning was used.
+The free text and its
+corresponding geolocation were obtained from wikidata. The evaluation of the model on
+both hierarchical and non-hierarchical metrics demonstrated the validity of the proposed
+approach for geolocation prediction.
+The free text at inference time sometimes only hints at a location. It is our intuition
+that by combining a huge decoder such as GPT3 [11] to produce more location-coherent
+text from an obscure location reference and then feeding it to T5 could improve the model’s
+performance. This evaluation is left for future work. In addition, we leave for future work
+the evaluation of our approach on a benchmark such as Wikipedia Toponym Retrieval [12].
+References
+[1] Sayali Kulkarni, Shailee Jain, Mohammad Javad Hosseini, Jason Baldridge, Eugene
+Ie, and Li Zhang. Spatial language representation with multi-level geocoding. arXiv
+preprint arXiv:2008.09236, 2020.
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+https://github.com/r-spatial/s2, https://s2geometry.io/.
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+Advances in neural information processing
+systems, 33:1877–1901, 2020.
+[12] Milan Gritta, Mohammad Taher Pilehvar, Nut Limsopatham, and Nigel Collier. What’s
+missing in geographical parsing? Language Resources and Evaluation, 52(2):603–623,
+2018.
+10
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf,len=232
+page_content='Transformer Based Geocoding  Yuval Solaz∗  yuval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='solaz@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='com  Vitaly Shalumov∗  vitaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='shalumov@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='com  January 4, 2023  Abstract  In this paper, we formulate the problem of predicting a geolocation from free text as  a sequence-to-sequence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Using this formulation, we obtain a geocoding model  by training a T5 encoder-decoder transformer model using free text as an input and  geolocation as an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The geocoding model was trained on geo-tagged wikidump  data with adaptive cell partitioning for the geolocation representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' All of the code  including Rest-based application, dataset and model checkpoints used in this work are  publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  1  Introduction  Social media such as Twitter and Wikipedia contains considerable amount of location-related  text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' In this paper, we develop a model that learns to predict spatial probabilities from  free text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Given a query sentence, the model outputs a discrete probability distribution over  the surface earth, by assigning each geographical cell a likelihood that the input text relates  to the location inside said cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The resulting model is capable of localizing a large variety  of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Viewing the task as a hierarchical classification problem allows the model to  express its uncertainty in the location associated with the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The resulting model can be  used for resolving ambiguity of the location references in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' This capability is central  to the success of finding exact location from free text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' For example, Paris can refer to more  than one possible location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' In a context such as: The International Olympic Committee  confirmed the city chosen to host the Olympic Games in 2024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The Games will be held in  Paris, geocoding models like the one proposed in this paper can help in the resolution of  the correct location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  This work introduces the following contributions:  Synthesizing a dataset for supervised learning, including adaptive cell partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Formulating the geocoding problem as a sequence-to-sequence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Training an end-to-end geocoding model using said formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Publicly releasing the curated dataset, a Rest-based application and the T5 geocoding  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  ∗Equal Contribution  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='01170v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='CL]  2 Jan 2023  2  Related works  The geolocation prediction from free text was extensively addressed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The  authors of [1] presented a multi-level geocoding model that learns to associate texts to  geographic locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The downstream task was formulated as a multi-level classification  problem based on multi-level S2 [2] cells as the output space from a multi-headed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  In [3], the surface of the earth was subdivided into thousands of multi-scale geographic  cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The authors trained a deep neural network using millions of geo-tagged images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' We  adopt the adaptive partition approach introduced in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The authors of [4] created  language models of locations using coordinates extracted from geo-tagged Twitter data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  The locations were modeled at varying levels of granularity, from zip code to the country  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' In [5], an end-to-end probabilistic model for geocoding text data was presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' In  addition, the authors collected a novel data set for evaluating the performance of geocoding  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The model-based solution, called ELECTRo-map was compared to the open source  system available at the time of publication for geocoding texts for event data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' An algorithm  for estimating a distribution over geographic locations from a single image using a purely  data-driven scene matching approach was provided in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  The aforementioned approaches to geocoding can be viewed as sequential tasks of token  classification (which is assumed to be solved perfectly) and geocoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Unfortunately, this  approach suffers from several drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  First, it cannot handle correctly inputs that  contain several descriptions of locations (for example: ”We live in country X city Y”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Second, it lacks the context of free text (for example: ”Country south of France”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  3  Text Geolocation with Transformer  We pose the task of text geolocation as a sequence-to-sequence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The model trans-  lates input text into hierarchy set of geographic cells which represent probability distribution  over the surface of the earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The model output is a sequence encoding of hierarchical cell  representation of the surface of the earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='1  Adaptive Cell Partitioning  We use Google’s open source S2 Geometry library [2] to partition the surface of the earth into  non-overlapping cells that define the target of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The S2 library defines hierarchical  partitioning of the surface of a sphere by projecting the surfaces of an enclosing cube on  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The six sides of the cube are subdivided hierarchically by six quad-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' A node in a  quad-tree defines a region on the sphere called an S2 cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 1 illustrates the S2 cells in  several resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 1 shows the resolution and number of cells for each S2 Geometry  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  There are several reasons for choosing this subdivision scheme over a simple subdivision  of latitude/longitude coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Firstly, the latitude/longitude regions get elongated near  the poles while S2 cells keep a close-to-quadratic shape, and secondly, S2 cells are mostly  uniform in size (the ratio between the largest and smallest S2 cell is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' A naive approach  to define a tiling of the earth would be to use all S2 cells at a certain fixed depth in the  hierarchy, resulting in a set of roughly equally sized cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' However, this would produce a  very imbalanced class distribution since the geographical distribution of wikidata items [7]  which was adopted in this paper, has strong peaks in densely populated areas see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  2  Figure 1: S2 Geometry hierarchical partitioning of the earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Table 1: S2 Geometry levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Level  Average area  Number of cells  00  85M km2  6  01  21M km2  24  02  5M km2  96  03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='3M km2  384  04  330K km2  1536  05  83K km2  6K  06  20K km2  24K  07  5K km2  98K  08  1297 km2  393K  09  324 km2  1573  10  81 km2  6M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='.  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='.  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='.  29  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='95 cm2  1729 ∗ 1015  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='74 cm2  7 ∗ 1018  We therefore perform adaptive subdivision, based on the dataset item location: starting  at the roots, we recursively descend each quad-tree and subdivide cells until no cell contains  more than a certain fixed number (max cell samples parameter) of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' By using  this approach, sparsely populated areas are covered by larger cells and densely populated  areas are covered by finer cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' This adaptive tiling has several advantages over a uniform  one: (i) training classes are more balanced and (ii) it makes effective use of the parameter  space because more model capacity is spent on densely populated areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 3 demonstrates  the S2 partitioning for our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  3  Poland  Germany  Ukraine  Kazakhstan  Frances  Mongolia  Spain  Italy  NO!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Turkey  China  Japan  Tantic  Afghanistan  SouthKon  cean  Aira  Iran  Algeria  Libya  Egypt  Pakistan  Saudi  Arabia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  India  Mali  Niger  Sudan  Chad  Venezuela  Nigeria  Ethiopi  Colombia  DRCongo  Kenya  Tanzania  Indonesia  Papua New  Brazil  Guinea  Peru  Angola  Bolivia  Namibia  Mad  agascar  indian  South  Botswana  Ocean  Chile  Atiantic  AUS  alta  Ocean  SouthAfrica  Argentina  New  ZealandFigure 2: Data points distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Figure 3: S2 Geometry adaptive partitioning of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='2  Dataset  This section describes the dataset used for training the model and for the evaluation ex-  periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The dataset was constructed from the wikidata archive [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The archive was  filtered as to select all records with location label (approximately 8M records).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' See Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 2  4  80 -  60  40-  to  20 -  40 -  60  150  100  50  50  100  150for samples of wikidata records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Table 2: Wikipedia Geo Data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Id  Latitude  Longitude  Text  01  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='08  historic county of England  02  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='0  country in Central Europe  03  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='88  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='5  sovereign state in Southern Europe  04  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='30  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='36  whisky distillery in Highland, Scotland, UK  05  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='69  city and commune in Indre-et-Loire, Centre-Val de Loire, France  06  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='0  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='0  sovereign state in South America  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='3  Data Labeling  Using the adaptive cells, we label each data sample with the cell id containing the sample  location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' In order to keep the hierarchical nature of the cells in the label we use the following  cell id encoding: the first digit represents the cell cube face with a digit between 0 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  The next digits represent for each level the corresponding node in the quad tree with a digit  between 0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' See Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 3 for a cell encoding example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Note that the label is a sequence  of digits with a variant length between 1 and 1+max level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Table 3: Cell encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Cell description  Cell representation  Face cell 2  2  Subcell 2 of face cell 1  12  Subcell 1 of subcell 3 of face 4  431  4  Model  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='1  Train  The model presented in this paper is based on the T5-base (220M) pre-trained sequence-  to-sequence model [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The model was fine-tuned on the wikidata dataset with the text  records as input and the location cell encoding as the output target sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' We chose to  utilize the standard cross-entropy loss in the fine-tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Evaluation of level-based  weighting in the loss function is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' We trained for 2M steps over 5 epochs  with batch size of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' We used the AdamW optimizer with learning rate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='5e-5 and linear  decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' We train the model on 80% of the wiki-data dataset and use the other 20% for in-  domain evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' A diagram of our text-to-location framework with a few input/output  examples is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  5  Figure 4: A diagram of our text-to-location framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Figure 5: Prediction results for the text ”Haifa”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The predicted S2 cell in blue and its  ancestor cells in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='2  Evaluation  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='1  Inference  The inference of our resulting model is as follows: given a sentence we predict the output  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Naturally it wold be 1 digit between 0 to 5 representing the predicted cube face,  followed by up to 9 digits between 0-3 representing the predicted s2cell on each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' This  sequence can easily converted to a s2 cell, which represents the probability distribution over  the earth for the input text location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' We used a beam search with size of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 5 for  inference example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='2  Evaluation Metric  The most straightforward classification metric is the ”accuracy measure” where only predic-  tions with a full match between the model’s output and the true label counts as a successful  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' We call this metric ”Flat Accuracy” (as oppose to ”Hierarchical Accuracy”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  6  "historic county of  England"  41023  \'sovereignstate in  Southern Europe\'  1221  T5Geo-Location  "sovereign state in  123331  South America"CYPRUS  ARRAQQF  MAGUSA  Halfa  Famagusta  Hamah  (Ammochostos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='Gazimagjusa)  SIACypray  Haifa  Salamiyah  AlMayadin  DHEKELIA (EASTERN)  TARTUS  S  RIA  LARNAKA  Tartus  UNITED KINGDOM  Hims  nassol  Tall Kalakh  AbuKamal  Tripoli  DAYR AZZAWR  LIBAN  NORD  AlQaryatayn  EL BEQAA  MON  LIBAN  AnNabk  Beirut  LEBANON  Yabrud  HIMS  BEYROUTH  Zahlah  Sidon  AL ANBAR  IMtHenpon  Damascus  2814m  DLMASHQ  Habbush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content="  LIBAN SU  Nabatiyee Tahta  Tyre  NABATIYE  QUNAYTIRAH  DAR'A  RIF DIMASHQ  ArRutbah  Haifa (98%)  Zefat  AS SUWAYDA'  Haifa  IAZAFON  Nawa  ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content="AsSuwayda'  Insid." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content="Dar'a  OFPALESTINE  HEFA  Hadera  IRBID  A  Nablus  rAjlen  R  TEL  (Shechem)  AJLLIN  Y  R  AVIM  E  ARAH  S  TelAvivYafo  ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='AzZara  Assalt  BANK  ALMAFRAQ  Amman  HAMERKAZ  TallAsu  D  1022mBALOA  Ashdod  Jerusalem (Yerushalayim,Al Quds)  YERUSHALAYINThis metric fails to capture the inherent hierarchical nature of the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Another measure  is the ”Mean distance error”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Mean distance error averages the distances between the pre-  dicted location (center of the predicted S2 cell in our case) and true location of the target  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' It too fails to capture the hierarchical nature of the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  For this reason we prefer a hierarchical classification metric described in Hierarchy-  Specific Variations on the Regular Classification Metrics [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Those are variations of the  well-known precision, recall and f-score metrics, specifically adapted to fit hierarchical clas-  sification:  hierarchical precision (hP):  hP =  �  i |Pi ∩ Ti|  �  i |Pi|  ,  (1)  hierarchical recall (hR):  hR =  �  i |Pi ∩ Ti|  �  i |Ti|  ,  (2)  and hierarchical f-measure (hF):  hF = 2 ∗ hP ∗ hR  hP + hR  (3)  where Pi is the set consisting of the most specific class predicted for each test example i, and  all of its ancestor classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Ti is the set consisting of the true most specific class of test example  i, and all its ancestor classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Each summation is computed over all of the test set examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  5  Results  In this section, the performance of the developed geocoding model is presented and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  To demonstrate the performance of the model, let us first present several inference examples  of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The examples of the predicted samples are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Table 4: Inference examples - true and predicted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Text  Predicted Label  True Label  townland in Drummaan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' County Clare,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Ireland  21002321  21002321  lake in Eksj¨o Municipality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Sweden  20302303  20302303  ancient monument in Denmark (2976)  20331122  20331122  school in Cheshire West and Chester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' UK  210033112  210033113  mountain in Iran  1333313  133302  railway stop in Harburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Germany  20331203  20331022  It is difficult to compare two sequences due to the fact that two adjacent cells can have  very different sequences in terms of string comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  This results from the fact that  two adjacent cells can originate from different parent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' To visualize the results, a web  demo application powered by the fune-tuned model was constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' This demo application  can facilitate an intuitive comparison between the predicted and the true label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Inference  examples using this application are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 6, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 7, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  7  Figure 6: Prediction results for the text ”Pacific ocean”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Figure 7: Prediction results for the text ”South”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' We can assume large amount of South  African samples biased the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Evaluation results on the wikidata test split using the metrics described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='2  are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Table 5: Evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Evaluation metric  Results  Flat accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='51547  Hierarchy accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='791  8  Pacific ocean  Q  Dhaka  ROFICOF  HongKong  Kokata  AOS  Haiphong  Cacutta)NeyPui Ta  ISLANDS  BHAILAND  South  Bay Raynay  China  Bengi  Sea  Manila  GUA  CAMB  VIETNAM  PHILIPPINES  Ho Chi  Pacific ocean (7%)  Minh City  SRILANK!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Kotte  KualaLumpur  BRUNEL  EOU  MALAYSIA  Celebes Sea  ngapore  Samarinda  Mamuju  INDONESIA  PAPUA  Surabaya  NEW  Dili  GUINEA  SOLOMONISLANDS  TIMOR  Honara  EASTTIMOR  Port Moresb  Timar  Arafurd  SeaOPIA  South  CAMEROON  Kampala  Wos  KENYA  South  DEMOCRATIC  GABON  REPUBLICOF  Kigali  Nairobi  BURUNDI  ANDPRINCIPE  THE CONGO  Braz  TANZANIA  Somali  Kinshasa  Mbuji-Mayi  Dodoma  Basin  Dares  Salaam  Luanda  ANGOLA  ZAMBIA  Lilongwe  Angola  MALAWI  Lusaka  Basin  Harare  SAINTHELEN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  ZIMBABWE  NAMIBIA  Windhoek  BOTSWANA  MADAGASCAR  TROFICCF CAFRI  (Tshwane)  Maputo  CRN  PortLou  Johannesburg  SWAZILAND  Bloenfontein  Natal  LESOTHDurban  Basin  SOUTH AFRICA  Mad  asir  agascar  Plat  CapeTown  Cape  Basin  South (14%)  Agulhas  Plateau  Agulhan  BasiFigure 8: Prediction results for the text ”Paris”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  6  Conclusions  In this paper we formulated the problem of geocoding as a sequence-to-sequence problem  and trained a transformer model for geocoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' To leverage the capabilities of a language  model, a pre-trained T5 model was used for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' For the sequence representation  of the geolocation label, an adaptive cell partitioning was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  The free text and its  corresponding geolocation were obtained from wikidata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' The evaluation of the model on  both hierarchical and non-hierarchical metrics demonstrated the validity of the proposed  approach for geolocation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  The free text at inference time sometimes only hints at a location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' It is our intuition  that by combining a huge decoder such as GPT3 [11] to produce more location-coherent  text from an obscure location reference and then feeding it to T5 could improve the model’s  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' This evaluation is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' In addition, we leave for future work  the evaluation of our approach on a benchmark such as Wikipedia Toponym Retrieval [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  References  [1] Sayali Kulkarni, Shailee Jain, Mohammad Javad Hosseini, Jason Baldridge, Eugene  Ie, and Li Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Spatial language representation with multi-level geocoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' arXiv  preprint arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='09236, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  [2] Dewey Dunnington, Edzer Pebesma, and Ege Rubak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  s2:  Spherical Geometry  Operators Using the S2 Geometry Library, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  https://r-spatial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='io/s2/,  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='com/r-spatial/s2, https://s2geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  [3] Tobias Weyand, Ilya Kostrikov, and James Philbin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Planet-photo geolocation with  convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' In European Conference on Computer Vision, pages  37–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  9  nnel  Amiens  Dieppe  Fecamp  HAUTE-NORMANDIE  P1  Cherbourg-Octeville  LeHavre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Rouen  Beauvais  DS  Helfer  Caeh  ERSEY  Lisieux  Saint-Lo  Mantes-l-Jolie  Evreux  Meau  Paris (97%)  Palis  BASSENORMANDIE  Versaille  Dreux  Avranches  t-Malo  Rambouillet  Me  Etampes  Alercon  Chartres[4] Sheila Kinsella, Vanessa Murdock, and Neil O’Hare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' ” i’m eating a sandwich in glasgow”  modeling locations with tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' In Proceedings of the 3rd international workshop on  Search and mining user-generated contents, pages 61–68, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  [5] Benjamin J Radford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Regressing location on text for probabilistic geocoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' arXiv  preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='00080, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  [6] James Hays and Alexei A Efros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Im2gps: estimating geographic information from a  single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' In 2008 ieee conference on computer vision and pattern recognition, pages  1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' IEEE, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  [7] Denny Vrandeˇci´c and Markus Kr¨otzsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Wikidata: a free collaborative knowledgebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Communications of the ACM, 57(10):78–85, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  [8] Denny Vrandeˇci´c and Markus Kr¨otzsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Wikidata: a free collaborative knowledgebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Communications of the ACM, 57(10):78–85, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  [9] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael  Matena, Yanqi Zhou, Wei Li, Peter J Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Exploring the limits of transfer learning  with a unified text-to-text transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=', 21(140):1–67, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  [10] Carlos Nascimento Silla and Alex Alves Freitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' A survey of hierarchical classification  across different application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Data Mining and Knowledge Discovery, 22:31–72,  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  [11] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla  Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  Advances in neural information processing  systems, 33:1877–1901, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  [12] Milan Gritta, Mohammad Taher Pilehvar, Nut Limsopatham, and Nigel Collier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' What’s  missing in geographical parsing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content=' Language Resources and Evaluation, 52(2):603–623,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
+page_content='  10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAzT4oBgHgl3EQfOvsw/content/2301.01170v1.pdf'}
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+Astronomy & Astrophysics manuscript no. nuv_HD189733b
+©ESO 2023
+January 31, 2023
+The Hubble/STIS Near-ultraviolet Transmission Spectrum of
+HD 189733 b
+Patricio E. Cubillos1, 2, Luca Fossati1, Tommi Koskinen3, Chenliang Huang3, A. G. Sreejith1, 4, Kevin France4, P.
+Wilson Cauley4, and Carole A. Haswell5
+1 Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, A-8042, Graz, Austria
+e-mail: patricio.cubillos@oeaw.ac.at
+2 INAF – Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese, Italy
+3 Lunar and Planetary Laboratory, University of Arizona, 1629 E. University Blvd., Tucson, AZ 85721, USA
+4 Laboratory for Atmospheric and Space Physics, University of Colorado, 600 UCB, Boulder, CO 80309, USA
+5 School of Physical Sciences, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK
+January 31, 2023
+ABSTRACT
+The benchmark hot Jupiter HD 189733 b has been a key target to lay out the foundations of comparative planetology for giant exo-
+planets. As such, HD 189733 b has been extensively studied across the electromagnetic spectrum. Here, we report the observation and
+analysis of three transit light curves of HD 189733 b obtained with Hubble/STIS in the near ultraviolet, the last remaining unexplored
+spectral window to be probed with present-day instrumentation for this planet. The NUV is a unique window for atmospheric mass-
+loss studies owing to the strong resonance lines and large photospheric flux. Overall, from a low-resolution analysis (R = 50) we
+found that the planet’s near-ultraviolet spectrum is well characterized by a relatively flat baseline, consistent with the optical-infrared
+transmission, plus two regions at ∼2350 and ∼2600 Å that exhibit a broad and significant excess absorption above the continuum.
+From an analysis at a higher resolution (R = 4700), we found that the transit depths at the core of the magnesium resonance lines are
+consistent with the surrounding continuum. We discarded the presence of Mg ii absorption in the upper atmosphere at a ∼2–4σ confi-
+dence level, whereas we could place no significant constraint for Mg i absorption. These broad absorption features coincide with the
+expected location of Fe ii bands; however, solar-abundance hydrodynamic models of the upper atmosphere are not able to reproduce
+the amplitude of these features with iron absorption. Such scenario would require a combination of little to no iron condensation in
+the lower-atmosphere, super-solar metallicities, and a mechanism to enhance the absorption features (such as zonal wind broadening).
+The true nature of this feature remains to be confirmed.
+Key words. Techniques: spectroscopic – Planets and satellites: atmospheres – Planets and satellites: individual: HD 189733 b
+1. Introduction
+The hot Jupiter HD 189733 b is one of the earliest transiting ex-
+oplanets to be detected (Bouchy et al. 2005), being one of the
+closest transiting planets to Earth. HD 189733 b’s large mass
+of 1.13 MJup, large radius of 1.13 RJup, and short-period 2.2-
+day orbit (Stassun et al. 2017) provide some of the most favor-
+able conditions for atmospheric characterization. Consequently,
+HD 189733 b has become one of the most extensively observed
+exoplanets, now constituting a benchmark target for atmospheric
+characterization via follow-up observations and theoretical mod-
+eling.
+From space, the Hubble Space Telescope (HST) has been in-
+strumental in probing the optical-to-infrared transmission spec-
+tra of this planet. Collective observations with the Advanced
+Camera for Surveys (ACS), the Space Telescope Imaging Spec-
+trograph (STIS), and the Wide Field Camera 3 (WFC3) have
+obtained nearly continuous spectra over the 0.3–1.7 µm range
+(Pont et al. 2008; Sing et al. 2011; Huitson et al. 2012; McCul-
+lough et al. 2014). Complementary infrared (IR) transmission
+photometry from the Spitzer Space Telescope between 3.6 and
+8.0 µm (Désert et al. 2009; Agol et al. 2010; Knutson et al.
+2012), and posterior re-analyses of the cumulative data (Pont
+et al. 2013; Sing et al. 2016) have revealed a steep Rayleigh scat-
+tering slope dominating the optical and near-infrared spectrum.
+These results are consistent with the presence of high-altitude
+atmospheric hazes, ruling out a clear atmosphere.
+Ground-based facilities have further allowed the detection of
+several spectral features; in particular, high-resolution observa-
+tions have lead to the detection of narrow absorption features
+above the haze layer from the H Balmer lines (Jensen et al. 2012;
+Cauley et al. 2015) and the Na D lines (Redfield et al. 2008; Wyt-
+tenbach et al. 2015). These observations constrained the tem-
+perature of the thermosphere, revealed strong wind velocities
+consistent with an eastward equatorial jet (Louden & Wheatley
+2015), and detected a transit early ingress, possibly caused by
+a hydrogen bow-shock ahead of the planet (Cauley et al. 2015).
+We note however that these detections have been contested in
+subsequent studies (see Barnes et al. 2016; Guilluy et al. 2020),
+arguing that the observed signal could be stellar in nature, rather
+than planetary. Similarly, stellar effects are also suspected to af-
+fect the steep UV slope observed by HST for this planet, although
+the magnitude of such contribution is not fully understood (see,
+e.g., McCullough et al. 2014; Rackham et al. 2019).
+High-resolution spectroscopy studies of HD 189733 b at
+near-infrared wavelengths via the cross-correlation method have
+yielded the detection of H2O absorption features (Birkby et al.
+2013; Brogi et al. 2018) and CO (Rodler et al. 2013; Brogi et al.
+Article number, page 1 of 17
+arXiv:2301.13025v1  [astro-ph.EP]  30 Jan 2023
+
+A&A proofs: manuscript no. nuv_HD189733b
+2016). These observations further constrain atmospheric compo-
+sition and suggest small wind speeds of ∼2 km s−1 (at pressures
+around the 10−2 − 10−3 bar levels).
+Infrared observations of the day-side emission during sec-
+ondary eclipse has revealed a lack of thermal inversion and a
+nearly solar H2O abundance in the lower planetary atmosphere
+(Crouzet et al. 2014; Todorov et al. 2014). Furthermore, in-
+frared photometry along the entire orbital phase has measured
+the brightness temperature variation, constraining the energy
+budget and the global circulation regime (Knutson et al. 2007,
+2009, 2012). These observations unveiled an efficient energy re-
+distribution of the incident stellar irradiation and an eastward off-
+set of the peak emission, relative to the sub-stellar point. As with
+the conclusions from optical observations, these measurements
+are consistent with circulation models exhibiting an eastward ad-
+vection by an equatorial super-rotating jet with wind speeds of
+the order of ∼km s−1 at pressures around the mbar level (e.g.,
+Showman et al. 2009, 2013).
+1.1. Upper-atmosphere Characterization of HD 189733 b
+Given the close planet-to-star distance, it is expected that
+HD 189733 b exhibits atmospheric escape. In fact, the host star,
+HD 189733, is an active K-dwarf star (radius of 0.75R⊙, mass of
+0.79M⊙, effective temperature of 5052 K, Stassun et al. 2017),
+showing clear evidence of activity and spots during transit events
+(e.g., Sing et al. 2011; Pont et al. 2013), which makes it a particu-
+larly interesting study case for star–planet interactions (e.g., Pop-
+penhaeger et al. 2013; Pillitteri et al. 2015; Cauley et al. 2018).
+Lyα transit observation in the far-ultraviolet with HST/STIS
+detected an extended hydrogen envelope that expands hydrody-
+namically and escapes to space, powered by absorption of the
+high-energy stellar flux (X-ray and extreme ultraviolet; Lecave-
+lier des Etangs et al. 2012; Bourrier et al. 2013). Strong temporal
+variations and the high velocity of the escaping gas suggest the
+presence of significant star–planet interactions (consistent with
+the active nature of the host star), which requires an accelera-
+tion mechanism on top of stellar radiation pressure, such as the
+stellar wind. Although later, Guo & Ben-Jaffel (2016) found that
+the Lyα variability can be simply explained by invoking a ther-
+mal population of H i in the planet’s upper atmosphere, without a
+need for complex dynamics involving stellar winds or radiation
+pressure. In this case, changes to the star’s EUV output would
+explain the different absorption signals. Far-ultraviolet observa-
+tions with the HST Cosmic Origins Spectrograph have also led
+to the detection of O i and C ii in the planetary upper atmosphere
+(Ben-Jaffel & Ballester 2013), indicating that heavy atoms are
+possibly entrained and dragged upwards in the hydrogen hydro-
+dynamic outflow of the atmosphere.
+The Hα absorption feature detection by Jensen et al. (2012)
+indicates that the upper atmosphere of HD 189733 b is neither in
+radiative equilibrium nor thermodynamic equilibrium, due to the
+large ultraviolet flux received by its active host star. In addition,
+Salz et al. (2018), Zhang et al. (2022), and Guilluy et al. (2020)
+detected a strong absorption signal from the He i triplet at 10830
+Å by means of ground-based high-resolution transmission spec-
+troscopy with CARMENES and GIARPS, respectively. The ob-
+served He absorption favors a compact helium atmosphere with
+a substantial column density and no evidence of He escape. This
+interpretation of the He observations would seem to be consis-
+tent with the aforementioned explanation of the Lyman alpha
+transit observations by Guo & Ben-Jaffel (2016).
+The many observations of HD 189733 b across the electro-
+magnetic spectrum have enabled an in-depth characterization of
+the composition and physical state of both the lower and up-
+per atmosphere of the planet. However, until recently, the near-
+ultraviolet (NUV) window, probing both the exosphere and ther-
+mosphere, escaped the spectral coverage probed by these ob-
+servations. King et al. (2021) analyzed 20 NUV photometric
+transits of HD 189733 b obtained from the XMM-Newton op-
+tical monitor. They detected a transit-depth signal consistent
+with that in the optical, ruling out extended broad-band absorp-
+tion towards or beyond the Roche lobe. However, to detect the
+NUV absorption signature from individual metallic lines, higher
+spectral-resolution observations are required, like those from
+HST/STIS, with a resolving power of R ≈ 30, 000 (see Sing et al.
+2019; Cubillos et al. 2020).
+For the slighly more strongly irradiated planet HD 209458 b,
+Cubillos et al. (2020) detected Fe ii absorption in a 100 Å-wide
+range around 2370 Å, lying beyond the planetary Roche lobe,
+while finding no evidence for absorption by Mg ii, neutral mag-
+nesium (Mg i), nor neutral iron (Fe i). These results suggest that
+hydrodynamic escape is strong enough to carry heavy atoms
+beyond the planetary Roche lobe; however, condensate forma-
+tion sequesters primarily Mg (rather than Fe) in the lower at-
+mosphere of the planet, consistent with microphysical cloud-
+formation models (Gao et al. 2020). For the ultra-hot Jupiter
+WASP-121b, Sing et al. (2019) reported the detection of strong
+narrow NUV absorption features, consistent with the singly ion-
+ized magnesium (Mg ii) doublet and several lines of ionized iron
+(Fe ii), emanating from altitudes that exceed the Roche lobe ra-
+dius of the planet. These results indicate that Mg ii and Fe ii
+species are not trapped into condensates at depth, and can hy-
+drodynamically escape the atmosphere.
+To complete the spectral survey of this benchmark planet
+and aid the quantitative assessment of the metallicity, ionization
+state, and outflow rate of the upper atmosphere of HD 189733 b,
+here we analyzed 15 HST orbits obtained from three STIS near-
+ultraviolet transit light curves. In Section 2 we describe the ob-
+servations in more detail. In Section 3 we describe our data-
+analysis procedure. In Section 4 we present the main results from
+the NUV data analysis. In Section 5 we place the observations
+in context by contrasting them with theoretical models of the
+planet. In Section 6 we speculate on the origin of the features
+seen in the NUV observations. Finally, in Section 7 we summa-
+rize our conclusions.
+2. Observations
+We observed three transits of the planet HD 189733 b, obtained
+with HST, using the Space Telescope Imaging Spectrograph
+(STIS, program GO #15338, PI L. Fossati). The observations
+were carried out on 2018-03-10, 2018-09-19, and 2019-08-27
+(visits 1, 2, and 3, respectively); each one using the NUV Multi-
+Anode Microchannel Array (NUV-MAMA) detector in ACCUM
+mode, E230M grating, and with an aperture of 0′′.2 × 0′′.2.
+Each transit observation (one HST visit) consists of five HST
+orbits. For each visit, the fourth orbit occurs during transit,
+whereas the first three and last HST orbits occur out of transit.
+The orbits are broken into exposures (frames) of ∼332 seconds
+each. The first two visits contain nine frames per orbit (except
+for the first orbit, which contains one fewer frame to accommo-
+date acquisition observations). The third visit contains one fewer
+frame per orbit, possibly due to a longer pointing adjustment
+time after HST returned to operations from a gyroscope failure.
+Article number, page 2 of 17
+
+Patricio E. Cubillos et al.: The HST NUV Transmission Spectrum of HD 189733 b
+2300
+2400
+2500
+2600
+2700
+2800
+2900
+3000
+Wavelength (Å)
+0
+2
+4
+6
+8
+Flux (10
+13 erg s
+1 cm
+2 Å
+1)
+2780
+2800
+2820
+0
+5
+10
+15
+20
+25
+Mg II
+2840
+2860
+2880
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Mg I
+0
+4
+8
+12
+16
+20
+Echelle order
+Fig. 1. HD 189733 system flux as observed by HST/STIS. This spectrum has been constructed from the mean time-series spectra obtained during
+the first visit, after discarding bad pixels. The spectrum is color coded by echelle order (see color bar on the right). The inset panels zoom-in
+around the location of the Mg ii and Mg i resonance lines.
+Each frame consists of an echelle spectrum comprising 23
+partially overlapping orders, covering the 2300–3100 Å spec-
+tral range. Each order contains 1024 wavelength samples, with
+a resolving power of R = λ/∆λ = 30 000, where the resolution
+element spans approximately two pixels (or equivalently ∼0.09
+Å, or ∼10 km s−1). Figure 1 shows a sample spectrum from visit
+1. As expected, the flux decreases toward shorter wavelengths,
+with a clear drop in signal-to-noise ratio (S/N) at wavelengths
+shorter than ∼ 2550 Å. The stellar spectrum shows many emis-
+sion features, most of them correspond to Fe ii lines rising from
+low energy levels, but also the Mg i and Mg ii resonance lines
+(the strongest line-emission feature in the data, as shown in the
+inset panels of Fig. 1). In contrast, stellar Fe i features appear in
+absorption.
+The input for our analysis consisted of the x1d CALSTIS-
+reduced fits files1 (version 3.4.2). The reduction comprises dark
+and flat-field correction, 1D spectral extraction of each echelle
+order, background subtraction, and wavelength and flux calibra-
+tion. The results are the 1D spectra (erg s−1cm−2Å−1) and their
+uncertainties.
+In addition, following Sing et al. (2019), we also considered
+the Engineering Data Processing System “jitter” files, which
+contain 28 different engineering measurements of HST’s Point-
+ing Control System for each exposure during an observation.
+3. Data Analysis
+The core of our analysis is based on the methodology of Cubillos
+et al. (2020). We separated our light-curve analysis into two main
+steps. First we constructed “white” lightcurves by integrating the
+flux per transit visit and per echelle order to boost the astro-
+physical and instrumental signals, from which we characterized
+the instrumental systematics. Then, we divided out the system-
+atics model from the raw data, and combined the systematics-
+1 https://hst-docs.stsci.edu/stisdhb/chapter-3-stis-calibration
+corrected data from all visits into spectral light curves binned at
+different resolving powers, from which we extracted the trans-
+mission spectra of the astrophysical signal.
+3.1. Analysis of Individual Visits
+We characterized the instrumental systematics to detrend them
+from the astrophysical signal and remove them from further
+analyses. To increase the S/N of the data, we integrated the flux
+over each echelle spectral order. The implicit assumption is that
+the instrumental systematics behave similarly over the integrated
+range (Kreidberg et al. 2014), in this case, over each echelle or-
+der.
+The analysis begins by masking bad and overly noisy data
+values. Following the standard practice for exoplanet HST/STIS
+time-series analyses, we discarded the first orbit from each visit
+and the first frame from each orbit, since they show significantly
+stronger instrumental systematics (see, e.g., Sing et al. 2011).
+Then, we used the overlap between echelle orders to discard data
+points where the fluxes differ from each other by more than 3σ.
+Finally, we manually inspected the edges of the echelle spec-
+tra to exclude data points with abnormally low fluxes (typically,
+a handful of data points at the ends of the echelle orders). We
+then produced raw “white” light curves by summing the flux
+over each spectral order and propagating the uncertainties ac-
+cordingly.
+We fit the raw light curves with parametric transit and sys-
+tematics models at each echelle order (denoted by their wave-
+length λ) as a function of time (t), HST orbital phase (φ), and
+jitter vector ( ji):
+Fλ(t) = Tλ(t) S λ(t, φ, ji),
+(1)
+where Tλ(t) is a Mandel & Agol (2002) transit model and
+S λ(t, φ, ji) is a model of the instrumental systematics. To ob-
+tain statistically robust parameter estimations, we computed
+best-fitting parameters and uncertainties from a Levenberg-
+Marquardt optimization and a Markov-chain Monte Carlo
+Article number, page 3 of 17
+
+A&A proofs: manuscript no. nuv_HD189733b
+(MCMC) sampling, respectively. For this matter, we employed
+the open-source mc3 statistical package2 (Cubillos et al. 2017),
+which implements the Snooker Differential-evolution MCMC
+algorithm of ter Braak & Vrugt (2008).
+The transit model parameterizes the astrophysical signal via
+the orbital parameters, the planet-to-star radius ratios Rp/Rs(λ),
+the out-of-transit stellar fluxes Fs(λ), and limb-darkening coef-
+ficients Ci(λ). For the orbital parameters we adopted previously
+measured values for the orbital period P = 2.21857567 ± 1.5 ×
+10−8 day, inclination i = 85◦.71, and semi-major-axis to stellar
+radius ratio a/Rs = 8.84 ± 0.27 (Stassun et al. 2017), which we
+kept fixed during the fitting and MCMC process. We fit the tran-
+sit mid-time epoch, applying a Gaussian prior according to the
+measured optical value (T0 = 53955.025551 ± 9 × 10−6 MJD,
+Bonomo et al. 2017), propagating uncertainties according to the
+epoch of the HST observations.
+For the limb-darkening parameters, we computed theoretical
+coefficients based on the stellar properties, since the low S/N ra-
+tios and gaps in the HST observations do not allow for a good
+estimation. For this calculation we used the open-source rou-
+tines of Espinoza & Jordán (2015) to compute limb-darkening
+coefficients over each wavelength range based on the PHOENIX
+stellar model (Husser et al. 2013) that best matches the proper-
+ties of HD 189733: effective temperature Teff = 5052 K, surface
+gravity log g = 4.49, and metallicity [M/H] = −0.02 (Stassun
+et al. 2017). We note that, in absence of any better known guess,
+the adopted limb-darkening coefficents correspond to the photo-
+sphere of the star. If a significant fraction of the stellar flux comes
+from its chromosphere and transition region, as suggested by the
+stellar line emission, then the used limb-darkening coefficients
+may not be as representative as expected.
+The HST/STIS time-series observations exhibit a number of
+well documented instrumental systematics: (1) periodic varia-
+tions in phase with the HST orbital period that are suspected to
+originate from the telescope’s thermal cycle (Brown et al. 2001),
+and (2) visit-long variations that might be attributed to stellar ac-
+tivity (e.g., Wakeford et al. 2016; Sing et al. 2019). Following the
+example of previous studies, we model these systematics consid-
+ering a family of polynomials in time (t) and HST orbital phase.
+Furthermore, Sing et al. (2019) showed that additional data from
+HST’s Pointing Control System (so called “jitter” data) can help
+decorrelate instrumental systematics from the astrophysical sig-
+nal. For each science exposure, the jitter data describe the HST’s
+fine guidance sensor (FGS) coordinates, FGS jitter, right ascen-
+sion, declination, position angle, latitude, longitude, angle be-
+tween HST zenith and target, Earth-limb angle, terminator an-
+gle, and magnetic field. HST stores the jitter measurements into
+time-tag ancillary engineering files, recorded at a high temporal
+resolution. For each science exposure we computed a value for
+each of the jitter parameters by calculating the median of the jit-
+ter values during the exposure. Therefore, our systematics model
+consists of a polynomial expression up to a quadratic degree in
+time, up to a quartic degree in HST phase, and a up to a linear
+degree in one of the jitter vectors:
+S λ(t, φ, ji) = 1 + a0(t − t0) + a1(t − t0)2
++ b0(φ − φ0) + b1(φ − φ0)2 + b2(φ − φ0)3 + b3(φ − φ0)4
++ c0( ji − ⟨ ji⟩),
+(2)
+where ak, bk, and ck are the polynomial coefficients to be fit. Note
+that we considered an individual set of fitting parameters for each
+echelle order (we dropped the λ-dependence of these coefficients
+2 https://mc3.readthedocs.io/
+for clarity). Regarding the jitter vectors ji, we considered each
+of the 21 vectors listed in Table 1 of Sing et al. (2019). t0 and φ0
+are reference values for the time and phase, which are fixed at
+the transit mid-time t0 = T0 and at the HST mid-phase φ0 = 0.2,
+respectively. ⟨ ji⟩ denotes the mean value of the jitter vector ji
+along the visit.
+3.1.1. Systematics Model Selection
+Given our limited understanding of the systematics affecting
+HST observations, we determined the optimal functional form
+for the systematics model S λ by comparing fits considering all
+possible combinations of polynomial orders in t and φ up to the
+degrees shown in Eq. (2), and each of the 21 jitter vectors, and
+fits with no jitter vectors. To this end we employed a Bayesian
+model selection approach (see, e.g., Trotta 2007), where one
+compares the posterior probability of two competing models—
+given the data—by computing the ratio of their Bayesian evi-
+dences (also known as Bayes factor). The main problem of this
+approach is that the evidence of a model is not readily calcu-
+lable. Numerically, calculating the evidence entails evaluating
+the model likelihood throughout its parameter space (e.g., via a
+nested-sampling algorithm). To avoid the computational cost of
+this task, we adopt a simpler model selection approach where
+we approximate the evidence ratio via the Bayesian Information
+Criterion (BIC, Liddle 2007):
+BIC = χ2
+min + k log N,
+(3)
+where χ2
+min is the goodness-of-fit parameter corresponding to the
+maximum likelihood for the given model, k is the number of
+free parameters, and N is the number of data points. The Bayes
+ratio is then simply calculated from the difference in BIC val-
+ues between the competing models B = exp{−(BIC1 − BIC2)/2}
+(Raftery 1995), which only requires to find the maximum likeli-
+hood for the given models. A Bayes factor B > 1 favors model
+1 over model 2, or alternatively, the favored model minimizes
+BIC.
+Certainly, these approximations are based on various as-
+sumptions, which may not be fulfilled to different extents,
+such as independent and identically distributed data or (nearly)
+Gaussian posterior distributions or that the forward model is
+correct, for example. Therefore, we also considered a second
+model-selection criterion based on information theory, that is the
+Akaike Information Criterion corrected for small sample sizes
+(AICc, Liddle 2007):
+AICc = χ2
+min +
+2kN
+N − k − 1.
+(4)
+Just like before, the favored model minimizes AICc.
+3.1.2. Model Selection Applied to the HST Observations
+For the HD 189733 b data we treated each HST visit indepen-
+dently of each other, where for each visit we fit simultaneously
+the astrophysical and systematics parameters. Also, for each visit
+we fit all 23 echelle orders simultaneously, jointly fitting the tran-
+sit mid-time epoch and adopting the same configuration of the
+systematics model for all orders. Table 1 reports the BIC and
+AICc of the global fit to all orders for the favored systematics
+model (with and without jitter decorrelation). Figure 2 shows an
+example of the light-curve fits for a selected echelle order.
+As seen from the raw light curves (top row of Fig. 2), the
+systematics are characterized by visit-long and HST-orbit-long
+Article number, page 4 of 17
+
+Patricio E. Cubillos et al.: The HST NUV Transmission Spectrum of HD 189733 b
+0.96
+0.98
+1.00
+Raw light curve
+visit 1
+0.08
+0.04
+0.00
+0.04
+Orbital phase
+0.96
+0.98
+1.00
+1.02
+Best-fit (Rp/Rs)2
+AIC
+BIC
+0.90
+0.95
+1.00
+1.05
+visit 2
+0.08
+0.04
+0.00
+0.04
+Orbital phase
+0.96
+0.98
+1.00
+1.02
+0.91
+0.94
+0.97
+1.00
+1.03
+visit 3
+0.08
+0.04
+0.00
+0.04
+Orbital phase
+0.96
+0.98
+1.00
+1.02
+Fig. 2. Sample transmission light-curve fitting of the HST/STIS observations of HD 189733 b (fourth echelle order, ¯λ = 2917Å). The top row shows
+the raw light curves (one column for each of the three visits). The markers with error bars denote the fluxes and their uncertainties integrated over
+the echelle order for each frame. The markers are color-coded according to the HST orbit, where the gray colors denote the frames discarded from
+the light-curve fitting. The bottom rows show the systematics-corrected light curves corresponding to the best-BIC (colored circles) and best-AICc
+statistics (empty black diamonds), including the jitter decorrelation. The solid lines denote the astrophysical model of the best-BIC (colored lines)
+and best-AICc models (black lines).
+Table 1. BIC/AICc Model Comparison
+Systematics model
+BIC
+AICc
+χ2
+red
+Comments
+(t-deg, φ-deg, jitter)
+Visit 1
+(1, 1)
+1362.2
+961.6
+1.16
+best BIC
+(1, 1, Lat)
+1434.5
+944.6
+1.08
+best AICc
+(1, 2)
+1441.5
+951.6
+1.09
+best AICc no-jit
+Visit 2
+(1, 1, Lat)
+1617.7
+1127.8
+1.37
+best BIC
+(1, 3, Lat)
+1692.9
+1039.7
+1.09
+best AICc
+(1, 2)
+1699.0
+1209.1
+1.50
+best BIC no-jit
+(2, 2)
+1740.3
+1166.1
+1.38
+best AICc no-jit
+Visit 3
+(1, 1, LOS Zenith)
+1665.0
+1198.2
+1.73
+best BIC
+(1, 3, LOS Zenith)
+1704.7
+1090.8
+1.36
+best AICc
+(1, 2)
+1798.4
+1331.6
+1.98
+best BIC no-jit
+(2, 2)
+1826.6
+1282.8
+1.84
+best AICc no-jit
+Notes. The first two terms in the systematics model column show the
+polynomial degree for time and orbital phase, the third term shows the
+name of the jitter vector (which is always modeled as a linear polyno-
+mial). The absence of a third term indicates that no jitter vector was
+included.
+variations. Overall, the second visit shows stronger systematics
+than the first visit, and the third visit shows stronger systematics
+than the second, which is reflected in the best-fitting BIC, AICc,
+and reduced χ2 values (χ2
+red). In particular, the second and third
+visit systematics vary on the order of 10% of the raw flux. If we
+only included the more “traditional” polynomials in t and φ in
+the fit, we found clear systematic residual trends, which vary in
+strength for each echelle order. The relatively new practice of
+jitter decorrelation introduced by Sing et al. (2019) significantly
+2300
+2400
+2500
+2600
+2700
+2800
+2900
+3000
+Wavelength (Å)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Rp/Rs
+visit 1
+visit 2
+visit 3
+King et al. (2021)
+Fig. 3. HST/STIS NUV transmission spectrum of HD 189733 b (indi-
+vidual visits). The circle markers with error bars denote the transmis-
+sion planet-to-star radius ratios from each echelle order and visit (see
+legend), corresponding to the best-AICc fits. The colored dashed lines
+denote the mean value for each visit. The square marker denotes the
+XMM-Newton/UVM2 photometric measurement by King et al. (2021).
+improves the fit for the second and third visits, minimizing both
+information criteria statistics and residual systematics (bottom
+row of Fig. 2). In these fits, the AICc imparts a weaker penalty
+per fitting parameter than the BIC, thus favoring more complex
+models, which notably improves the χ2
+red values. However, al-
+though the BIC and AICc statistics favor different models, the
+two criteria usually produced consistent transit depths. All fol-
+lowing analyses presented here will focus on the best AICc anal-
+ysis including jitter decorrelation.
+Figure 3 shows the resulting transmission spectra according
+to the best-AICc models for each visit and at each echelle order.
+One clear feature seen in the transmission spectra is an offset be-
+tween the third visit and the first two. This offset is consistent
+with the signature of unocculted stellar spots (see, e.g., Rack-
+Article number, page 5 of 17
+
+A&A proofs: manuscript no. nuv_HD189733b
+ham et al. 2018; Zellem et al. 2017), which is a likely cause
+given the well known active nature of HD 189733. For each visit,
+we found that the transit radius ratios vary closely proportional
+to the square root of the bolometric fluxes of the observations
+(Rp/Rs ∝ Fs−1/2), which is a direct consequence of observations
+arising from a heterogeneous stellar photosphere including spots
+(Rackham et al. 2018). The weighted mean of the Rp/Rs values
+for each individual visit are 0.1585, 0.1595, and 0.1701; whereas
+the relative bolometric fluxes between visits are Fv1
+s /Fv2
+s = 1.02
+and Fv1
+s /Fv3
+s = 1.16. Later, the combined analysis in Section 4.1
+will show an independent confirmation that we are observing
+stellar spot contamination in the data.
+Once corrected for stellar activity, the analysis of each of the
+three visits produces a consistent transmission spectrum across
+the NUV. We obtained a mostly flat transmission spectrum be-
+tween 2300 Å and 3100 Å, with a planet-to-star radius ratio of ∼
+0.16, which becomes significantly noisier at wavelengths shorter
+than 2550 Å, due to the much lower stellar flux. These results
+are also consistent with the previous NUV observation by King
+et al. (2021), which measured the transit depth of HD 189733 b
+in the UVM2 photometric band of the XMM-Newton Space Ob-
+servatory (effective wavelength 2310 Å, width 480 Å).
+As a reference, the estimated altitude of the Roche lobe ra-
+dius as probed during transit is RL1′ = 0.44 Rs (i.e., perpendicu-
+lar to the star–planet axis). This value is approximated by 2/3 the
+distance to the L1 Lagrange point (see, e.g., Vidal-Madjar et al.
+2008). All transit depths at each echelle order are well under the
+Roche lobe radius.
+3.2. Analysis of Combined Visits
+To search for the narrow metal line-absorption features in the
+transmission spectrum, we need to analyze the observations at a
+high resolving power, finding a compromise between sufficient
+S/N per wavelength channel and diluting the narrow absorption
+features. Therefore, we treated the data at multiple spectral re-
+solving powers ranging from R = 10−70 (coarse sampling anal-
+ysis), and then at a R = 4700 (fine sampling analysis). We pro-
+ceeded in the same fashion as in Cubillos et al. (2020), that is,
+calibrate the wavelength solution to ensure that the spectra are
+well aligned, use the modeling results from the previous analy-
+sis to divide out the instrumental systematics, combine the data
+from all three visits into a single spectrum, and fit the transit
+depths at each bin. All future analyses use the best-AICc dataset
+since it attained χ2
+red values closer to one.
+3.2.1. Wavelength Calibration
+We used a cross-correlation procedure to calibrate the wave-
+length solution of each echelle order in each frame and visit.
+First, we created a master stellar template (enhancing the S/N)
+by co-adding the spectra from all frames within an echelle order
+and visit. We then Doppler-shifted the individual-frame spectra
+and cross-correlated them with their template, searching for the
+relative velocity with respect to the template given by the shift
+that maximized the correlation function. Once we obtained the
+wavelength correction of the individual frames within an echelle
+order, we repeated the procedure between the template spectra
+of each visit to find the relative wavelength correction between
+all spectra. Finally, we compared our wavelength solution with a
+theoretical model spectrum of HD 189733 (Shulyak et al. 2004)
+to obtain an absolute wavelength solution at a rest reference
+frame. Once calibrated, we Doppler-shifted all frames within a
+given echelle order to a common wavelength solution using a cu-
+bic spline interpolation. To minimize the impact of the interpo-
+lation, we selected a reference point that minimized the shifts of
+the set of frames. For a given visit, we found offsets of less than
+half a pixel between the frames, and in general the wavelength
+solution for each echelle order follows a similar trend along the
+visit, with no outliers.
+3.2.2. Divide-white Spectral Extraction
+To
+extract
+a
+high-resolution
+transmission
+spectrum
+of
+HD 189733 b, we followed the “divide-white” spectral analysis
+by Kreidberg et al. (2014), as described in Cubillos et al. (2020).
+In this approach, we first constructed a non-parametric model of
+the instrumental systematics by dividing the white-light curves
+(for each echelle order and visit) by their best-fit astrophysical
+transit model (see Sect. 3.1). Assuming that the instrumental
+systematics vary weakly with wavelength, each raw spectral
+data point is divided by the systematics model corresponding to
+their orbital phase, echelle order, and visit. Data uncertainties
+are properly accounted for, i.e., using the white-light model’s
+posterior distribution to estimate the uncertainties systematics
+model and propagating the errors.
+In a second step, we split the data into constant resolving-
+power wavelength bins, where for each spectral channel we co-
+added all flux contributions from all echelle orders, and propa-
+gated the errors accordingly.
+To analyze the spectral light curves we fit a Mandel & Agol
+(2002) transit model to the systematics-corrected spectral light-
+curves in an MCMC run. As previously, we fixed the orbital
+a/Rs, cos(i), and P parameters, and applied a Gaussian prior for
+the mid-transit epoch based on the white-light fitting results. We
+also computed theoretical limb-darkening curves at each spectral
+channel, which we kept fixed during the fit. At each wavelength
+bin we fit a unique transit depth common to all three visits, but
+we fit a separate out-of-transit flux value for each visit. Addi-
+tionally, we incorporated a stellar-spot correction model that we
+detail in the following section.
+3.2.3. Stellar Spot Correction
+To account for the spectral transit-depth variations due to unoc-
+culted stellar spots, we modeled the stellar flux as a linear com-
+bination of a nominal photospheric region and a spotted region.
+We assumed that a fraction fspot of the observed photosphere is
+covered by spots with a characteristic temperature Tspot, whereas
+the rest of the photosphere has the nominal effective temper-
+ature of the star Teff = 5052 K (with Teff ≥ Tspot). We used
+the PHOENIX library of high-resolution synthetic stellar spec-
+tra (Husser et al. 2013) to compute the flux of the nominal stel-
+lar photosphere Fs(Teff) and the spotted region Fspot(Tspot). This
+leads to the following correction term for the observed transit
+depth dobs(λ):
+dobs(λ) =
+d(λ)
+1 − fspot
+�
+1 − Fspot(λ)/Fs(λ)
+� ≡ d(λ) ϵ(λ),
+(5)
+where d(λ) is the true transit depth and ϵ(λ) is the contamination
+spectrum due to stellar hetereogeneities (as defined in Rackham
+et al. 2018). To evaluate the correction factor at any given Tspot,
+we interpolated in temperature between all available PHOENIX
+models (ranging from 2300 K to 5100 K) at a fixed metallicity
+([M/H]=0.0) and surface gravity (log(g) = 4.5).
+Article number, page 6 of 17
+
+Patricio E. Cubillos et al.: The HST NUV Transmission Spectrum of HD 189733 b
+Fig. 4. NUV transmission spectrum of HD 189733 b from the combined HST/STIS observations. The red markers with error bars denote the
+systematics- and stellar-spot corrected transmission spectrum, their 1σ uncertainties, and the span of the spectral bins (the data has been binned in
+wavelength over a resolving power of R = 50). The green markers show our results when neglecting the stellar spot contamination in the analysis
+(markers have been slightly shifted in wavelength for visibility). The blue spectrum shows a theoretical model of the planet’s upper atmosphere
+(see Section 5.1) and the blue dots show the model integrated over the spectral bins. The orange curve shows an extrapolation from a fit to the
+optical-IR transmission spectrum of the planet (see Section 5.2).
+The stellar-spot correction model thus contains two free pa-
+rameters for each visit (fspot and Tspot), which we fit simultane-
+ously along with the other astrophysical parameters during the
+MCMC. We considered a uniform prior for the fspot values be-
+tween 0.0 and 0.5 (it would be unlikely to find unocculted spot
+covering fractions greater than 50%). For the spot temperature
+we considered a logarithmic-uniform prior in ∆Tspot = Teff−Tspot
+because the spectral variation of the spot correction is more pro-
+nounced at small ∆Tspot values (McCullough et al. 2014). When
+∆Tspot ≳ 1500 K the correction becomes nearly achromatic. Re-
+gardless of the parameterization, the main conclusions of our
+analysis do not change when we consider a uniform prior in Tspot.
+4. Planetary Atmospheric Signatures
+4.1. Coarse-binning Spectra
+To boost the characterization of the astrophysical signal, we
+started with the analysis of the combined data binning over a
+relatively coarse wavelength sampling. Figure 4 shows the NUV
+transmission spectrum of HD 189733 b sampled at a resolving
+power of R = 50. This resolving power resulted in the best com-
+promise between having a good S/N per bin and having sufficient
+spectral resolution to identify variability with wavelength. Ap-
+pendix A shows the corresponding systematics- and stellar-spot
+corrected lightcurves for this analysis. At this resolving power,
+we obtained a relatively flat transmission spectrum with the ex-
+ception of two regions where the transit depth increases (by more
+than ∼1σ relative to the surrounding wavelengths) located at
+λ ∼ 2350 Å and λ ∼ 2600 Å.
+For completeness, we also analyzed the data at other similar
+resolving powers of R = ∼10, 30, 70 finding consistent results.
+Likewise, analyses with the bins shifted by half a bin-width re-
+turned consistent results as well.
+The stellar spot contamination was well characterized, as
+shown in Figure 5. The first two visits show a negligible spot
+contamination (ϵ < 1.05), with the models finding either a small
+spot covering fraction or a spot temperature close to the nominal
+stellar effective temperature (∆Tspot ≲ 10). In contrast, the third
+visit shows a clearly significant spot contamination ranging from
+ϵ(λ) ≈ 1.20 to 1.15 from short to long wavelengths. The contam-
+ination shows a weak variation with wavelength, which is not
+larger than the 1σ credible intervals of the constraints. Accord-
+ingly, we constrained the spot covering fraction to fspot > 0.11
+and the spot temperature to 100 K ≲ ∆Tspot ≲ 500 K, values
+which are strongly correlated (see posterior pair-wise distribu-
+tion in Fig. 5). Analyzing the combined data without a stellar-
+spot model produced slightly larger transit depths (Fig. 4); how-
+ever, the shape of the spectrum remains consistent with that of
+the spot-corrected analysis, which is consistent with the nearly
+wavelength-independent stellar spot correction.
+All in all, the ratio between contamination spectra of the vis-
+its (e.g., ϵv3/ϵv1 = 0.15 − 0.2) matches precisely the ratio be-
+tween bolometric fluxes (e.g., Fv1
+s /Fv3
+s = 0.16), which reinforces
+the hypothesis that the variability between visits is caused by a
+varying stellar spot coverage at different epochs.
+4.2. Fine-binning Spectra: Magnesium Resonance Lines
+We attempted to analyze the data binning at much higher re-
+solving power such that we can capture absorption of individ-
+ual metallic lines, however, the high-resolution lightcurves were
+largely dominated by noise, and not much could be concluded
+from the resulting spectra. The only region where we were able
+to place significant constraints at high resolution is at the core
+of the magnesium resonance lines, which have fluxes nearly ten
+times larger than at other wavelengths, and thus a much better
+S/N. For this, we selected a bin resolving power of R = 4700 (65
+km s−1 wide bins), which encapsulates the core of the Mg i and
+Mg ii resonance lines.
+Figure 6 shows the high-resolution light curves centered at
+the Mg i and Mg ii resonance (top), and the stellar spectra (mid-
+dle) and transit radius-ratio spectra (bottom) around the mag-
+nesium lines. For the Mg ii bins, the light curves show a clear
+transit signature (albeit with some residual noise for the Mg ii h
+Article number, page 7 of 17
+
+HST/STIS NUV transits (R = 50)
+0.22
+Uncorrected stellar spots
+0.20
+0.18
+0.16
+0.14
+0.12
+2300
+2400
+2500
+2600
+2700
+2800
+2900
+3000
+3100
+Wavelength (A)A&A proofs: manuscript no. nuv_HD189733b
+1
+1
+3
+log10( Tspot)
+0.0
+0.2
+0.4
+fspot
+visit 1
+0.0
+0.2
+0.4
+fspot
+1
+1
+3
+log10( Tspot)
+0.0
+0.2
+0.4
+fspot
+visit 2
+0.0
+0.2
+0.4
+fspot
+1.0
+2.0
+3.0
+log10( Tspot)
+0.0
+0.2
+0.4
+fspot
+visit 3
+0.0
+0.2
+0.4
+fspot
+2300
+2400
+2500
+2600
+2700
+2800
+2900
+3000
+Wavelength (um)
+1.00
+1.05
+1.10
+1.15
+1.20
+1.25
+Contamination, (
+)
+visit 1
+visit 2
+visit 3
+Fig. 5. Stellar spot contamination. The top panels show the posterior
+distribution of the spot parameters derived from the combined-data
+analysis. The blue shaded areas denote the 1σ highest posterior den-
+sity credible region. Visits 1 and 2 show a negligible spot contamination
+with either a small spot covering fraction or a negligible difference be-
+tween the spot and photospheric temperatures. Visit 3 in contrast shows
+a non-negligible contamination of ϵ(λ) ∼1.15–1.2, with a strong cor-
+relation between the spot covering fraction and spot temperature. The
+bottom panel shows the respective contamination spectra. The colored
+markers and vertical spans show the median and 1σ uncertainty of the
+contamination, respectively.
+line), and have S/Ns on the same order of magnitude as those
+of the low-resolution analysis. The light curve at the Mg i line,
+having much lower flux, is significantly noisier than the Mg ii
+lines. For all three magnesium lines, the transit depths do not
+show any significant excess above the low-resolution continuum
+values (R = 50 analysis), and are consistent with the optical-IR
+transmission spectrum of HD 189733 b.
+For context, in the bottom panels of Fig. 6 we also show the
+theoretical upper-atmospheric model for this planet (see Sect.
+5.1 for details). While HD 189733 b’s escape rate is not high
+enough to generate magnesium absorption signatures close to
+the the Roche-lobe altitude (RL1′/Rs = 0.44), the model predicts
+a noticeable excess absorption above the continuum. Thus, we
+used the binned model estimate to place non-detection limits for
+the Mg ii measurements. We found that the observations sit 2–4σ
+below the prediction for a solar-abundance atmospheric model
+containing Mg in the upper atmosphere (Table 2). Due to the
+higher noise at the Mg i line, we cannot reject nor confirm the
+presence of Mg i absorption from our observations.
+5. The NUV-to-Infrared Transmission Spectrum of
+HD 189733 b
+For context, the Hei infrared metastable triplet and Hi Lyman-α
+transit observations of HD 189733 b can be interpreted in terms
+of the planet hosting a compact upper atmosphere; extended, but
+not as much as that of other exoplanets observed in the NUV,
+Table 2. HD 189733 b Transit Measurements at Magnesium Lines
+Line
+λ0 (µm)
+Rp/Rs
+Non-detection significance†
+Mg ii k
+2796.4
+0.158+0.006
+−0.006
+3.7σ
+Mg ii h
+2803.5
+0.153+0.008
+−0.007
+2.3σ
+Mg i
+2853.0
+0.138+0.056
+−0.032
+<1σ
+† number of standard deviations below the expected Rp/Rs
+from the solar-abundance theoretical model.
+e.g., HD 209458 b. Metals like magnesium or iron may condense
+in the lower atmosphere and even if they make it to the upper
+atmosphere, a lower escape rate would not allow their density
+profiles to stretch out to high altitudes. At longer wavelengths,
+the optical-IR observations show a strong slope towards the blue,
+which can be interpreted as absorption from high-altitude hazes.
+To test these hypotheses and to get a better understanding
+of the interaction between the upper and lower atmosphere of
+HD 189733 b, we compare the observations to theoretical mod-
+els. Since it is not possible to model all required upper- and
+lower-atmosphere properties at once, we model individually the
+upper and lower regions, and then study how they relate to each
+other.
+5.1. Upper-atmosphere Modeling
+To interpret the NUV transmission spectrum of HD 189733 b, we
+employed the physical model of the upper atmospheres of close-
+in planets (Koskinen et al. 2013a,b, 2022). Since the parts of the
+NUV transmission spectrum that probe the upper atmosphere are
+dominated by signatures of ionized metals Fe ii and Mg ii (Sing
+et al. 2019; Cubillos et al. 2020), we added the heavy elements
+C, O, N, Mg, Si, Fe, S, Ca, Na, and K with solar abundances to
+the model, in addition to H and He, with related chemistry and
+physics adapted from Huang et al. (2017). We included Mg, Si,
+Fe, Ca with first and second ionization states, due to the rela-
+tively low ionization potential of the first ionization state, and
+the rest of the heavy elements with their first ionization state.
+The electron density is equal to the sum of the ion densities un-
+der the assumption of quasineutrality. The lower boundary of the
+escape model is at 0.1 µbar where the temperature is sufficient
+to dissociate molecules that are not included in the model.
+We used the results from the photochemical model of Lavvas
+& Koskinen (2017) to obtain the mean molecular weight profile
+in the lower and middle atmosphere to set the lower boundary
+altitude and temperature of the escape model. The temperature
+profile in the lower and middle atmosphere was not calculated
+self-consistently by the photochemical model and was instead
+adapted from Moses et al. (2011). Here, we used the MK profile
+from Lavvas & Koskinen (2017). The application of the photo-
+chemical model to HD189733b did not include Mg and Fe that
+are detectable in the NUV. In order to crudely model the full pro-
+files of the absorption lines, we added Mg and Fe to the lower
+and middle atmosphere, assuming solar abundances and thermal
+ionization according to the Saha equation (Menou 2012). This
+approximation does not affect our conclusions because our focus
+in this section is on the possible upper atmosphere signatures.
+Given the temperature profile assumed for HD 189733 b here
+and assuming equilibrium condensation, Mg, Fe, Si, and Ca are
+expected to condense to form mineral clouds and rain out from
+the atmosphere (e.g., Wakeford & Sing 2015). We included Mg
+and Fe in the escape model simply to explore if the modeled ab-
+Article number, page 8 of 17
+
+Patricio E. Cubillos et al.: The HST NUV Transmission Spectrum of HD 189733 b
+Fig. 6. Transit light curves (top) and spectrum (bottom) of HD 189733 b of the combined HST/STIS transits at the location of the magnesium
+resonance lines, and binned at a resolving power of R = 4700 (i.e., roughly the width of the magnesium lines). The top panels show the systematics-
+corrected light curves for the spectral channels centered at the core of the Mg ii and Mg i lines (see labels). The solid red curve and orange area
+denote the best-fitting transit depths and span of their 1σ uncertainties. The middle panels show the system flux around the magnesium lines, with
+the vertical dashed lines marking the edges of the spectral bins. The bottom panels show the transit planet-to-star radius ratio for each of the bins
+around the magnesium resonance lines (red markers with 1σ error bars). The blue curve shows the upper-atmosphere model (the diamond markers
+show the model binned at R = 4700). The peach-colored area shows the span of the low-resolution transit radius ratio (R = 50).
+sorption lines of Mg II and Fe II would be detectable in the NUV
+data. The rest of the heavy elements were retained for consis-
+tency with the no-condensation scenario. Although efficient mix-
+ing in the lower atmosphere could interfere with mineral cloud
+formation and allow for higher abundances of the relevant heavy
+elements in the upper atmosphere (Spiegel et al. 2009; Koskinen
+et al. 2013b), we do not hold a view on whether this is possible
+on HD 189733 b or not at this point.
+Figures 4 and 7 show the transmission spectrum from the
+upper atmosphere model in blue. Here, we calibrated the model
+spectrum so that the measured base Rp/Rs (in the near-IR) cor-
+responds to 2.3 mbar in the atmosphere (Lavvas & Koskinen
+2017). So as not to bias the fit to any particular interpretation at
+this point, we did not include extinction by the high-altitude haze
+modeled by Lavvas & Koskinen (2017) in this forward model.
+Instead, the transit continuum is due to Rayleigh scattering by
+H2, H, and He. It is immediately clear that the model continuum
+falls below the extrapolation based on the optical-IR transmis-
+sion spectrum of the planet, due to the lack of the high-altitude
+haze in the forward model. Since transit depth is not a cumulative
+quantity, however, the model still correctly represents absorption
+by Fe II and Mg II in the upper atmosphere in the absence of
+condensation, even if high-altitude hazes are present.
+We ran the upper atmosphere model assuming globally av-
+eraged radiative forcing with the XUV spectrum constructed
+from ϵ Eridani as an input, due to the well known similarity to
+HD 189733 (Lavvas & Koskinen 2017). The reference simula-
+tion, on which the spectrum in Figure 4 is based on, assumes
+an isolated planet. We also ran a second simulation that includes
+Roche lobe overflow (see, Koskinen et al. 2022) but this had neg-
+ligible effect on the modeled spectrum, in line with the expecta-
+tion that Roche lobe overflow does not significantly enhance the
+escape rate or upper atmosphere densities on HD 189733 b.
+We note that NUV observations have the potential to of-
+fer valuable new insights to the properties of possible high-
+altitude hazes and/or clouds. Unfortunately, S/N is an issue for
+HD 189733 b that precludes clear conclusions on the origin of
+the continuum in the transit observations. The visible spectrum
+shows a strong slope indicative of small particle hazes (Sing et al.
+2016; Lavvas & Koskinen 2017) that appears to be supported
+to some degree by observations by the Stratospheric Observa-
+tory for Infrared Astronomy (SOFIA) (Angerhausen et al. 2015);
+however, the latter found a slope offsetted to lower depths by
+∆Rp/Rs =∼ 0.001 (see Appendix A). The slope can, however,
+be enhanced by stellar activity (star spots) (McCullough et al.
+2014). Also, a general circulation model (GCM) that simulates
+the production and transport of hazes predicts a shallower slope
+unless mixing by sub-grid scale eddies exceeds mixing by global
+circulation (Steinrueck et al. 2021). The NUV observations are
+statistically consistent with the model that includes no haze, and
+generally the observed transmission falls below the extrapola-
+tion based on the visible slope (Figure 7), with the exception of
+the 2350 Å region. Due to low S/N in the NUV, however, the
+observations are also consistent with this extrapolation. Thus,
+Article number, page 9 of 17
+
+1.06
+1.06
+Mg k, 入 = 2796.4 A
+Mg h, ^= 2803.5 A
+1.2 FMg1, 入= 2853.0 A
+1.03
+1.03
+(%)
+1.1
+1.00
+1.00
+1o.0
+1.0
+R
+0.97
+0.97
+visit 1
+0.9
+visit
+2
+visit
+0.94
+0.94
+0.8
+-0.06
+-0.03
+0.00
+0.03
+-0.06
+-0.03
+0.00
+0.03
+-0.06
+-0.03
+0.00
+0.03
+Orbital phase
+Orbital phase
+Orbital phase
+1.5
+20
+1.0
+10
+0.5
+0-13
+0
+0.0
+三 0.30
+Mg Il
+Model
+0.30
+Mgi
+Binned model
+Data (R=50)
+0.25
+0.25
+R
+Data (R=4700)
+0.20
+0.20
+R
+R
+0.15
+0.15
+0.10
+0.10
+2790
+2792
+2794
+2796
+2798280028022804280628082810
+2847
+2850
+2853
+2856
+2859
+Wavelength (A)
+Wavelength (A)A&A proofs: manuscript no. nuv_HD189733b
+these NUV observations neither confirm or exclude the presence
+of high-altitude hazes, especially if the visible slope is enhanced
+by star spots.
+5.2. Lower-atmosphere Modeling
+To constrain the lower-atmosphere properties of HD 189733 b
+we used the open-source Pyrat Bay framework3 (Cubillos &
+Blecic 2021). We modeled the atmosphere from 100 to 10−9 bar,
+adopting a parametric temperature profile from Madhusudhan &
+Seager (2009), hydrostatic equilibrium, and a H/He-dominated
+composition considering constant-with-altitude volume mixing
+ratios.
+The Pyrat Bay radiative transfer computed the transmis-
+sion spectrum between 0.2 and 33.0 µm, at a constant resolving
+power of R = λ/∆λ = 15, 000. The opacities included line-by-
+line data from HITEMP for CO2, CO, and CH4 (Rothman et al.
+2010; Li et al. 2015; Hargreaves et al. 2020), and from ExoMol
+for HCN, H2O, NH3, and SiO (Harris et al. 2006, 2008; Polyan-
+sky et al. 2018; Coles et al. 2019; Yurchenko 2015; Yurchenko
+et al. 2022). We extracted the dominant transitions from these
+line-lists using the repack algorithm (Cubillos 2017), and then
+computed tabulated opacities by sampling over temperature and
+wavelength grid. Additionally, the model included collision-
+induced opacities for H2–H2 pairs (Borysow et al. 2001; Bo-
+rysow 2002) and H2–He pairs (Borysow et al. 1988, 1989; Bo-
+rysow & Frommhold 1989); Rayleigh-scattering opacity for H,
+H2, and He (Kurucz 1970); and a gray cloud deck model. Finally,
+the model considered a parametric haze absorber model (Lecave-
+lier Des Etangs et al. 2008) with arbitrary slope and strength
+(αhaze and κhaze), which are fit in the retrieval.
+The
+atmospheric
+retrieval
+employed
+the
+differential-
+evolution Markov-chain Monte Carlo sampler (ter Braak &
+Vrugt 2008), implemented via the open-source code mc3 (Cubil-
+los et al. 2017), guided by the transit observations from HST’s
+STIS G430 and G750, WFC3 G102 and G141 spectrographs;
+and from Spitzer’s IRAC 3.6, 4.5, 5.8, 8.0 µm and MIPS 24 µm
+photometric bands (Pont et al. 2013; Sing et al. 2016).
+Figure 7 shows the fit to the optical-IR transit observations.
+Appendix B presents the retrieval posterior-distribution results
+for all model parameters. The optical-IR observations point to
+a prominent haze slope, steeper than a Rayleigh slope (αhaze ≈
+−9.3±2.3). This is a similar conclusion to that found by previous
+retrieval studies on this planet (e.g., Pinhas et al. 2019; Barstow
+2020). From a physical perspective, the steep haze slope can be
+created by a combination of particle growth and efficient eddy
+(turbulent) mixing (Ohno & Kawashima 2020) or by stellar in-
+homogeneities (Rackham et al. 2018), effects that were captured
+by the parametric haze model (rather than being explicitly ac-
+counted for). The optical absorption is also stronger than that
+produced by hydrogen (as in the upper-atmosphere model), lead-
+ing to a higher continuum level at optical wavelengths, as well
+when extrapolated to NUV wavelengths. The NUV observations
+are consistent with the optical-IR continuum model, with the ex-
+ception of the few data points that extend above the continuum
+(mainly the 2350 Å region). Lastly, we can see that the constraint
+on the temperature profile becomes less precise above the µbar
+level since optical/IR data is mostly probing the lower layers of
+the atmosphere. At the same time, at this high altitudes the re-
+trieved temperature profile starts to differ from the higher tem-
+peratures predicted by self-consistent models that account for
+high-energy flux that heats the thermosphere of the planet (Lav-
+3 https://pyratbay.readthedocs.io/
+vas & Koskinen 2017), i.e., the models used as the base of our
+upper-atmosphere model.
+6. On the Nature of the NUV Absorption
+6.1. Fe ii Absorption
+It is interesting that the excess NUV absorption matches pre-
+cisely the locations of the two strongest Fe ii bands in the region,
+while there does not seem to be any absorption corresponding to
+the Fe i band shortly blueward of the magnesium doublet. Unfor-
+tunately, the low S/N achieved when binning the data at higher
+resolution does not allow us to identify absorption from individ-
+ual lines.
+In fact, a detection of Fe ii would be very controversial in this
+planet for multiple reasons. In first place, iron is not expected to
+survive the cold trap at the lower layers. Even if we neglected
+the iron condensation, as is the case for our upper-atmosphere
+model, we cannot match the large amplitude of the observed fea-
+tures when we bin the model at the resolution of the observation.
+Another argument against Fe ii is that given the relatively cool
+expected temperature of the planet and the late spectral type of
+the star, one would also expect to observe Fe i if Fe ii were de-
+tected.
+If the NUV absorption indeed corresponded to Fe ii, it would
+imply that our hydrodynamic model severely underestimates the
+absorption signature, and we would be in need of a mechanism
+to enhance the lines such that they become detectable in broader
+bins. This would require a significant broadening of the iron lines
+and super-solar abundances, even when assuming no condensa-
+tion in the lower atmosphere. However, global circulation mod-
+els (GCM) predict winds of the order of ±8 km s−1 at the ter-
+minator (Steinrueck et al. 2021). These models include regions
+of significant size with flows both away from and towards the
+Earth due to the complexity of the circulation and rotation ef-
+fects. These kinds of flows could effectively broaden metal lines
+in spatially unresolved transit observations.
+6.2. Absorption from an Unknown Absorber or other Source
+The broad-band absorption suggests the presence of an extra ab-
+sorber, but the fact that the absorption matched the location of
+Fe ii bands might be deceptive. It is certainly possible that the ab-
+sorption comes from another unidentified species, possibly from
+either a haze or from the electronic band system of a molecu-
+lar species, given the width of the absorption and the transition
+energies involved.
+Following Lothringer et al. (2022), we explored whether
+SiO, say vaporized from strongly mixed cloud particles, could
+appear in the upper atmosphere. However, the atmospheric re-
+trievals do not favor significant SiO absorption when it was
+included in the model. From a further heuristic exploration,
+we could not find any configuration of temperatures and abun-
+dances producing SiO bands at the observed strength nor loca-
+tion. We finally tested for other candidates available in the Ex-
+oMol database and that extended into the NUV, which led us to
+discard both HS and OH.
+One might consider that the absorption originates from an
+external source such as accreted material from trojan or circum-
+planetary satellites (Kislyakova et al. 2016; Oza et al. 2019).
+However, this is a highly speculative scenario, since to date there
+is no model that demonstrates that material from an external
+source can produce such large absorption features, nor that it
+Article number, page 10 of 17
+
+Patricio E. Cubillos et al.: The HST NUV Transmission Spectrum of HD 189733 b
+Fig. 7. Infrared-to-ultraviolet transmission spectrum of HD 189733 b. Top and bottom panels show the same data, focused on different spectral
+regions for better visibility. The colored markers with error bars show the transit observations, 1σ uncertainties, and wavelength span of the
+observations (see legend). Our data points have been computed from the data binned at a resolving power of R=33 (see Appendix A). The red
+curve and orange shaded areas shows the spectrum posterior distribution of the optical-IR fit (median, 68%, and 95% quantiles of the distribution,
+see Appendix B). The blue curve shows the upper-atmosphere model (shaded in a slightly clear color when sitting below the optical/infrared
+continuum).
+could generate exclusively an Fe ii signature but not that from
+other particles like magnesium.
+It seems also unlikely that the absorption features come from
+a yet-undetermined aerosol. To our knowledge there is no self-
+consistent GCM that couples haze production and circulation to
+a degree that it can state with confidence that particles could
+form and exist at the required high altitudes (∼0.1 nbar). It is
+hard then to explain how aerosols would form and remain aloft
+at such altitudes.
+We also tested whether the stellar spot correction could gen-
+erate the observed features (Rackham et al. 2018, 2019), but on
+close inspection of the spectral dependency of the spot contam-
+ination at different temperatures, we found no indication that it
+could mimic extra absorption at the location of the Fe ii bands.
+Finally, we considered the impact of stellar center-to-limb
+flux variation. As the stellar Fe ii lines show in emission (Fig.
+1), the observations should be probing the (hotter) stellar chro-
+mosphere and transition region rather than the (cooler) photo-
+sphere. Therefore the lines may be showing limb brightening
+instead of limb darkening, leading to a quite different transit
+signature (see, e.g., Schlawin et al. 2010). Unfortunately, the
+limb-darkening/brightening law for these lines is unknown, and
+it cannot be reasonably estimated with current instrumentation.
+In any case, while the unknown limb-darkening law of the emis-
+sion lines introduces a degree of uncertainty in the NUV transit
+depths, unaccounted limb brightening would lead to underesti-
+mates transit depths, and thus does not explain the excess ab-
+sorption seen at 2350 Å and 2600 Å.
+6.3. Exoplanet NUV Observations in Context
+To date three exoplanets have been spectroscopically ob-
+served during transit in the NUV with HST/STIS: WASP-121b
+(Sing et al. 2019), HD 209458 b (Cubillos et al. 2020), and
+HD 189733 b (this work). These three planets span a wide
+range of irradiation regimes, receiving a stellar flux of 364×
+(HD 189733 b), 769× (HD 209458 b), and 5624× (WASP-121b)
+that of the one received by the Earth. The analysis of this aggre-
+gated data provides a first glance at the physical processes that
+shape the properties of the planets’ upper atmospheres.
+The upper panel of Fig. 8 shows the transmission spectra
+of these three planets at a resolution of R = 1300 in units of
+r/Rp, with Rp being the optical radius. We can see that WASP-
+121b exhibits enhanced transit depths coinciding with Fe ii and
+Mg ii lines, whereas the other two planets show subtler features.
+When comparing the transmission spectra at a resolving power
+of R = 130 in units of r/RRL, with RRL being the Roche-lobe
+terminator radius, (lower panel of Fig. 8) we see that the pseudo-
+continuum extends to the Roche-lobe boundary, which indicates
+that the middle atmosphere/lower thermosphere of WASP-121b
+undergoes Roche-lobe overflow. In clear contrast, this is not the
+case for HD 189733 b nor HD 209458 b. WASP-121b is there-
+fore unique among these planets, and must have a much higher
+mass loss rate than the others, as expected. This broad behav-
+ior is qualitatively consistent with expectations based on model
+predictions.
+Article number, page 11 of 17
+
+0.160
+0.158
+R
+R
+0.156
+0.154
+0.2
+0.3
+0.5
+1.0
+2.0
+5.0
+10.0
+20.0
+Upper-atmosphere model
+0.20
+Lower-atmosphere retrieval
+HST NUV data
+s
+Sing et al. (2015)
+R
+0.18
+King et al. (2021)
+R
+0.16
+888888888
+0.2
+0.3
+0.4
+0.6
+1.0
+1.5
+Wavelength (um)A&A proofs: manuscript no. nuv_HD189733b
+Fig.
+8.
+Comparison
+HST/STIS
+NUV
+transmission
+spectra
+of
+HD 189733 b, HD 209458 b, and WASP-121b scaled by the optical tran-
+sit radius Rp (top panel) and by the Roche-lobe terminator radius RRL of
+each planet (bottom panel). The shaded regions denote the wavelength
+span of the dominant Fe ii and Mg ii NUV bands and lines.
+The lower panel of Fig. 8 illustrates the dramatic difference
+between WASP-121b and the two other planets. The NUV spec-
+tra of HD 209458 b and HD 189733 b look nearly featureless
+when compared to that of WASP-121b. Certainly, WASP-121b is
+not an ordinary planet—it must have substantially larger mass-
+loss rates than the other two planets. Neither HD 209458 b nor
+HD 189733 b show the Mg ii absorption feature that survives in
+the spectrum of WASP-121b. As we already know, the spectrum
+of HD 189733 b shows a strong feature close to the blue Fe ii ab-
+sorption band. This feature does not extend to the Roche lobe,
+but the peak is ∼150 surface scale heights above the optical ra-
+dius. Given that heating in the thermosphere increases the scale
+height by a factor of ∼20 above the ∼1 µbar level, the peak prob-
+ably probes the 1 nbar level.
+Finally, we note that the feature in the spectrum of
+HD 189733 b is not identical to the Fe ii feature in the spectrum
+of WASP-121b though. In particular, the point at ∼2400 Å coin-
+cides with a strong Fe ii band and should show maximum absorp-
+tion towards the longer-wavelength edge. This is true for WASP-
+121b, but in the spectrum of HD 189733 b absorption drops in-
+stead. This poses a conundrum, however, given the uncertainties
+in the data analysis, noise could be responsible for this apparent
+inconsistency.
+7. Conclusions
+HD 189733 b is the coolest hot Jupiter that has been spectrosopi-
+cally observed in the NUV so far. Transmission observations in-
+dicate that this is a very hazy planet, but the composition of the
+hazes are not really know, some of them may present additional
+absorption in the NUV.
+By analyzing the combined NUV HST/STIS transmission
+spectra (3 transits) at a relatively coarse resolution (R = 50),
+we obtained better S/Ns and spectral resolution than previous
+NUV observation of HD 189733 b in the NUV by King et al.
+(2021). The transit signature in the light curves is clear and with
+no significant residual systematics. We were able to constrain
+a continuum level that sits in between the extrapolation of the
+optical haze slope and the continuum of a pure-H2/He upper at-
+mosphere model. Thus, given the uncertainties of our continuum
+data points, the NUV data is statistically consistent with either a
+hazy or clear atmosphere scenario. In addition, the NUV trans-
+mission spectrum presents strong and broad absorption features
+that coincide with the location of two strong Fe ii bands, while
+there is no apparent excess absorption correlated with Fe i bands.
+When analyzing the data at a higher resolution (R = 4700), we
+found that most of the lightcurves are dominated by noise, which
+does not allow us to confirm nor discard iron as the source of the
+absorption. However, thanks to the much higher stellar flux at the
+location of the magnesium resonance lines, we were able to rule
+out the presense of Mg ii excess absorption above the continuum.
+When compared to a solar-metallicity model with magnesium in
+the upper atmosphere, the measured transmission Rp/Rs values at
+the core of the Mg ii lines are 2–4σ below the model predictions.
+The S/Ns of the data at the Mg i line is not high enough to reject
+or confirm excess absorption due to magnesium.
+HD 189733 b has a surface gravitational acceleration twice
+that of HD 209458 b and unlike WASP-121b, Roche-lobe over-
+flow effects are not expected to be that significant. Its atmo-
+sphere is also cooler, suggesting that the formation of clouds
+is expected. This is well in agreement with optical observa-
+tions, where the data are consistent with a high-altitude haze,
+which could have a soot-like precursor. HD 189733 b upper-
+atmosphere models assuming that magnesium did not condense
+in the lower atmosphere show considerable excess absorption
+at the core of the Mg ii resonance lines. Thus, the absense of
+excess absorption suggests that magnesium is not escaping on
+HD 189733 b and that Mg-Si clouds likely form in the lower at-
+mosphere.
+At this point, the nature of the broad NUV absorption fea-
+tures is not known. An iron origin raises significant challenges
+for existing upper-atmosphere models. In first place, given the
+relatively low temperatures of HD 189733 b, condensation is ex-
+pected to sequester heavy metals into the lower layers of the at-
+mosphere, though it is possible that condensates form primarily
+from other metals instead, such as magnesium (Gao et al. 2020;
+Woitke et al. 2020). However, even if iron is not strongly de-
+pleted, one would still need a mechanism to enhance its absorp-
+tion beyond that predicted by our upper-atmosphere models. The
+two most immediate possibilities would be to consider higher
+metallicities and zonal-wind velocity broadening of the absorp-
+tion lines. GCM estimations of HD 189733 b predict wind veloc-
+ities at ∼8 km s−1 at the terminator, the question for future stud-
+ies is whether these mechanisms are enough to boost the signal.
+Heavy metals play a key role in the modelling and interpretation
+of NUV transit observations and hence in our understanding of
+exoplanetary atmospheres as a whole. The detection of metals
+at Roche lobe distances would be surprising instead of expected
+Article number, page 12 of 17
+
+HD 209458b
+Fell
+Mgll
+3.0
+HD 189733b
+WASP-121b
+2.5
+2.0
+a
+1.5
+1.0
+0.5
+2300
+2400
+2500
+2600
+2700
+2800
+2900
+3000
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+2400
+2500
+2600
+2700
+2800
+2900
+3000
+Wavelength (A)Patricio E. Cubillos et al.: The HST NUV Transmission Spectrum of HD 189733 b
+for HD 189733 b. Here we presented the detection of a broad
+absorption feature in the NUV that matches the location of Fe ii
+bands, but cannot fully confirm the nature of the absorber, both
+future observational and theoretical work are needed to under-
+stand more precisely the origin of this absorption.
+Acknowledgements. We thank the anonymous referee for his/her time and valu-
+able comments. We thank contributors to the Python Programming Language and
+the free and open-source community, including: Pyrat Bay (Cubillos & Blecic
+2021), mc3 (Cubillos et al. 2017), repack (Cubillos 2017), Numpy (Harris et al.
+2020), SciPy (Virtanen et al. 2020), Matplotlib (Hunter 2007), IPython (Pérez
+& Granger 2007), and bibmanager (Cubillos 2020). This research has made use
+of NASA’s Astrophysics Data System Bibliographic Services. This project was
+funded in part by the Austrian Science Fund (FWF) Erwin Schroedinger Fellow-
+ship, programs J4595-N and J4596-N. A.G.S. and L.F. acknowledge financial
+support from the FFG project 865968. Based on observations made with the
+NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science
+Institute, which is operated by the Association of Universities for Research in
+Astronomy, Inc., under NASA contract NAS 5-26555. These observations are
+associated with program #15338.
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+
+A&A proofs: manuscript no. nuv_HD189733b
+Appendix A: HD 189733 b NUV Transmission
+Spectra
+Fig A.1 shows the NUV transmission spectrum binned at a
+coarse resolving power of R = 10 and shows a SOFIA-like offset
+of the transit depth for comparison.
+Figure A.2 shows the systematics- and stellar-spot corrected
+lightcurves of the combined HST/STIS HD 189733 b observa-
+tions, when binned over a coarse wavelength sampling with a
+resolving power of R = 50. We used the posterior-distribution
+median and the central 68% percentiles statistics to estimate the
+parameter values and their credible intervals.
+Tables A.1, A.2, A.3 present the estimated values of the NUV
+transmission planet-to-star radius ratio when analyzed at resolv-
+ing powers of 50, 33, and 10, respectively.
+Fig. A.1. NUV transmission spectrum of HD 189733 b from the com-
+bined HST/STIS observations. The red markers with error bars denote
+the systematics- and stellar-spot corrected transmission spectrum, their
+1σ uncertainties, and the span of the spectral bins for a resolving power
+of R = 10. The blue spectrum shows a theoretical model of the planet’s
+upper atmosphere. The orange curve shows the fit to the optical-IR
+transmission spectrum extrapolated into the NUV (the shaded areas de-
+note the 1σ and 2σ uncertainties). The black dashed curve shows the
+optical-IR fit shifted downwards according to a SOFIA-like offset of
+∆Rp/Rs = 0.001.
+Appendix B: Lower-atmosphere Transmission
+Retrieval
+Table B.1 shows the parameterization, priors, and retrieved pos-
+terior values for the analysis of the HD 189733 b optical-IR
+transmission spectrum. Figure B.1 shows the posterior distribu-
+tions of the model parameters and temperature profile.
+Table A.1. HD 189733 b NUV Transmission Spectrum at R=50
+λ (µm)
+Bin half-width (µm)
+Rp/Rs
+2297.5
+23.1
+0.139+0.036
+−0.022
+2344.2
+23.6
+0.189+0.013
+−0.011
+2391.9
+24.1
+0.212+0.009
+−0.009
+2440.5
+24.6
+0.150+0.014
+−0.012
+2490.1
+25.1
+0.154+0.011
+−0.010
+2540.7
+25.6
+0.155+0.009
+−0.008
+2592.4
+26.1
+0.166+0.006
+−0.005
+2645.1
+26.6
+0.165+0.004
+−0.004
+2698.8
+27.2
+0.156+0.004
+−0.004
+2753.7
+27.7
+0.157+0.004
+−0.004
+2809.6
+28.3
+0.159+0.003
+−0.003
+2866.7
+28.8
+0.160+0.004
+−0.003
+2925.0
+29.4
+0.150+0.003
+−0.002
+2984.5
+30.0
+0.161+0.003
+−0.003
+3043.7
+29.2
+0.158+0.003
+−0.003
+Table A.2. HD 189733 b NUV Transmission Spectrum at R=33
+λ (µm)
+Bin half-width (µm)
+Rp/Rs
+2309.4
+35.0
+0.172+0.016
+−0.014
+2380.5
+36.1
+0.202+0.008
+−0.008
+2453.7
+37.2
+0.157+0.010
+−0.009
+2529.2
+38.3
+0.155+0.008
+−0.007
+2607.1
+39.5
+0.168+0.004
+−0.004
+2687.3
+40.7
+0.158+0.004
+−0.003
+2770.0
+42.0
+0.158+0.003
+−0.003
+2855.2
+43.3
+0.155+0.003
+−0.003
+2943.0
+44.6
+0.156+0.002
+−0.002
+3030.3
+42.7
+0.158+0.003
+−0.002
+Table A.3. HD 189733 b NUV Transmission Spectrum at R=10
+λ (µm)
+Bin half-width (µm)
+Rp/Rs
+2405.1
+130.7
+0.1734+0.0055
+−0.0052
+2681.6
+145.7
+0.1607+0.0022
+−0.0018
+2950.1
+122.8
+0.1564+0.0017
+−0.0013
+Article number, page 14 of 17
+
+0.180
+Upper-atmosphere model
+HST optical-IR transit depth
+SOFIA Optical-IR transit depth
+0.175
+HST/STiS NUV transits (R = 1O)
+0.170
+R
+0.165
+0.160
+0.155
+2300
+2400
+2500
+2600
+2700
+2800
+2900
+3000
+3100
+Wavelength (A)Patricio E. Cubillos et al.: The HST NUV Transmission Spectrum of HD 189733 b
+0.06
+0.03
+0.00
+0.03
+Orbital phase
+0.6
+0.7
+0.8
+0.9
+1.0
+(Rp/Rs)2 (%)
+= 2298 A
+= 2344 A
+= 2392 A
+= 2441 A
+= 2490 A
+0.06
+0.03
+0.00
+0.03
+Orbital phase
+0.80
+0.85
+0.90
+0.95
+1.00
+= 2541 A
+= 2592 A
+= 2645 A
+= 2699 A
+= 2754 A
+0.06
+0.03
+0.00
+0.03
+Orbital phase
+0.88
+0.91
+0.94
+0.97
+1.00
+= 2810 A
+= 2867 A
+= 2925 A
+= 2984 A
+= 3044 A
+visit  1
+visit  2
+visit  3
+Fig. A.2. Transit light curves of the combined HST/STIS observations of HD 189733 b binned at a resolving power of R = 50. The colored markers
+with error bars denote the systematics- and stellar-spot corrected measurements, color coded for each visit (see legend). The solid red curve and
+orange area denote the best-fitting transit depths and span of their 1σ uncertainties. The lightcurve of each bin has been shifted in the vertical axis
+for visualization. The labels next to each ligtcurve denotes the mean wavelength of the spectral bin. Note that the scale of the vertical axis changes
+from left to right as the S/N of the data improves with increasing wavelength.
+Article number, page 15 of 17
+
+A&A proofs: manuscript no. nuv_HD189733b
+Fig. B.1. Pairwise distribution (lower-left corner panels) and marginal histograms (diagonal) of the retrieved posterior parameters of the
+HD 189733 b optical-IR retrieval. The dashed lines indicate the marginal maximum likelihood of the parameters (the mode of the histograms,
+or median when the parameter not well constrained). The shaded areas denote the 68% highest posterior density. The top-right panel shows the
+posterior distribution of the temperature-profile models (the median and the span of the 68% and 95% central credible intervals).
+Article number, page 16 of 17
+
+log(p1) = - 8.17:98
+10-9
+log(p2) = - 2.803:1
+median
+log(p2)
+log(p3) = - 2.02:§
+10-7
+(Ed)60l
+-5
+a1 = 0.79±8:18
+0.75
+ 0.50
+Pressure (bar)
+10-
+0.25
+a2 = 0.34±8:13
+0.75
+0.50
+0.25
+To = 1211.5+4:6
+10
+3000
+2000
+1000
+Rp (Rjup) = 1.1363±0:8835
+1.14
+10-1
+３４
+log(Xco) = - 6.4:i
+101
+log(Xco2) = - 5.17±9:92
+500
+1000 1500 2000 2500
+3000
+Temperature (K)
+log(XcH4) = - 8.5±:2
+-10
+log(XNH3) = - 8.8±2:
+log(XHCN) = - 3.82±9:17
+log(Xsio) = 
+- 7.5±3:3
+log(Xna) = - 1.17+8:31
+log(XNa) 
+log(Xk) = - 3.53:
+log(Xk)
+05
+log(fray) = 5.17+8:72
+log(fray)
+5.0
+Xray
+aray
+-15
+-1.283:48
+(dod)60l
+0.0
+5
+0
+0
+055055
+C
+000T
+log(p1) log(p2)
+log(p3)
+0g(XH20) 1
+a1
+a2
+Rp (Rjup)
+log(xco)log(x
+αray
+To
+log(ptop)Patricio E. Cubillos et al.: The HST NUV Transmission Spectrum of HD 189733 b
+Table B.1. HD 189733b Optical-IR Retrieval Parameters Summary
+Parameter
+Priors†
+Retrieved value‡
+log10(p1/bar)
+U(−9, 2)
+−8.17+2.00
+−0.78
+log10(p2/bar)
+U(−9, 2)
+−2.80+3.30
+−0.81
+log10(p3/bar)
+U(−9, 2) and p3 > p1
+−2.0+3.6
+−2.5
+a1 (K−0.5)
+U(0.02, 2.0)
+0.79+0.19
+−0.38
+a2 (K−0.5)
+U(0.02, 2.0)
+0.34+0.33
+−0.10
+T0 (K)
+U(500, 3500)
+1210.0+720.0
+−490.0
+Rplanet (RJup)
+U(0.5, 2.0)
+1.1363+0.0025
+−0.0033
+log10(XH2O)
+U(−12, −1)
+−3.31+0.39
+−0.47
+log10(XCO)
+U(−12, −1)
+−6.4+4.6
+−5.1
+log10(XCO2)
+U(−12, −1)
+−5.17+0.99
+−3.92
+log10(XCH4)
+U(−12, −1)
+−8.5+2.4
+−3.2
+log10(XNH3)
+U(−12, −1)
+−8.8+2.1
+−2.7
+log10(XHCN)
+U(−12, −1)
+−3.82+0.87
+−3.17
+log10(XSiO)
+U(−12, −1)
+−7.5+2.8
+−3.3
+log10(XNa)
+U(−12, −1)
+−1.17+0.17
+−0.91
+log10(XK)
+U(−12, −1)
+−3.5+1.6
+−3.3
+log10( fray)
+U(−4, 9)
+5.17+0.55
+−0.72
+αray
+U(−20, 0)
+−9.3+2.2
+−2.3
+log10(Ptop/bar)
+U(−7, 2)
+−1.28+3.08
+−0.40
+† U(a, b) stands for a uniform distribution between a and b.
+‡ The values correspond to the maximum likelihood of
+the marginal posterior distributions. The uncertainties corre-
+spond to the span of the 68% highest-posterior-density cred-
+ible interval (Andrae 2010).
+Article number, page 17 of 17
+
diff --git a/ptFPT4oBgHgl3EQfMDTj/content/tmp_files/load_file.txt b/ptFPT4oBgHgl3EQfMDTj/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1734 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf,len=1733
+page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' nuv_HD189733b  ©ESO 2023  January 31, 2023  The Hubble/STIS Near-ultraviolet Transmission Spectrum of  HD 189733 b  Patricio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos1, 2, Luca Fossati1, Tommi Koskinen3, Chenliang Huang3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Sreejith1, 4, Kevin France4, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Wilson Cauley4, and Carole A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Haswell5  1 Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, A-8042, Graz, Austria  e-mail: patricio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='cubillos@oeaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='at  2 INAF – Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese, Italy  3 Lunar and Planetary Laboratory, University of Arizona, 1629 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' University Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Tucson, AZ 85721, USA  4 Laboratory for Atmospheric and Space Physics, University of Colorado, 600 UCB, Boulder, CO 80309, USA  5 School of Physical Sciences, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK  January 31, 2023  ABSTRACT  The benchmark hot Jupiter HD 189733 b has been a key target to lay out the foundations of comparative planetology for giant exo-  planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' As such, HD 189733 b has been extensively studied across the electromagnetic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Here, we report the observation and  analysis of three transit light curves of HD 189733 b obtained with Hubble/STIS in the near ultraviolet, the last remaining unexplored  spectral window to be probed with present-day instrumentation for this planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The NUV is a unique window for atmospheric mass-  loss studies owing to the strong resonance lines and large photospheric flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Overall, from a low-resolution analysis (R = 50) we  found that the planet’s near-ultraviolet spectrum is well characterized by a relatively flat baseline, consistent with the optical-infrared  transmission, plus two regions at ∼2350 and ∼2600 Å that exhibit a broad and significant excess absorption above the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  From an analysis at a higher resolution (R = 4700), we found that the transit depths at the core of the magnesium resonance lines are  consistent with the surrounding continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We discarded the presence of Mg ii absorption in the upper atmosphere at a ∼2–4σ confi-  dence level, whereas we could place no significant constraint for Mg i absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These broad absorption features coincide with the  expected location of Fe ii bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' however, solar-abundance hydrodynamic models of the upper atmosphere are not able to reproduce  the amplitude of these features with iron absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Such scenario would require a combination of little to no iron condensation in  the lower-atmosphere, super-solar metallicities, and a mechanism to enhance the absorption features (such as zonal wind broadening).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The true nature of this feature remains to be confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Techniques: spectroscopic – Planets and satellites: atmospheres – Planets and satellites: individual: HD 189733 b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Introduction  The hot Jupiter HD 189733 b is one of the earliest transiting ex-  oplanets to be detected (Bouchy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2005), being one of the  closest transiting planets to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' HD 189733 b’s large mass  of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='13 MJup, large radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='13 RJup, and short-period 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2-  day orbit (Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2017) provide some of the most favor-  able conditions for atmospheric characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Consequently,  HD 189733 b has become one of the most extensively observed  exoplanets, now constituting a benchmark target for atmospheric  characterization via follow-up observations and theoretical mod-  eling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  From space, the Hubble Space Telescope (HST) has been in-  strumental in probing the optical-to-infrared transmission spec-  tra of this planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Collective observations with the Advanced  Camera for Surveys (ACS), the Space Telescope Imaging Spec-  trograph (STIS), and the Wide Field Camera 3 (WFC3) have  obtained nearly continuous spectra over the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='7 µm range  (Pont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Huitson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' McCul-  lough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Complementary infrared (IR) transmission  photometry from the Spitzer Space Telescope between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6 and  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0 µm (Désert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Agol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Knutson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  2012), and posterior re-analyses of the cumulative data (Pont  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2016) have revealed a steep Rayleigh scat-  tering slope dominating the optical and near-infrared spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  These results are consistent with the presence of high-altitude  atmospheric hazes, ruling out a clear atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Ground-based facilities have further allowed the detection of  several spectral features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' in particular, high-resolution observa-  tions have lead to the detection of narrow absorption features  above the haze layer from the H Balmer lines (Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Cauley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2015) and the Na D lines (Redfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Wyt-  tenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These observations constrained the tem-  perature of the thermosphere, revealed strong wind velocities  consistent with an eastward equatorial jet (Louden & Wheatley  2015), and detected a transit early ingress, possibly caused by  a hydrogen bow-shock ahead of the planet (Cauley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  We note however that these detections have been contested in  subsequent studies (see Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Guilluy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2020),  arguing that the observed signal could be stellar in nature, rather  than planetary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Similarly, stellar effects are also suspected to af-  fect the steep UV slope observed by HST for this planet, although  the magnitude of such contribution is not fully understood (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', McCullough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Rackham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  High-resolution spectroscopy studies of HD 189733 b at  near-infrared wavelengths via the cross-correlation method have  yielded the detection of H2O absorption features (Birkby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Brogi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2018) and CO (Rodler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Brogi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Article number, page 1 of 17  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='13025v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='EP]  30 Jan 2023  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' nuv_HD189733b  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These observations further constrain atmospheric compo-  sition and suggest small wind speeds of ∼2 km s−1 (at pressures  around the 10−2 − 10−3 bar levels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Infrared observations of the day-side emission during sec-  ondary eclipse has revealed a lack of thermal inversion and a  nearly solar H2O abundance in the lower planetary atmosphere  (Crouzet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Todorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Furthermore, in-  frared photometry along the entire orbital phase has measured  the brightness temperature variation, constraining the energy  budget and the global circulation regime (Knutson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2007,  2009, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These observations unveiled an efficient energy re-  distribution of the incident stellar irradiation and an eastward off-  set of the peak emission, relative to the sub-stellar point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' As with  the conclusions from optical observations, these measurements  are consistent with circulation models exhibiting an eastward ad-  vection by an equatorial super-rotating jet with wind speeds of  the order of ∼km s−1 at pressures around the mbar level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=',  Showman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2009, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Upper-atmosphere Characterization of HD 189733 b  Given the close planet-to-star distance, it is expected that  HD 189733 b exhibits atmospheric escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In fact, the host star,  HD 189733, is an active K-dwarf star (radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='75R⊙, mass of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='79M⊙, effective temperature of 5052 K, Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2017),  showing clear evidence of activity and spots during transit events  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Pont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2013), which makes it a particu-  larly interesting study case for star–planet interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Pop-  penhaeger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Pillitteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cauley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Lyα transit observation in the far-ultraviolet with HST/STIS  detected an extended hydrogen envelope that expands hydrody-  namically and escapes to space, powered by absorption of the  high-energy stellar flux (X-ray and extreme ultraviolet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Lecave-  lier des Etangs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Strong temporal  variations and the high velocity of the escaping gas suggest the  presence of significant star–planet interactions (consistent with  the active nature of the host star), which requires an accelera-  tion mechanism on top of stellar radiation pressure, such as the  stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Although later, Guo & Ben-Jaffel (2016) found that  the Lyα variability can be simply explained by invoking a ther-  mal population of H i in the planet’s upper atmosphere, without a  need for complex dynamics involving stellar winds or radiation  pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In this case, changes to the star’s EUV output would  explain the different absorption signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Far-ultraviolet observa-  tions with the HST Cosmic Origins Spectrograph have also led  to the detection of O i and C ii in the planetary upper atmosphere  (Ben-Jaffel & Ballester 2013), indicating that heavy atoms are  possibly entrained and dragged upwards in the hydrogen hydro-  dynamic outflow of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The Hα absorption feature detection by Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2012)  indicates that the upper atmosphere of HD 189733 b is neither in  radiative equilibrium nor thermodynamic equilibrium, due to the  large ultraviolet flux received by its active host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In addition,  Salz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2018), Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2022), and Guilluy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2020)  detected a strong absorption signal from the He i triplet at 10830  Å by means of ground-based high-resolution transmission spec-  troscopy with CARMENES and GIARPS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The ob-  served He absorption favors a compact helium atmosphere with  a substantial column density and no evidence of He escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This  interpretation of the He observations would seem to be consis-  tent with the aforementioned explanation of the Lyman alpha  transit observations by Guo & Ben-Jaffel (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The many observations of HD 189733 b across the electro-  magnetic spectrum have enabled an in-depth characterization of  the composition and physical state of both the lower and up-  per atmosphere of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' However, until recently, the near-  ultraviolet (NUV) window, probing both the exosphere and ther-  mosphere, escaped the spectral coverage probed by these ob-  servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2021) analyzed 20 NUV photometric  transits of HD 189733 b obtained from the XMM-Newton op-  tical monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' They detected a transit-depth signal consistent  with that in the optical, ruling out extended broad-band absorp-  tion towards or beyond the Roche lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' However, to detect the  NUV absorption signature from individual metallic lines, higher  spectral-resolution observations are required, like those from  HST/STIS, with a resolving power of R ≈ 30, 000 (see Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  For the slighly more strongly irradiated planet HD 209458 b,  Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2020) detected Fe ii absorption in a 100 Å-wide  range around 2370 Å, lying beyond the planetary Roche lobe,  while finding no evidence for absorption by Mg ii, neutral mag-  nesium (Mg i), nor neutral iron (Fe i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These results suggest that  hydrodynamic escape is strong enough to carry heavy atoms  beyond the planetary Roche lobe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' however, condensate forma-  tion sequesters primarily Mg (rather than Fe) in the lower at-  mosphere of the planet, consistent with microphysical cloud-  formation models (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For the ultra-hot Jupiter  WASP-121b, Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2019) reported the detection of strong  narrow NUV absorption features, consistent with the singly ion-  ized magnesium (Mg ii) doublet and several lines of ionized iron  (Fe ii), emanating from altitudes that exceed the Roche lobe ra-  dius of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These results indicate that Mg ii and Fe ii  species are not trapped into condensates at depth, and can hy-  drodynamically escape the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  To complete the spectral survey of this benchmark planet  and aid the quantitative assessment of the metallicity, ionization  state, and outflow rate of the upper atmosphere of HD 189733 b,  here we analyzed 15 HST orbits obtained from three STIS near-  ultraviolet transit light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In Section 2 we describe the ob-  servations in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In Section 3 we describe our data-  analysis procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In Section 4 we present the main results from  the NUV data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In Section 5 we place the observations  in context by contrasting them with theoretical models of the  planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In Section 6 we speculate on the origin of the features  seen in the NUV observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Finally, in Section 7 we summa-  rize our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Observations  We observed three transits of the planet HD 189733 b, obtained  with HST, using the Space Telescope Imaging Spectrograph  (STIS, program GO #15338, PI L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Fossati).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The observations  were carried out on 2018-03-10, 2018-09-19, and 2019-08-27  (visits 1, 2, and 3, respectively);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' each one using the NUV Multi-  Anode Microchannel Array (NUV-MAMA) detector in ACCUM  mode, E230M grating, and with an aperture of 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2 × 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Each transit observation (one HST visit) consists of five HST  orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For each visit, the fourth orbit occurs during transit,  whereas the first three and last HST orbits occur out of transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The orbits are broken into exposures (frames) of ∼332 seconds  each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The first two visits contain nine frames per orbit (except  for the first orbit, which contains one fewer frame to accommo-  date acquisition observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The third visit contains one fewer  frame per orbit, possibly due to a longer pointing adjustment  time after HST returned to operations from a gyroscope failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Article number, page 2 of 17  Patricio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' : The HST NUV Transmission Spectrum of HD 189733 b  2300  2400  2500  2600  2700  2800  2900  3000  Wavelength (Å)  0  2  4  6  8  Flux (10  13 erg s  1 cm  2 Å  1)  2780  2800  2820  0  5  10  15  20  25  Mg II  2840  2860  2880  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  Mg I  0  4  8  12  16  20  Echelle order  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' HD 189733 system flux as observed by HST/STIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This spectrum has been constructed from the mean time-series spectra obtained during  the first visit, after discarding bad pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The spectrum is color coded by echelle order (see color bar on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The inset panels zoom-in  around the location of the Mg ii and Mg i resonance lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Each frame consists of an echelle spectrum comprising 23  partially overlapping orders, covering the 2300–3100 Å spec-  tral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Each order contains 1024 wavelength samples, with  a resolving power of R = λ/∆λ = 30 000, where the resolution  element spans approximately two pixels (or equivalently ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='09  Å, or ∼10 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Figure 1 shows a sample spectrum from visit  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' As expected, the flux decreases toward shorter wavelengths,  with a clear drop in signal-to-noise ratio (S/N) at wavelengths  shorter than ∼ 2550 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The stellar spectrum shows many emis-  sion features, most of them correspond to Fe ii lines rising from  low energy levels, but also the Mg i and Mg ii resonance lines  (the strongest line-emission feature in the data, as shown in the  inset panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In contrast, stellar Fe i features appear in  absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The input for our analysis consisted of the x1d CALSTIS-  reduced fits files1 (version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The reduction comprises dark  and flat-field correction, 1D spectral extraction of each echelle  order, background subtraction, and wavelength and flux calibra-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The results are the 1D spectra (erg s−1cm−2Å−1) and their  uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  In addition, following Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2019), we also considered  the Engineering Data Processing System “jitter” files, which  contain 28 different engineering measurements of HST’s Point-  ing Control System for each exposure during an observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Data Analysis  The core of our analysis is based on the methodology of Cubillos  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We separated our light-curve analysis into two main  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' First we constructed “white” lightcurves by integrating the  flux per transit visit and per echelle order to boost the astro-  physical and instrumental signals, from which we characterized  the instrumental systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Then, we divided out the system-  atics model from the raw data, and combined the systematics-  1 https://hst-docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='edu/stisdhb/chapter-3-stis-calibration  corrected data from all visits into spectral light curves binned at  different resolving powers, from which we extracted the trans-  mission spectra of the astrophysical signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Analysis of Individual Visits  We characterized the instrumental systematics to detrend them  from the astrophysical signal and remove them from further  analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' To increase the S/N of the data, we integrated the flux  over each echelle spectral order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The implicit assumption is that  the instrumental systematics behave similarly over the integrated  range (Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2014), in this case, over each echelle or-  der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The analysis begins by masking bad and overly noisy data  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Following the standard practice for exoplanet HST/STIS  time-series analyses, we discarded the first orbit from each visit  and the first frame from each orbit, since they show significantly  stronger instrumental systematics (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Then, we used the overlap between echelle orders to discard data  points where the fluxes differ from each other by more than 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Finally, we manually inspected the edges of the echelle spec-  tra to exclude data points with abnormally low fluxes (typically,  a handful of data points at the ends of the echelle orders).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We  then produced raw “white” light curves by summing the flux  over each spectral order and propagating the uncertainties ac-  cordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  We fit the raw light curves with parametric transit and sys-  tematics models at each echelle order (denoted by their wave-  length λ) as a function of time (t), HST orbital phase (φ), and  jitter vector ( ji):  Fλ(t) = Tλ(t) S λ(t, φ, ji),  (1)  where Tλ(t) is a Mandel & Agol (2002) transit model and  S λ(t, φ, ji) is a model of the instrumental systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' To ob-  tain statistically robust parameter estimations, we computed  best-fitting parameters and uncertainties from a Levenberg-  Marquardt optimization and a Markov-chain Monte Carlo  Article number, page 3 of 17  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' nuv_HD189733b  (MCMC) sampling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For this matter, we employed  the open-source mc3 statistical package2 (Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2017),  which implements the Snooker Differential-evolution MCMC  algorithm of ter Braak & Vrugt (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The transit model parameterizes the astrophysical signal via  the orbital parameters, the planet-to-star radius ratios Rp/Rs(λ),  the out-of-transit stellar fluxes Fs(λ), and limb-darkening coef-  ficients Ci(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For the orbital parameters we adopted previously  measured values for the orbital period P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='21857567 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5 ×  10−8 day, inclination i = 85◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='71, and semi-major-axis to stellar  radius ratio a/Rs = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='27 (Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2017), which we  kept fixed during the fitting and MCMC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We fit the tran-  sit mid-time epoch, applying a Gaussian prior according to the  measured optical value (T0 = 53955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='025551 ± 9 × 10−6 MJD,  Bonomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2017), propagating uncertainties according to the  epoch of the HST observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  For the limb-darkening parameters, we computed theoretical  coefficients based on the stellar properties, since the low S/N ra-  tios and gaps in the HST observations do not allow for a good  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For this calculation we used the open-source rou-  tines of Espinoza & Jordán (2015) to compute limb-darkening  coefficients over each wavelength range based on the PHOENIX  stellar model (Husser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2013) that best matches the proper-  ties of HD 189733: effective temperature Teff = 5052 K, surface  gravity log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='49, and metallicity [M/H] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='02 (Stassun  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We note that, in absence of any better known guess,  the adopted limb-darkening coefficents correspond to the photo-  sphere of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' If a significant fraction of the stellar flux comes  from its chromosphere and transition region, as suggested by the  stellar line emission, then the used limb-darkening coefficients  may not be as representative as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The HST/STIS time-series observations exhibit a number of  well documented instrumental systematics: (1) periodic varia-  tions in phase with the HST orbital period that are suspected to  originate from the telescope’s thermal cycle (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2001),  and (2) visit-long variations that might be attributed to stellar ac-  tivity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Wakeford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Following the  example of previous studies, we model these systematics consid-  ering a family of polynomials in time (t) and HST orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Furthermore, Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2019) showed that additional data from  HST’s Pointing Control System (so called “jitter” data) can help  decorrelate instrumental systematics from the astrophysical sig-  nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For each science exposure, the jitter data describe the HST’s  fine guidance sensor (FGS) coordinates, FGS jitter, right ascen-  sion, declination, position angle, latitude, longitude, angle be-  tween HST zenith and target, Earth-limb angle, terminator an-  gle, and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' HST stores the jitter measurements into  time-tag ancillary engineering files, recorded at a high temporal  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For each science exposure we computed a value for  each of the jitter parameters by calculating the median of the jit-  ter values during the exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Therefore, our systematics model  consists of a polynomial expression up to a quadratic degree in  time, up to a quartic degree in HST phase, and a up to a linear  degree in one of the jitter vectors:  S λ(t, φ, ji) = 1 + a0(t − t0) + a1(t − t0)2  + b0(φ − φ0) + b1(φ − φ0)2 + b2(φ − φ0)3 + b3(φ − φ0)4  + c0( ji − ⟨ ji⟩),  (2)  where ak, bk, and ck are the polynomial coefficients to be fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Note  that we considered an individual set of fitting parameters for each  echelle order (we dropped the λ-dependence of these coefficients  2 https://mc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='io/  for clarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Regarding the jitter vectors ji, we considered each  of the 21 vectors listed in Table 1 of Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' t0 and φ0  are reference values for the time and phase, which are fixed at  the transit mid-time t0 = T0 and at the HST mid-phase φ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' ⟨ ji⟩ denotes the mean value of the jitter vector ji  along the visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Systematics Model Selection  Given our limited understanding of the systematics affecting  HST observations, we determined the optimal functional form  for the systematics model S λ by comparing fits considering all  possible combinations of polynomial orders in t and φ up to the  degrees shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2), and each of the 21 jitter vectors, and  fits with no jitter vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' To this end we employed a Bayesian  model selection approach (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Trotta 2007), where one  compares the posterior probability of two competing models—  given the data—by computing the ratio of their Bayesian evi-  dences (also known as Bayes factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The main problem of this  approach is that the evidence of a model is not readily calcu-  lable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Numerically, calculating the evidence entails evaluating  the model likelihood throughout its parameter space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', via a  nested-sampling algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' To avoid the computational cost of  this task, we adopt a simpler model selection approach where  we approximate the evidence ratio via the Bayesian Information  Criterion (BIC, Liddle 2007):  BIC = χ2  min + k log N,  (3)  where χ2  min is the goodness-of-fit parameter corresponding to the  maximum likelihood for the given model, k is the number of  free parameters, and N is the number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The Bayes  ratio is then simply calculated from the difference in BIC val-  ues between the competing models B = exp{−(BIC1 − BIC2)/2}  (Raftery 1995), which only requires to find the maximum likeli-  hood for the given models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' A Bayes factor B > 1 favors model  1 over model 2, or alternatively, the favored model minimizes  BIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Certainly, these approximations are based on various as-  sumptions, which may not be fulfilled to different extents,  such as independent and identically distributed data or (nearly)  Gaussian posterior distributions or that the forward model is  correct, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Therefore, we also considered a second  model-selection criterion based on information theory, that is the  Akaike Information Criterion corrected for small sample sizes  (AICc, Liddle 2007):  AICc = χ2  min +  2kN  N − k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  (4)  Just like before, the favored model minimizes AICc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Model Selection Applied to the HST Observations  For the HD 189733 b data we treated each HST visit indepen-  dently of each other, where for each visit we fit simultaneously  the astrophysical and systematics parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Also, for each visit  we fit all 23 echelle orders simultaneously, jointly fitting the tran-  sit mid-time epoch and adopting the same configuration of the  systematics model for all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Table 1 reports the BIC and  AICc of the global fit to all orders for the favored systematics  model (with and without jitter decorrelation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Figure 2 shows an  example of the light-curve fits for a selected echelle order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  As seen from the raw light curves (top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2), the  systematics are characterized by visit-long and HST-orbit-long  Article number, page 4 of 17  Patricio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' : The HST NUV Transmission Spectrum of HD 189733 b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='04  Orbital phase  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='02  Best-fit (Rp/Rs)2  AIC  BIC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='04  Orbital phase  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='04  Orbital phase  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='02  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Sample transmission light-curve fitting of the HST/STIS observations of HD 189733 b (fourth echelle order, ¯λ = 2917Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The top row shows  the raw light curves (one column for each of the three visits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The markers with error bars denote the fluxes and their uncertainties integrated over  the echelle order for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The markers are color-coded according to the HST orbit, where the gray colors denote the frames discarded from  the light-curve fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The bottom rows show the systematics-corrected light curves corresponding to the best-BIC (colored circles) and best-AICc  statistics (empty black diamonds), including the jitter decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The solid lines denote the astrophysical model of the best-BIC (colored lines)  and best-AICc models (black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' BIC/AICc Model Comparison  Systematics model  BIC  AICc  χ2  red  Comments  (t-deg, φ-deg, jitter)  Visit 1  (1, 1)  1362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='16  best BIC  (1, 1, Lat)  1434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='08  best AICc  (1, 2)  1441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='09  best AICc no-jit  Visit 2  (1, 1, Lat)  1617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='7  1127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='37  best BIC  (1, 3, Lat)  1692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='9  1039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='09  best AICc  (1, 2)  1699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='50  best BIC no-jit  (2, 2)  1740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3  1166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='38  best AICc no-jit  Visit 3  (1, 1, LOS Zenith)  1665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  1198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='73  best BIC  (1, 3, LOS Zenith)  1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='7  1090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='36  best AICc  (1, 2)  1798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='98  best BIC no-jit  (2, 2)  1826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6  1282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='84  best AICc no-jit  Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The first two terms in the systematics model column show the  polynomial degree for time and orbital phase, the third term shows the  name of the jitter vector (which is always modeled as a linear polyno-  mial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The absence of a third term indicates that no jitter vector was  included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Overall, the second visit shows stronger systematics  than the first visit, and the third visit shows stronger systematics  than the second, which is reflected in the best-fitting BIC, AICc,  and reduced χ2 values (χ2  red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In particular, the second and third  visit systematics vary on the order of 10% of the raw flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' If we  only included the more “traditional” polynomials in t and φ in  the fit, we found clear systematic residual trends, which vary in  strength for each echelle order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The relatively new practice of  jitter decorrelation introduced by Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2019) significantly  2300  2400  2500  2600  2700  2800  2900  3000  Wavelength (Å)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='35  Rp/Rs  visit 1  visit 2  visit 3  King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2021)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' HST/STIS NUV transmission spectrum of HD 189733 b (indi-  vidual visits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The circle markers with error bars denote the transmis-  sion planet-to-star radius ratios from each echelle order and visit (see  legend), corresponding to the best-AICc fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The colored dashed lines  denote the mean value for each visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The square marker denotes the  XMM-Newton/UVM2 photometric measurement by King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  improves the fit for the second and third visits, minimizing both  information criteria statistics and residual systematics (bottom  row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In these fits, the AICc imparts a weaker penalty  per fitting parameter than the BIC, thus favoring more complex  models, which notably improves the χ2  red values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' However, al-  though the BIC and AICc statistics favor different models, the  two criteria usually produced consistent transit depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' All fol-  lowing analyses presented here will focus on the best AICc anal-  ysis including jitter decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Figure 3 shows the resulting transmission spectra according  to the best-AICc models for each visit and at each echelle order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  One clear feature seen in the transmission spectra is an offset be-  tween the third visit and the first two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This offset is consistent  with the signature of unocculted stellar spots (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Rack-  Article number, page 5 of 17  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' nuv_HD189733b  ham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Zellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2017), which is a likely cause  given the well known active nature of HD 189733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For each visit,  we found that the transit radius ratios vary closely proportional  to the square root of the bolometric fluxes of the observations  (Rp/Rs ∝ Fs−1/2), which is a direct consequence of observations  arising from a heterogeneous stellar photosphere including spots  (Rackham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The weighted mean of the Rp/Rs values  for each individual visit are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1585, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1595, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1701;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' whereas  the relative bolometric fluxes between visits are Fv1  s /Fv2  s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='02  and Fv1  s /Fv3  s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Later, the combined analysis in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1  will show an independent confirmation that we are observing  stellar spot contamination in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Once corrected for stellar activity, the analysis of each of the  three visits produces a consistent transmission spectrum across  the NUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We obtained a mostly flat transmission spectrum be-  tween 2300 Å and 3100 Å, with a planet-to-star radius ratio of ∼  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='16, which becomes significantly noisier at wavelengths shorter  than 2550 Å, due to the much lower stellar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These results  are also consistent with the previous NUV observation by King  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2021), which measured the transit depth of HD 189733 b  in the UVM2 photometric band of the XMM-Newton Space Ob-  servatory (effective wavelength 2310 Å, width 480 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  As a reference, the estimated altitude of the Roche lobe ra-  dius as probed during transit is RL1′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='44 Rs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', perpendicu-  lar to the star–planet axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This value is approximated by 2/3 the  distance to the L1 Lagrange point (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Vidal-Madjar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' All transit depths at each echelle order are well under the  Roche lobe radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Analysis of Combined Visits  To search for the narrow metal line-absorption features in the  transmission spectrum, we need to analyze the observations at a  high resolving power, finding a compromise between sufficient  S/N per wavelength channel and diluting the narrow absorption  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Therefore, we treated the data at multiple spectral re-  solving powers ranging from R = 10−70 (coarse sampling anal-  ysis), and then at a R = 4700 (fine sampling analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We pro-  ceeded in the same fashion as in Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2020), that is,  calibrate the wavelength solution to ensure that the spectra are  well aligned, use the modeling results from the previous analy-  sis to divide out the instrumental systematics, combine the data  from all three visits into a single spectrum, and fit the transit  depths at each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' All future analyses use the best-AICc dataset  since it attained χ2  red values closer to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Wavelength Calibration  We used a cross-correlation procedure to calibrate the wave-  length solution of each echelle order in each frame and visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  First, we created a master stellar template (enhancing the S/N)  by co-adding the spectra from all frames within an echelle order  and visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We then Doppler-shifted the individual-frame spectra  and cross-correlated them with their template, searching for the  relative velocity with respect to the template given by the shift  that maximized the correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Once we obtained the  wavelength correction of the individual frames within an echelle  order, we repeated the procedure between the template spectra  of each visit to find the relative wavelength correction between  all spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Finally, we compared our wavelength solution with a  theoretical model spectrum of HD 189733 (Shulyak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2004)  to obtain an absolute wavelength solution at a rest reference  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Once calibrated, we Doppler-shifted all frames within a  given echelle order to a common wavelength solution using a cu-  bic spline interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' To minimize the impact of the interpo-  lation, we selected a reference point that minimized the shifts of  the set of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For a given visit, we found offsets of less than  half a pixel between the frames, and in general the wavelength  solution for each echelle order follows a similar trend along the  visit, with no outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Divide-white Spectral Extraction  To  extract  a  high-resolution  transmission  spectrum  of  HD 189733 b, we followed the “divide-white” spectral analysis  by Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2014), as described in Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  In this approach, we first constructed a non-parametric model of  the instrumental systematics by dividing the white-light curves  (for each echelle order and visit) by their best-fit astrophysical  transit model (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Assuming that the instrumental  systematics vary weakly with wavelength, each raw spectral  data point is divided by the systematics model corresponding to  their orbital phase, echelle order, and visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Data uncertainties  are properly accounted for, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', using the white-light model’s  posterior distribution to estimate the uncertainties systematics  model and propagating the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  In a second step, we split the data into constant resolving-  power wavelength bins, where for each spectral channel we co-  added all flux contributions from all echelle orders, and propa-  gated the errors accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  To analyze the spectral light curves we fit a Mandel & Agol  (2002) transit model to the systematics-corrected spectral light-  curves in an MCMC run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' As previously, we fixed the orbital  a/Rs, cos(i), and P parameters, and applied a Gaussian prior for  the mid-transit epoch based on the white-light fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We  also computed theoretical limb-darkening curves at each spectral  channel, which we kept fixed during the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' At each wavelength  bin we fit a unique transit depth common to all three visits, but  we fit a separate out-of-transit flux value for each visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Addi-  tionally, we incorporated a stellar-spot correction model that we  detail in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Stellar Spot Correction  To account for the spectral transit-depth variations due to unoc-  culted stellar spots, we modeled the stellar flux as a linear com-  bination of a nominal photospheric region and a spotted region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  We assumed that a fraction fspot of the observed photosphere is  covered by spots with a characteristic temperature Tspot, whereas  the rest of the photosphere has the nominal effective temper-  ature of the star Teff = 5052 K (with Teff ≥ Tspot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We used  the PHOENIX library of high-resolution synthetic stellar spec-  tra (Husser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2013) to compute the flux of the nominal stel-  lar photosphere Fs(Teff) and the spotted region Fspot(Tspot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This  leads to the following correction term for the observed transit  depth dobs(λ):  dobs(λ) =  d(λ)  1 − fspot  �  1 − Fspot(λ)/Fs(λ)  � ≡ d(λ) ϵ(λ),  (5)  where d(λ) is the true transit depth and ϵ(λ) is the contamination  spectrum due to stellar hetereogeneities (as defined in Rackham  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' To evaluate the correction factor at any given Tspot,  we interpolated in temperature between all available PHOENIX  models (ranging from 2300 K to 5100 K) at a fixed metallicity  ([M/H]=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0) and surface gravity (log(g) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Article number, page 6 of 17  Patricio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' : The HST NUV Transmission Spectrum of HD 189733 b  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' NUV transmission spectrum of HD 189733 b from the combined HST/STIS observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The red markers with error bars denote the  systematics- and stellar-spot corrected transmission spectrum, their 1σ uncertainties, and the span of the spectral bins (the data has been binned in  wavelength over a resolving power of R = 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The green markers show our results when neglecting the stellar spot contamination in the analysis  (markers have been slightly shifted in wavelength for visibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The blue spectrum shows a theoretical model of the planet’s upper atmosphere  (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1) and the blue dots show the model integrated over the spectral bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The orange curve shows an extrapolation from a fit to the  optical-IR transmission spectrum of the planet (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The stellar-spot correction model thus contains two free pa-  rameters for each visit (fspot and Tspot), which we fit simultane-  ously along with the other astrophysical parameters during the  MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We considered a uniform prior for the fspot values be-  tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5 (it would be unlikely to find unocculted spot  covering fractions greater than 50%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For the spot temperature  we considered a logarithmic-uniform prior in ∆Tspot = Teff−Tspot  because the spectral variation of the spot correction is more pro-  nounced at small ∆Tspot values (McCullough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' When  ∆Tspot ≳ 1500 K the correction becomes nearly achromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Re-  gardless of the parameterization, the main conclusions of our  analysis do not change when we consider a uniform prior in Tspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Planetary Atmospheric Signatures  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Coarse-binning Spectra  To boost the characterization of the astrophysical signal, we  started with the analysis of the combined data binning over a  relatively coarse wavelength sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Figure 4 shows the NUV  transmission spectrum of HD 189733 b sampled at a resolving  power of R = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This resolving power resulted in the best com-  promise between having a good S/N per bin and having sufficient  spectral resolution to identify variability with wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Ap-  pendix A shows the corresponding systematics- and stellar-spot  corrected lightcurves for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' At this resolving power,  we obtained a relatively flat transmission spectrum with the ex-  ception of two regions where the transit depth increases (by more  than ∼1σ relative to the surrounding wavelengths) located at  λ ∼ 2350 Å and λ ∼ 2600 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  For completeness, we also analyzed the data at other similar  resolving powers of R = ∼10, 30, 70 finding consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Likewise, analyses with the bins shifted by half a bin-width re-  turned consistent results as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The stellar spot contamination was well characterized, as  shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The first two visits show a negligible spot  contamination (ϵ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='05), with the models finding either a small  spot covering fraction or a spot temperature close to the nominal  stellar effective temperature (∆Tspot ≲ 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In contrast, the third  visit shows a clearly significant spot contamination ranging from  ϵ(λ) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='20 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='15 from short to long wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The contam-  ination shows a weak variation with wavelength, which is not  larger than the 1σ credible intervals of the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Accord-  ingly, we constrained the spot covering fraction to fspot > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='11  and the spot temperature to 100 K ≲ ∆Tspot ≲ 500 K, values  which are strongly correlated (see posterior pair-wise distribu-  tion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Analyzing the combined data without a stellar-  spot model produced slightly larger transit depths (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' how-  ever, the shape of the spectrum remains consistent with that of  the spot-corrected analysis, which is consistent with the nearly  wavelength-independent stellar spot correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  All in all, the ratio between contamination spectra of the vis-  its (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', ϵv3/ϵv1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='15 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2) matches precisely the ratio be-  tween bolometric fluxes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Fv1  s /Fv3  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='16), which reinforces  the hypothesis that the variability between visits is caused by a  varying stellar spot coverage at different epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Fine-binning Spectra: Magnesium Resonance Lines  We attempted to analyze the data binning at much higher re-  solving power such that we can capture absorption of individ-  ual metallic lines, however, the high-resolution lightcurves were  largely dominated by noise, and not much could be concluded  from the resulting spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The only region where we were able  to place significant constraints at high resolution is at the core  of the magnesium resonance lines, which have fluxes nearly ten  times larger than at other wavelengths, and thus a much better  S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For this, we selected a bin resolving power of R = 4700 (65  km s−1 wide bins), which encapsulates the core of the Mg i and  Mg ii resonance lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Figure 6 shows the high-resolution light curves centered at  the Mg i and Mg ii resonance (top), and the stellar spectra (mid-  dle) and transit radius-ratio spectra (bottom) around the mag-  nesium lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For the Mg ii bins, the light curves show a clear  transit signature (albeit with some residual noise for the Mg ii h  Article number, page 7 of 17  HST/STIS NUV transits (R = 50)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='22  Uncorrected stellar spots  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='12  2300  2400  2500  2600  2700  2800  2900  3000  3100  Wavelength (A)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' nuv_HD189733b  1  1  3  log10( Tspot)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  fspot  visit 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  fspot  1  1  3  log10( Tspot)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  fspot  visit 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  fspot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  log10( Tspot)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  fspot  visit 3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  fspot  2300  2400  2500  2600  2700  2800  2900  3000  Wavelength (um)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='25  Contamination, (  )  visit 1  visit 2  visit 3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Stellar spot contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The top panels show the posterior  distribution of the spot parameters derived from the combined-data  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The blue shaded areas denote the 1σ highest posterior den-  sity credible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Visits 1 and 2 show a negligible spot contamination  with either a small spot covering fraction or a negligible difference be-  tween the spot and photospheric temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Visit 3 in contrast shows  a non-negligible contamination of ϵ(λ) ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='15–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2, with a strong cor-  relation between the spot covering fraction and spot temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The  bottom panel shows the respective contamination spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The colored  markers and vertical spans show the median and 1σ uncertainty of the  contamination, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  line), and have S/Ns on the same order of magnitude as those  of the low-resolution analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The light curve at the Mg i line,  having much lower flux, is significantly noisier than the Mg ii  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' For all three magnesium lines, the transit depths do not  show any significant excess above the low-resolution continuum  values (R = 50 analysis), and are consistent with the optical-IR  transmission spectrum of HD 189733 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  For context, in the bottom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 6 we also show the  theoretical upper-atmospheric model for this planet (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' While HD 189733 b’s escape rate is not high  enough to generate magnesium absorption signatures close to  the the Roche-lobe altitude (RL1′/Rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='44), the model predicts  a noticeable excess absorption above the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Thus, we  used the binned model estimate to place non-detection limits for  the Mg ii measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We found that the observations sit 2–4σ  below the prediction for a solar-abundance atmospheric model  containing Mg in the upper atmosphere (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Due to the  higher noise at the Mg i line, we cannot reject nor confirm the  presence of Mg i absorption from our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The NUV-to-Infrared Transmission Spectrum of  HD 189733 b  For context, the Hei infrared metastable triplet and Hi Lyman-α  transit observations of HD 189733 b can be interpreted in terms  of the planet hosting a compact upper atmosphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' extended, but  not as much as that of other exoplanets observed in the NUV,  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' HD 189733 b Transit Measurements at Magnesium Lines  Line  λ0 (µm)  Rp/Rs  Non-detection significance†  Mg ii k  2796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='158+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='006  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='006  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='7σ  Mg ii h  2803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='153+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='008  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='007  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3σ  Mg i  2853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='138+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='056  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='032  <1σ  † number of standard deviations below the expected Rp/Rs  from the solar-abundance theoretical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', HD 209458 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Metals like magnesium or iron may condense  in the lower atmosphere and even if they make it to the upper  atmosphere, a lower escape rate would not allow their density  profiles to stretch out to high altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' At longer wavelengths,  the optical-IR observations show a strong slope towards the blue,  which can be interpreted as absorption from high-altitude hazes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  To test these hypotheses and to get a better understanding  of the interaction between the upper and lower atmosphere of  HD 189733 b, we compare the observations to theoretical mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Since it is not possible to model all required upper- and  lower-atmosphere properties at once, we model individually the  upper and lower regions, and then study how they relate to each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Upper-atmosphere Modeling  To interpret the NUV transmission spectrum of HD 189733 b, we  employed the physical model of the upper atmospheres of close-  in planets (Koskinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2013a,b, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Since the parts of the  NUV transmission spectrum that probe the upper atmosphere are  dominated by signatures of ionized metals Fe ii and Mg ii (Sing  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2020), we added the heavy elements  C, O, N, Mg, Si, Fe, S, Ca, Na, and K with solar abundances to  the model, in addition to H and He, with related chemistry and  physics adapted from Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We included Mg, Si,  Fe, Ca with first and second ionization states, due to the rela-  tively low ionization potential of the first ionization state, and  the rest of the heavy elements with their first ionization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The electron density is equal to the sum of the ion densities un-  der the assumption of quasineutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The lower boundary of the  escape model is at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1 µbar where the temperature is sufficient  to dissociate molecules that are not included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  We used the results from the photochemical model of Lavvas  & Koskinen (2017) to obtain the mean molecular weight profile  in the lower and middle atmosphere to set the lower boundary  altitude and temperature of the escape model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The temperature  profile in the lower and middle atmosphere was not calculated  self-consistently by the photochemical model and was instead  adapted from Moses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Here, we used the MK profile  from Lavvas & Koskinen (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The application of the photo-  chemical model to HD189733b did not include Mg and Fe that  are detectable in the NUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In order to crudely model the full pro-  files of the absorption lines, we added Mg and Fe to the lower  and middle atmosphere, assuming solar abundances and thermal  ionization according to the Saha equation (Menou 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This  approximation does not affect our conclusions because our focus  in this section is on the possible upper atmosphere signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Given the temperature profile assumed for HD 189733 b here  and assuming equilibrium condensation, Mg, Fe, Si, and Ca are  expected to condense to form mineral clouds and rain out from  the atmosphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Wakeford & Sing 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We included Mg  and Fe in the escape model simply to explore if the modeled ab-  Article number, page 8 of 17  Patricio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' : The HST NUV Transmission Spectrum of HD 189733 b  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Transit light curves (top) and spectrum (bottom) of HD 189733 b of the combined HST/STIS transits at the location of the magnesium  resonance lines, and binned at a resolving power of R = 4700 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', roughly the width of the magnesium lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The top panels show the systematics-  corrected light curves for the spectral channels centered at the core of the Mg ii and Mg i lines (see labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The solid red curve and orange area  denote the best-fitting transit depths and span of their 1σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The middle panels show the system flux around the magnesium lines, with  the vertical dashed lines marking the edges of the spectral bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The bottom panels show the transit planet-to-star radius ratio for each of the bins  around the magnesium resonance lines (red markers with 1σ error bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The blue curve shows the upper-atmosphere model (the diamond markers  show the model binned at R = 4700).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The peach-colored area shows the span of the low-resolution transit radius ratio (R = 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  sorption lines of Mg II and Fe II would be detectable in the NUV  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The rest of the heavy elements were retained for consis-  tency with the no-condensation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Although efficient mix-  ing in the lower atmosphere could interfere with mineral cloud  formation and allow for higher abundances of the relevant heavy  elements in the upper atmosphere (Spiegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Koskinen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2013b), we do not hold a view on whether this is possible  on HD 189733 b or not at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Figures 4 and 7 show the transmission spectrum from the  upper atmosphere model in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Here, we calibrated the model  spectrum so that the measured base Rp/Rs (in the near-IR) cor-  responds to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3 mbar in the atmosphere (Lavvas & Koskinen  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' So as not to bias the fit to any particular interpretation at  this point, we did not include extinction by the high-altitude haze  modeled by Lavvas & Koskinen (2017) in this forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Instead, the transit continuum is due to Rayleigh scattering by  H2, H, and He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' It is immediately clear that the model continuum  falls below the extrapolation based on the optical-IR transmis-  sion spectrum of the planet, due to the lack of the high-altitude  haze in the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Since transit depth is not a cumulative  quantity, however, the model still correctly represents absorption  by Fe II and Mg II in the upper atmosphere in the absence of  condensation, even if high-altitude hazes are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  We ran the upper atmosphere model assuming globally av-  eraged radiative forcing with the XUV spectrum constructed  from ϵ Eridani as an input, due to the well known similarity to  HD 189733 (Lavvas & Koskinen 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The reference simula-  tion, on which the spectrum in Figure 4 is based on, assumes  an isolated planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We also ran a second simulation that includes  Roche lobe overflow (see, Koskinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2022) but this had neg-  ligible effect on the modeled spectrum, in line with the expecta-  tion that Roche lobe overflow does not significantly enhance the  escape rate or upper atmosphere densities on HD 189733 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  We note that NUV observations have the potential to of-  fer valuable new insights to the properties of possible high-  altitude hazes and/or clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Unfortunately, S/N is an issue for  HD 189733 b that precludes clear conclusions on the origin of  the continuum in the transit observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The visible spectrum  shows a strong slope indicative of small particle hazes (Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Lavvas & Koskinen 2017) that appears to be supported  to some degree by observations by the Stratospheric Observa-  tory for Infrared Astronomy (SOFIA) (Angerhausen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  however, the latter found a slope offsetted to lower depths by  ∆Rp/Rs =∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='001 (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The slope can, however,  be enhanced by stellar activity (star spots) (McCullough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Also, a general circulation model (GCM) that simulates  the production and transport of hazes predicts a shallower slope  unless mixing by sub-grid scale eddies exceeds mixing by global  circulation (Steinrueck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The NUV observations are  statistically consistent with the model that includes no haze, and  generally the observed transmission falls below the extrapola-  tion based on the visible slope (Figure 7), with the exception of  the 2350 Å region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Due to low S/N in the NUV, however, the  observations are also consistent with this extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Thus,  Article number, page 9 of 17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='06  Mg k, 入 = 2796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4 A  Mg h, ^= 2803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5 A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2 FMg1, 入= 2853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0 A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  (%)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  Orbital phase  Orbital phase  Orbital phase  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  0-13  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  三 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='30  Mg Il  Model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='30  Mgi  Binned model  Data (R=50)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='25  R  Data (R=4700)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='20  R  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='10  2790  2792  2794  2796  2798280028022804280628082810  2847  2850  2853  2856  2859  Wavelength (A)  Wavelength (A)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' nuv_HD189733b  these NUV observations neither confirm or exclude the presence  of high-altitude hazes, especially if the visible slope is enhanced  by star spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Lower-atmosphere Modeling  To constrain the lower-atmosphere properties of HD 189733 b  we used the open-source Pyrat Bay framework3 (Cubillos &  Blecic 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We modeled the atmosphere from 100 to 10−9 bar,  adopting a parametric temperature profile from Madhusudhan &  Seager (2009), hydrostatic equilibrium, and a H/He-dominated  composition considering constant-with-altitude volume mixing  ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The Pyrat Bay radiative transfer computed the transmis-  sion spectrum between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2 and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0 µm, at a constant resolving  power of R = λ/∆λ = 15, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The opacities included line-by-  line data from HITEMP for CO2, CO, and CH4 (Rothman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Hargreaves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2020), and from ExoMol  for HCN, H2O, NH3, and SiO (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2006, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Polyan-  sky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Coles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Yurchenko 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Yurchenko  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We extracted the dominant transitions from these  line-lists using the repack algorithm (Cubillos 2017), and then  computed tabulated opacities by sampling over temperature and  wavelength grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Additionally, the model included collision-  induced opacities for H2–H2 pairs (Borysow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Bo-  rysow 2002) and H2–He pairs (Borysow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 1988, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Bo-  rysow & Frommhold 1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Rayleigh-scattering opacity for H,  H2, and He (Kurucz 1970);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' and a gray cloud deck model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Finally,  the model considered a parametric haze absorber model (Lecave-  lier Des Etangs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2008) with arbitrary slope and strength  (αhaze and κhaze), which are fit in the retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The  atmospheric  retrieval  employed  the  differential-  evolution Markov-chain Monte Carlo sampler (ter Braak &  Vrugt 2008), implemented via the open-source code mc3 (Cubil-  los et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2017), guided by the transit observations from HST’s  STIS G430 and G750, WFC3 G102 and G141 spectrographs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  and from Spitzer’s IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='8, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0 µm and MIPS 24 µm  photometric bands (Pont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Figure 7 shows the fit to the optical-IR transit observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Appendix B presents the retrieval posterior-distribution results  for all model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The optical-IR observations point to  a prominent haze slope, steeper than a Rayleigh slope (αhaze ≈  −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This is a similar conclusion to that found by previous  retrieval studies on this planet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Pinhas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Barstow  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' From a physical perspective, the steep haze slope can be  created by a combination of particle growth and efficient eddy  (turbulent) mixing (Ohno & Kawashima 2020) or by stellar in-  homogeneities (Rackham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2018), effects that were captured  by the parametric haze model (rather than being explicitly ac-  counted for).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The optical absorption is also stronger than that  produced by hydrogen (as in the upper-atmosphere model), lead-  ing to a higher continuum level at optical wavelengths, as well  when extrapolated to NUV wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The NUV observations  are consistent with the optical-IR continuum model, with the ex-  ception of the few data points that extend above the continuum  (mainly the 2350 Å region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Lastly, we can see that the constraint  on the temperature profile becomes less precise above the µbar  level since optical/IR data is mostly probing the lower layers of  the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' At the same time, at this high altitudes the re-  trieved temperature profile starts to differ from the higher tem-  peratures predicted by self-consistent models that account for  high-energy flux that heats the thermosphere of the planet (Lav-  3 https://pyratbay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='io/  vas & Koskinen 2017), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', the models used as the base of our  upper-atmosphere model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' On the Nature of the NUV Absorption  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Fe ii Absorption  It is interesting that the excess NUV absorption matches pre-  cisely the locations of the two strongest Fe ii bands in the region,  while there does not seem to be any absorption corresponding to  the Fe i band shortly blueward of the magnesium doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Unfor-  tunately, the low S/N achieved when binning the data at higher  resolution does not allow us to identify absorption from individ-  ual lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  In fact, a detection of Fe ii would be very controversial in this  planet for multiple reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In first place, iron is not expected to  survive the cold trap at the lower layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Even if we neglected  the iron condensation, as is the case for our upper-atmosphere  model, we cannot match the large amplitude of the observed fea-  tures when we bin the model at the resolution of the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Another argument against Fe ii is that given the relatively cool  expected temperature of the planet and the late spectral type of  the star, one would also expect to observe Fe i if Fe ii were de-  tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  If the NUV absorption indeed corresponded to Fe ii, it would  imply that our hydrodynamic model severely underestimates the  absorption signature, and we would be in need of a mechanism  to enhance the lines such that they become detectable in broader  bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This would require a significant broadening of the iron lines  and super-solar abundances, even when assuming no condensa-  tion in the lower atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' However, global circulation mod-  els (GCM) predict winds of the order of ±8 km s−1 at the ter-  minator (Steinrueck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These models include regions  of significant size with flows both away from and towards the  Earth due to the complexity of the circulation and rotation ef-  fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These kinds of flows could effectively broaden metal lines  in spatially unresolved transit observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Absorption from an Unknown Absorber or other Source  The broad-band absorption suggests the presence of an extra ab-  sorber, but the fact that the absorption matched the location of  Fe ii bands might be deceptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' It is certainly possible that the ab-  sorption comes from another unidentified species, possibly from  either a haze or from the electronic band system of a molecu-  lar species, given the width of the absorption and the transition  energies involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Following Lothringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2022), we explored whether  SiO, say vaporized from strongly mixed cloud particles, could  appear in the upper atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' However, the atmospheric re-  trievals do not favor significant SiO absorption when it was  included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' From a further heuristic exploration,  we could not find any configuration of temperatures and abun-  dances producing SiO bands at the observed strength nor loca-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We finally tested for other candidates available in the Ex-  oMol database and that extended into the NUV, which led us to  discard both HS and OH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  One might consider that the absorption originates from an  external source such as accreted material from trojan or circum-  planetary satellites (Kislyakova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Oza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  However, this is a highly speculative scenario, since to date there  is no model that demonstrates that material from an external  source can produce such large absorption features, nor that it  Article number, page 10 of 17  Patricio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' : The HST NUV Transmission Spectrum of HD 189733 b  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Infrared-to-ultraviolet transmission spectrum of HD 189733 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Top and bottom panels show the same data, focused on different spectral  regions for better visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The colored markers with error bars show the transit observations, 1σ uncertainties, and wavelength span of the  observations (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Our data points have been computed from the data binned at a resolving power of R=33 (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The red  curve and orange shaded areas shows the spectrum posterior distribution of the optical-IR fit (median, 68%, and 95% quantiles of the distribution,  see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The blue curve shows the upper-atmosphere model (shaded in a slightly clear color when sitting below the optical/infrared  continuum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  could generate exclusively an Fe ii signature but not that from  other particles like magnesium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  It seems also unlikely that the absorption features come from  a yet-undetermined aerosol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' To our knowledge there is no self-  consistent GCM that couples haze production and circulation to  a degree that it can state with confidence that particles could  form and exist at the required high altitudes (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1 nbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' It is  hard then to explain how aerosols would form and remain aloft  at such altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  We also tested whether the stellar spot correction could gen-  erate the observed features (Rackham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2018, 2019), but on  close inspection of the spectral dependency of the spot contam-  ination at different temperatures, we found no indication that it  could mimic extra absorption at the location of the Fe ii bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Finally, we considered the impact of stellar center-to-limb  flux variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' As the stellar Fe ii lines show in emission (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  1), the observations should be probing the (hotter) stellar chro-  mosphere and transition region rather than the (cooler) photo-  sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Therefore the lines may be showing limb brightening  instead of limb darkening, leading to a quite different transit  signature (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Schlawin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Unfortunately, the  limb-darkening/brightening law for these lines is unknown, and  it cannot be reasonably estimated with current instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  In any case, while the unknown limb-darkening law of the emis-  sion lines introduces a degree of uncertainty in the NUV transit  depths, unaccounted limb brightening would lead to underesti-  mates transit depths, and thus does not explain the excess ab-  sorption seen at 2350 Å and 2600 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Exoplanet NUV Observations in Context  To date three exoplanets have been spectroscopically ob-  served during transit in the NUV with HST/STIS: WASP-121b  (Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2019), HD 209458 b (Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2020), and  HD 189733 b (this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These three planets span a wide  range of irradiation regimes, receiving a stellar flux of 364×  (HD 189733 b), 769× (HD 209458 b), and 5624× (WASP-121b)  that of the one received by the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The analysis of this aggre-  gated data provides a first glance at the physical processes that  shape the properties of the planets’ upper atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 8 shows the transmission spectra  of these three planets at a resolution of R = 1300 in units of  r/Rp, with Rp being the optical radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We can see that WASP-  121b exhibits enhanced transit depths coinciding with Fe ii and  Mg ii lines, whereas the other two planets show subtler features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  When comparing the transmission spectra at a resolving power  of R = 130 in units of r/RRL, with RRL being the Roche-lobe  terminator radius, (lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 8) we see that the pseudo-  continuum extends to the Roche-lobe boundary, which indicates  that the middle atmosphere/lower thermosphere of WASP-121b  undergoes Roche-lobe overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In clear contrast, this is not the  case for HD 189733 b nor HD 209458 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' WASP-121b is there-  fore unique among these planets, and must have a much higher  mass loss rate than the others, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This broad behav-  ior is qualitatively consistent with expectations based on model  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Article number, page 11 of 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='160  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='158  R  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='156  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='154  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  Upper-atmosphere model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='20  Lower-atmosphere retrieval  HST NUV data  s  Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2015)  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='18  King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' (2021)  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='16  888888888  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  Wavelength (um)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' nuv_HD189733b  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Comparison  HST/STIS  NUV  transmission  spectra  of  HD 189733 b, HD 209458 b, and WASP-121b scaled by the optical tran-  sit radius Rp (top panel) and by the Roche-lobe terminator radius RRL of  each planet (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The shaded regions denote the wavelength  span of the dominant Fe ii and Mg ii NUV bands and lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 8 illustrates the dramatic difference  between WASP-121b and the two other planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The NUV spec-  tra of HD 209458 b and HD 189733 b look nearly featureless  when compared to that of WASP-121b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Certainly, WASP-121b is  not an ordinary planet—it must have substantially larger mass-  loss rates than the other two planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Neither HD 209458 b nor  HD 189733 b show the Mg ii absorption feature that survives in  the spectrum of WASP-121b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' As we already know, the spectrum  of HD 189733 b shows a strong feature close to the blue Fe ii ab-  sorption band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This feature does not extend to the Roche lobe,  but the peak is ∼150 surface scale heights above the optical ra-  dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Given that heating in the thermosphere increases the scale  height by a factor of ∼20 above the ∼1 µbar level, the peak prob-  ably probes the 1 nbar level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Finally, we note that the feature in the spectrum of  HD 189733 b is not identical to the Fe ii feature in the spectrum  of WASP-121b though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In particular, the point at ∼2400 Å coin-  cides with a strong Fe ii band and should show maximum absorp-  tion towards the longer-wavelength edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This is true for WASP-  121b, but in the spectrum of HD 189733 b absorption drops in-  stead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This poses a conundrum, however, given the uncertainties  in the data analysis, noise could be responsible for this apparent  inconsistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Conclusions  HD 189733 b is the coolest hot Jupiter that has been spectrosopi-  cally observed in the NUV so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Transmission observations in-  dicate that this is a very hazy planet, but the composition of the  hazes are not really know, some of them may present additional  absorption in the NUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  By analyzing the combined NUV HST/STIS transmission  spectra (3 transits) at a relatively coarse resolution (R = 50),  we obtained better S/Ns and spectral resolution than previous  NUV observation of HD 189733 b in the NUV by King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The transit signature in the light curves is clear and with  no significant residual systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We were able to constrain  a continuum level that sits in between the extrapolation of the  optical haze slope and the continuum of a pure-H2/He upper at-  mosphere model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Thus, given the uncertainties of our continuum  data points, the NUV data is statistically consistent with either a  hazy or clear atmosphere scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In addition, the NUV trans-  mission spectrum presents strong and broad absorption features  that coincide with the location of two strong Fe ii bands, while  there is no apparent excess absorption correlated with Fe i bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  When analyzing the data at a higher resolution (R = 4700), we  found that most of the lightcurves are dominated by noise, which  does not allow us to confirm nor discard iron as the source of the  absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' However, thanks to the much higher stellar flux at the  location of the magnesium resonance lines, we were able to rule  out the presense of Mg ii excess absorption above the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  When compared to a solar-metallicity model with magnesium in  the upper atmosphere, the measured transmission Rp/Rs values at  the core of the Mg ii lines are 2–4σ below the model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  The S/Ns of the data at the Mg i line is not high enough to reject  or confirm excess absorption due to magnesium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  HD 189733 b has a surface gravitational acceleration twice  that of HD 209458 b and unlike WASP-121b, Roche-lobe over-  flow effects are not expected to be that significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Its atmo-  sphere is also cooler, suggesting that the formation of clouds  is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This is well in agreement with optical observa-  tions, where the data are consistent with a high-altitude haze,  which could have a soot-like precursor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' HD 189733 b upper-  atmosphere models assuming that magnesium did not condense  in the lower atmosphere show considerable excess absorption  at the core of the Mg ii resonance lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Thus, the absense of  excess absorption suggests that magnesium is not escaping on  HD 189733 b and that Mg-Si clouds likely form in the lower at-  mosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  At this point, the nature of the broad NUV absorption fea-  tures is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' An iron origin raises significant challenges  for existing upper-atmosphere models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' In first place, given the  relatively low temperatures of HD 189733 b, condensation is ex-  pected to sequester heavy metals into the lower layers of the at-  mosphere, though it is possible that condensates form primarily  from other metals instead, such as magnesium (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Woitke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' However, even if iron is not strongly de-  pleted, one would still need a mechanism to enhance its absorp-  tion beyond that predicted by our upper-atmosphere models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The  two most immediate possibilities would be to consider higher  metallicities and zonal-wind velocity broadening of the absorp-  tion lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' GCM estimations of HD 189733 b predict wind veloc-  ities at ∼8 km s−1 at the terminator, the question for future stud-  ies is whether these mechanisms are enough to boost the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Heavy metals play a key role in the modelling and interpretation  of NUV transit observations and hence in our understanding of  exoplanetary atmospheres as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The detection of metals  at Roche lobe distances would be surprising instead of expected  Article number, page 12 of 17  HD 209458b  Fell  Mgll  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  HD 189733b  WASP-121b  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  a  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5  2300  2400  2500  2600  2700  2800  2900  3000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2  2400  2500  2600  2700  2800  2900  3000  Wavelength (A)Patricio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' : The HST NUV Transmission Spectrum of HD 189733 b  for HD 189733 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Here we presented the detection of a broad  absorption feature in the NUV that matches the location of Fe ii  bands, but cannot fully confirm the nature of the absorber, both  future observational and theoretical work are needed to under-  stand more precisely the origin of this absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We thank the anonymous referee for his/her time and valu-  able comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We thank contributors to the Python Programming Language and  the free and open-source community, including: Pyrat Bay (Cubillos & Blecic  2021), mc3 (Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2017), repack (Cubillos 2017), Numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  2020), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2020), Matplotlib (Hunter 2007), IPython (Pérez  & Granger 2007), and bibmanager (Cubillos 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This research has made use  of NASA’s Astrophysics Data System Bibliographic Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' This project was  funded in part by the Austrian Science Fund (FWF) Erwin Schroedinger Fellow-  ship, programs J4595-N and J4596-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' acknowledge financial  support from the FFG project 865968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Based on observations made with the  NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science  Institute, which is operated by the Association of Universities for Research in  Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', under NASA contract NAS 5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' These observations are  associated with program #15338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2016, ApJ, 817, 106, ADS, 1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='05175  Brogi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='10369  Crouzet, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2014, ApJ,  795, 166, ADS, 1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4000  Cubillos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='01336  Cubillos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2547042, Zenodo, ADS  Cubillos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2015, MNRAS, 450, 1879, ADS, 1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='1536  Koskinen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2017, ApJ, 847, 32, ADS  Lecavelier des Etangs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=', Pont, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2002, ApJ, 580, L171, ADS, astro-ph/0210099  McCullough, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2014, ApJ,  791, 55, ADS, 1407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2462  Menou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2012, ApJ, 745, 138, ADS, 1108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2011, ApJ, 737, 15, ADS, 1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2020, ApJ, 895, L47, ADS, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2019, ApJ, 885, 168, ADS, 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2015, ApJ, 805, 52, ADS, 1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='05590  Pinhas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2019, MNRAS,  482, 1485, ADS, 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='00011  Polyansky, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2018, MNRAS, 480, 2597, ADS, 1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='04529  Pont, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2008,  MNRAS, 385, 109, ADS, 0712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1374  Pont, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' 2013, MNRAS, 432, 2917, ADS, 1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4163  Poppenhaeger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=', Helling, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', & Gunn, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2020, A&A, 634, A23, ADS, 1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03777  Wyttenbach, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Ehrenreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Lovis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', Udry, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', & Pepe, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2015, A&A, 577,  A62, ADS, 1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='05581  Yurchenko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2015, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Spectr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=', 152, 28, ADS, 1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2022, MNRAS, 510, 903, ADS, 2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='04859  Zellem, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2017, ApJ, 844, 27, ADS, 1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='04708  Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' 2022, AJ, 164, 237, ADS, 2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='02985  Article number, page 13 of 17  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' nuv_HD189733b  Appendix A: HD 189733 b NUV Transmission  Spectra  Fig A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1 shows the NUV transmission spectrum binned at a  coarse resolving power of R = 10 and shows a SOFIA-like offset  of the transit depth for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2 shows the systematics- and stellar-spot corrected  lightcurves of the combined HST/STIS HD 189733 b observa-  tions, when binned over a coarse wavelength sampling with a  resolving power of R = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' We used the posterior-distribution  median and the central 68% percentiles statistics to estimate the  parameter values and their credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3 present the estimated values of the NUV  transmission planet-to-star radius ratio when analyzed at resolv-  ing powers of 50, 33, and 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' NUV transmission spectrum of HD 189733 b from the com-  bined HST/STIS observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The red markers with error bars denote  the systematics- and stellar-spot corrected transmission spectrum, their  1σ uncertainties, and the span of the spectral bins for a resolving power  of R = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The blue spectrum shows a theoretical model of the planet’s  upper atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The orange curve shows the fit to the optical-IR  transmission spectrum extrapolated into the NUV (the shaded areas de-  note the 1σ and 2σ uncertainties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The black dashed curve shows the  optical-IR fit shifted downwards according to a SOFIA-like offset of  ∆Rp/Rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Appendix B: Lower-atmosphere Transmission  Retrieval  Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1 shows the parameterization, priors, and retrieved pos-  terior values for the analysis of the HD 189733 b optical-IR  transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1 shows the posterior distribu-  tions of the model parameters and temperature profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content=' HD 189733 b NUV Transmission Spectrum at R=33  λ (µm)  Bin half-width (µm)  Rp/Rs  2309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='003  2770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='003  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='003  2943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='156+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='002  3030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='158+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='003  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='002  Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' HD 189733 b NUV Transmission Spectrum at R=10  λ (µm)  Bin half-width (µm)  Rp/Rs  2405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1  130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1734+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0055  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0052  2681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6  145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='0018  2950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1  122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1564+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0017  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0013  Article number, page 14 of 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='180  Upper-atmosphere model  HST optical-IR transit depth  SOFIA Optical-IR transit depth  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='175  HST/STiS NUV transits (R = 1O)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='170  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='165  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='160  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='155  2300  2400  2500  2600  2700  2800  2900  3000  3100  Wavelength (A)Patricio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' : The HST NUV Transmission Spectrum of HD 189733 b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  Orbital phase  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  (Rp/Rs)2 (%)  = 2298 A  = 2344 A  = 2392 A  = 2441 A  = 2490 A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  Orbital phase  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='00  = 2541 A  = 2592 A  = 2645 A  = 2699 A  = 2754 A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='03  Orbital phase  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='97  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='00  = 2810 A  = 2867 A  = 2925 A  = 2984 A  = 3044 A  visit  1  visit  2  visit  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Transit light curves of the combined HST/STIS observations of HD 189733 b binned at a resolving power of R = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The colored markers  with error bars denote the systematics- and stellar-spot corrected measurements, color coded for each visit (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The solid red curve and  orange area denote the best-fitting transit depths and span of their 1σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The lightcurve of each bin has been shifted in the vertical axis  for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The labels next to each ligtcurve denotes the mean wavelength of the spectral bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Note that the scale of the vertical axis changes  from left to right as the S/N of the data improves with increasing wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Article number, page 15 of 17  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' nuv_HD189733b  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Pairwise distribution (lower-left corner panels) and marginal histograms (diagonal) of the retrieved posterior parameters of the  HD 189733 b optical-IR retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The dashed lines indicate the marginal maximum likelihood of the parameters (the mode of the histograms,  or median when the parameter not well constrained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The shaded areas denote the 68% highest posterior density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' The top-right panel shows the  posterior distribution of the temperature-profile models (the median and the span of the 68% and 95% central credible intervals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='  Article number, page 16 of 17  log(p1) = - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='17:98  10-9  log(p2) = - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='803:1  median  log(p2)  log(p3) = - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='02:§  10-7  (Ed)60l  5  a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='79±8:18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='50  Pressure (bar)  10-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='25  a2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='34±8:13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='25  To = 1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5+4:6  10  3000  2000  1000  Rp (Rjup) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1363±0:8835  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='14  10-1  ３４  log(Xco) = - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='4:i  101  log(Xco2) = - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='17±9:92  500  1000 1500 2000 2500  3000  Temperature (K)  log(XcH4) = - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5±:2  10  log(XNH3) = - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='8±2:  log(XHCN) = - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='82±9:17  log(Xsio) =  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='5±3:3  log(Xna) = - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='17+8:31  log(XNa)  log(Xk) = - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='53:  log(Xk)  05  log(fray) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='17+8:72  log(fray)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  Xray  aray  15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='283:48  (dod)60l  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='0  5  0  0  055055  C  000T  log(p1) log(p2)  log(p3)  0g(XH20) 1  a1  a2  Rp (Rjup)  log(xco)log(x  αray  To  log(ptop)Patricio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' : The HST NUV Transmission Spectrum of HD 189733 b  Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
+page_content=' HD 189733b Optical-IR Retrieval Parameters Summary  Parameter  Priors†  Retrieved value‡  log10(p1/bar)  U(−9, 2)  −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+page_content='81  log10(p3/bar)  U(−9, 2) and p3 > p1  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQfMDTj/content/2301.13025v1.pdf'}
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+ 
+1 
+Photodissociation of H! on Ag and Au 
+Nanoparticles: Effect of Size, and Plasmon 
+versus Inter-Band Transitions on Threshold 
+Intensities for Dissociation  
+Sajal Kumar Giri and George C. Schatz* 
+Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, 
+Illinois 60208-3113, United States 
+E-mail: g-schatz@northwestern.edu 
+ 
+ 
+This paper provides new insights concerning the simulation of plasmon driven chemical reactions 
+using real-time TDDFT based on the tight-binding electronic structure code DFTB+, with 
+applications to the dissociation of H! on octahedral silver and gold nanoparticles with 19-489 
+atoms.  A new component of these calculations involves sampling a 300 K canonical ensemble to 
+determine the distribution of possible outcomes of the calculations, and with this approach we are 
+able to determine the threshold for dissociation as a function of laser intensity, wavelength, and 
+nanocluster size.  We show that the threshold intensity varies as an inverse power of nanocluster 
+size, which makes it possible to extrapolate the results to sizes that are more typical of experimental 
+studies.  The intensities obtained from this extrapolation are around a factor of 100 above powers 
+used in the pulsed experiments. This is a closer comparison of theory and experiment than has 
+
+ 
+2 
+been obtained in  previous real-time simulations, and the remaining discrepancy can be understood 
+in terms of electromagnetic hot spots that are associated with cluster formation.  We also compare 
+the influence of plasmon excitation versus inter-band excitation on reaction thresholds, revealing 
+that for silver clusters plasmon excitation leads to lower thresholds, but for gold clusters, inter-
+band excitation is more effective.  Our study also includes an analysis of charge transfer to and 
+from the H! molecule, and a determination of orbital populations during and after the pulse, 
+showing the correlation between metal excitations and the location of the antibonding level of H!. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Introduction 
+Plasmon-driven chemistry (PDC) has recently gained substantial interest as it holds great 
+promise for dramatically boosting the efficiency of sunlight absorption and solar energy 
+conversion.1-3 It is becoming a fast-growing field of surface photocatalysis where chemical 
+reaction rates can be enhanced by several orders of magnitude, harvesting the electromagnetic field 
+
+14
+10
+0.5 nm
+H, dissociation
+13
+10
+1 nm
+AunH,
+2 0o = 2.3 eV.
+12
+AunH2 Wo = 4 eV
+10
+1.7 nm
+0o = 2.7 eV
+11
+10
+5
+10
+15
+20
+0
+2.2 nm
+Nanoparticle size n [nm] 
+3 
+energy associated with localized surface plasmon resonance (LSPR) excitation of a metal 
+nanoparticle (NP).4-6 Within a few femtoseconds after plasmonic excitation, dephasing leads to 
+highly excited electron-hole pairs, that subsequently can undergo scattering processes to become 
+hot carriers that provide an elevated electronic temperature in the nanoparticle compared to the 
+lattice temperature.7,8 In addition, these hot carriers can be transferred to an electron (or hole)-
+accepting orbital in adsorbed molecules, leading to dissociation or other chemical reactions, either 
+by directly populating antibonding electronic states, or by providing energy to overcome the 
+barriers needed for dissociation, isomerization or other processes.9,10 The electronic subsystem can 
+also equilibrate with the lattice subsystem on the picosecond timescale. Another process involves 
+direct photoexcitation of electron on the metal to the lowest unoccupied molecular orbital (LUMO) 
+of the adsorbed molecules, which can also lead to dissociation dynamics. This can circumvent the 
+problems associated with hot-carrier generation and decay but efficiency is limited by the 
+requirement that the metal atoms be very near the nanoparticle surface, and that the energy of the 
+charge transfer matches the excitation energy.10,11 LSPR-induced dissociation of small important 
+molecules including H! dissociation,12-15 H!O splitting,16,17 CO! reduction,18,19 NH" 
+decomposition,20 and others,21,22 has been reported. The efficiency of plasmon-enhanced 
+dissociation is determined by several factors that influence hot-carrier generation and charge 
+transfer dynamics, including the shape and size of NPs, and pulse properties like wavelength, pulse 
+duration and intensity of the photoexcitation.23,24 
+The generation of hot carriers and charge transfer into surface molecules has been 
+described using various theoretical models25,26 and ab-initio calculations.27,28 Real-time (RT) 
+dynamics of NPs provide an explicit time-dependent perspective and a natural choice when a time-
+dependent electric field interacts with the system to generate hot carriers, in contrast to the more 
+
+ 
+4 
+limited picture obtained from linear response frequency domain calculations.29-31 The RT 
+dynamics deals with the propagation of a single particle density matrix with an explicit time-
+dependent electric field. Some earlier studies have addressed RT-TDDFT simulations for NPs but 
+mostly are restricted to small (few atom) systems or with serious approximations due to the 
+systems’ complexity and computational expense.32-35 Schatz and co-workers36 have demonstrated 
+plasmon-mediated H! dissociation in a nanocavity performing RT-TDDFT calculations with a 
+Gaussian pulse envelope but this required simplifications based on the jellium model. Dissociation 
+was observed only for very high intensities (~10#$ W/cm2) and extremely short pulse durations 
+(1fs). Other theoretical works have reported the use of 10#$ − 10#% W/cm2  intensities and short 
+pulse durations for dissociation to take place.37-39 Such powers are close to the limit where 
+relativistic electrodynamics would be required.40 Pulsed laser-driven dissociation of molecules in 
+presence of plasmonic NPs has been demonstrated experimentally with intensities and durations 
+of 1-100 MW/cm2 and 100-250 fs respectively.41 It is of particular interest to explore the real-time 
+dynamics of larger nanoparticles with sizes on the order of 10-20 nm and to establish the 
+dependence of threshold intensity on particle size. Recently tight-binding density functional theory 
+has been developed,42 enabling the study of the RT dynamics of relatively large particles of Ag 
+and Au atoms with dimensions up to 2-3 nm, but this is still far below the sizes studied in the 
+experiments.   
+In this work, we investigate the dissociation of  H! on Ag and Au nanoparticles induced 
+by laser pulses using the RT-TDDFTB method implemented in the DFTB+ code. By introducing 
+the canonical ensemble sampling of the initial vibrational coordinates and momenta of the 
+molecules from an initial temperature of 300 K (but neglecting zero-point energies), we explore 
+the effect of NPs size (from 0.5 nm to 2.2 nm), as well as driving frequencies, on the dissociation 
+
+ 
+5 
+intensity threshold. This enables a determination of how the threshold intensity scales with particle 
+size, providing a meaningful comparison with experimental intensity values for pulse durations in 
+the range 100-250 fs. Two different driving frequencies are used to explore intra-band vs inter-
+band transition-assisted dissociations, and for each frequency, intensity, cluster size and element 
+(Ag or Au) we perform trajectory simulations to determine dissociation probabilities. Furthermore, 
+we have calculated orbital populations and time-dependent charge transfer between NPs and 
+molecules along the trajectories to demonstrate how electron transfer drives the dissociation. The 
+pulse amplitude vanishes after 25 fs for the pulse parameters we have chosen, but we propagate 
+the density matrix up to 200 fs to observe the subsequent dynamics. Larger pulse durations might 
+be more effective as more interaction time enables additional energy transfer, however, pulses 
+longer than 200 fs for fixed fluence are not beneficial as plasmon decay processes (that are not 
+fully described in DFTB+) convert electronic excitation into vibrational excitation.  
+The paper is organized as follows. In Section II we provide details of dynamical equations, 
+pulses, calculations of spectra and nuclear trajectories.  Then we discuss all the results in Section 
+III and finally conclude in Section IV. 
+ 
+Theory and Computation 
+Under the real-time dynamics used in DFTB+, the single-particle density matrix 𝜌(𝑡) is 
+evolved in time using an initial condition defined by the ground state Hamiltonian 𝐻&, overlap 
+matrix 𝑆, and density matrix 𝜌&. The density matrix is propagated according to the Liouville-von 
+Neumann equation within the Ehrenfest ansatz to include electron-nuclear dynamics43 using 
+'
+'( 𝜌(𝑡) = −i(𝑆)#𝐻𝜌 − 𝜌𝐻𝑆)#) −	(	𝑆)#𝐷𝜌 + 𝜌𝐷*𝑆)#),                                                              (1) 
+ 
+
+ 
+6 
+where 𝐻 is the Hamilton matrix, 𝑆)# is the inverse of the overlap matrix, and 𝐷+, = 〈𝜙+|𝜙̇,〉 are 
+the non-adiabatic coupling matrix elements with localized atomic orbitals |𝜙⟩. Note that the 
+derivative coupling term 𝐷 arises from nuclear motion, so in contrast to past work with frequency 
+domain calculations for frozen structures to determine optical spectra, the RT approach leads rather 
+naturally to dissociation of metal cluster and adsorbed molecule. The first term in the above 
+equation is only for electron dynamics whereas the second term is responsible for coupling 
+between electron and nuclear motions. To describe photoexcitation within this framework, the 
+Hamilton matrix 𝐻 includes a time-dependent electric field through the interaction 𝑉(𝑡) = −𝝁 ⋅
+𝑬(𝑡), with an external electric field 𝑬(𝑡) and transition dipole 𝝁, where the latter is obtained 
+summing over partial charges on the atoms. We note that both 𝐷 and 𝑉(𝑡) couple electronic states 
+but in different ways. 𝐷 involves derivative couplings, while	 𝑉(𝑡) includes the field-matter 
+couplings. The force experienced by each atom is computed from the evolved non-equilibrium 
+density matrix 𝜌(𝑡) and is determined by averaging over the distribution of electronic states:31,42 
+𝑀𝑹̈ =	−𝛁-[Tr(𝜌(𝑡)𝐻(𝑡))/Tr(𝜌(𝑡))],                                                                                               (2) 
+where 𝑹 denotes the atomic coordinates and 𝑀 specifies the masses of the atoms. Atomic charges 
+are calculated using the Mulliken approximation31,42, 𝑞.(𝑡) = Tr.[𝜌(𝑡)𝑆], where Tr. indicates 
+the trace over the orbitals centered on atom A. 
+ 
+We consider the following initial perturbation (delta function in time) in all possible spatial 
+directions to calculate the real-time absorption spectra: 
+𝑬(𝑡) = 𝐸&𝛿(𝑡)𝒆,                                                                                                                           (3) 
+with the field amplitude 𝐸&, and a unit vector 𝒆 ∈ {𝑥, 𝑦, 𝑧}. In the frequency domain (the Fourier 
+transform of 𝑬(𝑡) in Eq.3), this perturbation corresponds to a constant value (𝐸&) independent of 
+frequency, and therefore can include all possible frequency components in driving the system. 
+
+ 
+7 
+From the Fourier transform of the resultant time-dependent dipole along each spatial direction, we 
+compute absorption spectrum as44,45 
+𝜎(𝜔) =
+$/0
+"1 ℑ[Tr[𝜶(𝜔)]],                                                                                                            (4) 
+ 
+where 𝑐 is the speed of light, and 𝜶 is the polarizability tensor,  𝜶(𝜔) = (𝝁U(𝜔) − 𝝁&)/𝐸&, with 
+the frequency-dependent and initial (before the perturbation) dipoles 𝝁U(𝜔) and 𝝁& respectively. 
+Alternatively, we compute spectra in the frequency domain using the linear response TDDFTB 
+method through the calculation of oscillator strengths between the states. We use equivalent 
+damping factors of the Lorentzian lineshape in the time and frequency domain to obtain the final 
+smoothed spectra in order to compare spectra from both domains.   
+ 
+For laser pulse driven molecular dissociation dynamics, we have used an electric field of 
+the following form 
+𝐸(𝑡) = 𝐸&e)(()(!)"/5"sin	(𝜔&(𝑡 − 𝑡1)),                                                                              (5) 
+ 
+with peak amplitude 𝐸&, driving frequency 𝜔&, peak position of the pulse 𝑡1, and pulse duration 𝑇. 
+During the calculation of nuclear trajectories, we consider different initial vibrational coordinates 
+and momenta sampled from a canonical ensemble and calculate an ensemble of trajectories to 
+generate a distribution of final outcomes. This type of calculation has not been considered in 
+previous studies, but it is known to provide a more accurate treatment of the dynamics than the 
+zero temperature approach involving a single trajectory, as has been used in the past. From the 
+trajectories, we compute dissociation probabilities as a function of time by using a Gaussian 
+distribution along inter-atomic distances to weight the reactive probabilities associated with each 
+trajectory i.e., identifying the fraction of dissociative trajectories. A Gaussian function centered 
+around 6 Å with a width of 1.3 Å is used, along with the assumption that as soon as the inter-
+atomic distance passes the center of the Gaussian (6 Å), the probability is kept at the peak value. 
+
+ 
+8 
+See the supporting information for the dependence of dissociation probability on Gaussian widths. 
+We note that the explicit nuclear motion of all atoms has been included during the trajectory 
+calculation.  
+ 
+We have performed all the electronic structure calculations and solved the real-time 
+dynamics using the DFTB+ code.42 A step size of 0.002 fs is used for the time evolution of the 
+density matrix up to 200 fs. We have checked the propagation convergence using a smaller step 
+size (smaller than 0.002 fs) and it turns out that a step size of 0.002 fs is reasonable to keep a 
+balance between the propagation error and computational cost. For pulsed excitations, the laser 
+pulse amplitude is chosen to vanish after 25 fs, but we further propagate the density matrix to 
+observe the after-excitation dynamics, leading to charge transfer and dissociation of the molecules.   
+ 
+Results and Discussion 
+ 
+In this work, we have studied octahedral geometries of Ag and Au nanoparticles (NPs) of 
+various numbers of atoms. We chose octahedrons for this work as this provides a simple motif for 
+varying NP size over a wide range, and octahedron-shaped gold NPs have been studied 
+experimentally,46 having plasmon resonance properties that are similar  (somewhat red-shifted) to 
+what are found for spheres. Of course, the sharp points should provide hot spots with larger field 
+enhancements than for spheres.  Four different sizes have been used for each of the Ag and Au 
+NPs with the number of atoms 19, 85, 231, and 489 corresponding to 0.5, 1, 1.7, and 2.2 nm 
+dimensions respectively. Also, these geometries have been considered in some previous 
+experimental47-50 and theoretical51,52 works.  We have optimized each NP geometry using the 
+DFTB method with hyb-0-2 parameters set for Ag NPs and auorg-1-1 parameters set for Au NPs.53 
+A H! molecule is attached next to the tip of the optimized NP and then we further optimize the full 
+system (NP with molecule). All the optimized geometries are shown in Fig.1 where the molecular 
+
+ 
+9 
+axis is perpendicular to the z-axis. The inter-atomic distance of the hydrogen molecule is 
+approximately 0.85 Å. We note that the gas phase equilibrium bond length of the molecule is 
+approximately 0.74 Å but due to the adsorption on the NP surfaces here the bond length is slightly 
+larger. The distance between the tip atom and molecule for systems involving Ag (Au) atoms is 
+around 2.5 Å (1.8 Å). These optimized system parameters are comparable to previous DFT 
+calculations for a tetrahedral Au6H! system.15 
+ 
+ 
+Figure 1: 𝐻! molecule adsorbed on octahedral Ag and Au nanoparticle surfaces of different 
+dimensions determined by the number of atoms 𝑛, (𝑛 = 19, 85, 231, and 489). All the geometries 
+are optimized with the DFTB method using the parameters set hyb-0-2 and auorg-1-1 for Ag and 
+Au NPs respectively.   
+As a test of the RT code, we compute absorption spectra solving both the real-time 
+dynamics (RT-TDDFTB) through the computation of time-dependent dipoles and linear response 
+TDDFTB frequency domain calculations. In the linear response limit (at low intensities involving 
+a one photon process), both spectra should match each other. For the time domain spectra, we 
+
+AgnH2
+00
+n= 19
+n = 85
+n = 489
+n = 231
+2.2 nm
+0.5 nm
+1.7 nm
+1 nm
+AunH2 
+10 
+solve the Liouville-von Neumann equation (Eq.1) with a delta kick perturbation described in Eq.3. 
+From the resultant time-dependent dipole along each spatial direction, we calculate absorption 
+spectra through Eq.4. We set the peak amplitude of the perturbation (𝐸&) to a small value 0.001 
+V/Å which corresponds to a peak intensity of 1.3 × 107 W/cm2. We use the TDDFTB method in 
+the Casida formulation in computing oscillator strengths between ground and excited states. 
+Spectra from both methods agree very well with each other. Throughout we have used a damping 
+factor of 0.16 eV in computing spectra. See the supporting information for the comparison between 
+the time and frequency domain spectra for Ag#8H! and Au#8H! NPs. 
+ 
+ 
+ 
+Figure 2: Real-time absorption spectra for (a) 𝐴𝑔9𝐻! and (b) 𝐴𝑢9𝐻! systems. The two frequencies 
+used for the dissociation dynamics are shown by vertical dashed lines. 
+Real-time absorption spectra for all systems are shown in Fig.2. The absorption intensity 
+increases gradually with the size of the NPs as the density of states (DOS) increases. For Ag NPs, 
+the LSPR peak appears between [2.5, 3] eV and shows a prominent red shift with increasing NPs 
+size. The peak shifts towards lower energy by around 0.25 eV when we increase the number of 
+
+(a)
+n= 489
+0.8
+n=231
+n=85
+0.6
+n=19
+0.4
+[arb. units]
+0.2
+ Absorbance [
+-
+(b)
+-
+-
+0.8
+-
+-
+0.6
+0.4
+0.2
+0
+1
+4
+2
+3
+5
+6
+Energy [eV] 
+11 
+atoms from 19 (0.5 nm) to 489 (2.2 nm). This trend was observed in previous studies of 0.5-2.0 
+nm clusters54 where it was attributed to quantum size effects, and is distinct from the red shift that 
+is seen for much larger octahedral NPs49 which arises from electromagnetic depolarization 
+effects.55 The high energy flat signals (above 3 eV) originate from inter-band transitions i.e., from 
+d to sp band transitions. For Au NPs, the plasmon peaks are relatively broader due to the short 
+lifetime of the excited states compared to Ag NPs. As a result, it becomes difficult to distinguish 
+the plasmon peaks from high-energy inter-band transitions for the smaller particles. The plasmon 
+peak for the Au$:8H! system arises at around 2.3 eV, which is similar to the plasmon in larger 
+gold NPs with roughly spherical shape.55,56 Here the plasmon peak is more pronounced and is 
+distinct from the high-energy inter-band transitions. This also agrees with the previous study by 
+Berdakin and co-workers57 where plasmon peaks start to appear distinct from inter-band transitions 
+for icosahedral Au NPs involving more than 300 atoms. We note that the broad oscillating 
+structures in the Au NP spectra in Fig.2 at energies above 2.3 eV were also observed by Schatz 
+and co-workers58,59 for tetrahedral Au NPs using the TDDFT calculations. 
+We use two different driving frequencies 𝜔& = 2.7 eV (2.3 eV) and 𝜔& = 4 eV for Ag (Au) 
+NPs to explore the dissociation dynamics of H! due to plasmon and inter-band transitions. These 
+two frequencies are labelled in Fig.2 with vertical dashed lines and the width around each line is 
+inversely proportional to the pulse duration 𝑇. The driving field described in Eq.4 is used in the 
+simulation. We vary the laser peak intensity through the parameter 𝐸&. We consider the 𝑧-axis as 
+the laser polarization direction for all systems. Note that along this direction the tip atom of the 
+cluster points towards the molecule and appears to be more suitable for bond dissociation 
+compared to other possible directions. We have included an example pulse and its frequency 
+content (Fourier transform) in the supporting information.  
+
+ 
+12 
+ 
+Figure 3: Trajectories showing inter-atomic distances and dissociation probabilities for (a-d) 
+𝐴𝑔9𝐻! and (e-h) 𝐴𝑢9𝐻! systems with driving frequencies 2.7 eV and 2.3 eV, respectively, 
+corresponding to plasmon-driven dissociation. From top to bottom, the size of the particle 
+increases, 𝑛	= 19, 85, 231, and 489 respectively. Threshold intensities are given for all plots, 
+denoting intensities where dissociation starts. The red lines are the ensemble-averaged dissociation 
+probability (obtained through the Gaussian binning along bond distances) as a function of time. 
+Here we have calculated 20 trajectories for each plot and sorted the color code from green to blue 
+for non-dissociative and dissociative trajectories.  
+Laser pulse-driven dissociative trajectories for Ag9H! (Au9H!) systems with 𝜔& = 2.7 eV 
+(2.3 eV) are shown in Fig.3 where we have plotted inter-atomic distances 𝑑;);  for an ensemble 
+of trajectories and dissociation probabilities (averaging the trajectories) as a function of time. Note 
+that these frequencies correspond to plasmon-driven dynamics. We employ different pulse 
+
+AgnH.
+Au,H,
+6
+I = 2.3 × 1013 W/cm²
+0.5
+3
+I = 1.6 × 1013
+0
+6
+Dissociation probability
+(b)
+3
+0.5
+I = 6.5 × 1012 W/cm
+A
+I = 1.6 × 1013 W/cm²
+6
+I = 3.3 × 1012 W/cm2
+I = 1.1 × 1013 W/cm2
+3
+0.5
+0
+6
+h
+3
+0.5
+I = 2.1 × 1012 W/cm²
+I = 8.5 × 1012 W/cml
+0
+0
+50
+0
+100
+2000
+50
+100
+150
+200
+150
+Time [fs] 
+13 
+intensities for each plot, which are the lowest intensities necessary to initiate dissociation. We find 
+that the dissociation probability increases linearly with the pulse intensity, from a threshold 
+intensity up to a plateau where the probability is 1. The linear increase is because more hot carriers 
+are generated with increasing photon flux and a one photon transition is sufficient to enable hot 
+carrier generation leading to dissociation near threshold. The linear dependence was also observed 
+in the previous experimental study of H! dissociation on a Au NP surface.12,13 Most of the 
+dissociations take place during interaction with the pulse i.e., within 25 fs. At low intensities, 
+dissociation strongly depends on the initial vibrational phases, and therefore only a fraction of 
+trajectories dissociate. For the remaining trajectories, the inter-atomic distance 𝑑;); shows  
+coherent oscillation around the equilibrium bond length. We note that for most of these non-
+dissociative trajectories, molecules desorb from the surface very quickly within 20-30 fs. At high 
+intensities, the dissociation probability increases as shown in Figs.4(a, b) for the 𝑛 = 231 systems 
+with intensities chosen as the threshold intensity for the 𝑛	= 19 systems.  We found that beyond 
+certain intensities, all the trajectories dissociate. The time-dependent dissociation probability 
+calculated from an ensemble of trajectories through Gaussian binning along inter-atomic distances 
+increases during the pulse interaction and saturates to a finite value within 40 fs. We note that there 
+are interactions between dissociated atoms and the NPs as shown in Fig.5.  Therefore, the 
+dissociated outgoing trajectories are not completely smooth at the threshold intensities while at 
+high intensities the dissociated atoms gain sufficient velocities to leave the surface as soon as the 
+molecule dissociates, resulting in smoothed dissociative trajectories as shown in Figs.4(a, b).   
+
+ 
+14 
+ 
+Figure 4: Same as Figs.3(c, g) but for (a, b) high intensities, and (c, d) 𝜔& = 4	eV. 
+ 
+Figure 5: Geometries along the trajectories (a single trajectory for each system 𝐴𝑔!"#𝐻! or 
+𝐴𝑢#!&𝐻!) at 20, 50, 100, and 200 fs from Figs.3(c, g). Here  𝑑<)< = 23.3	Å (22.8 Å) for Ag (Au) 
+NPs at 200 fs.  
+Trajectories for the Ag!"#H! and Au!"#H! systems with 𝜔& = 4 eV are shown in Figs.4(c, 
+d). High-frequency driving shows similar dissociation but involves different intensities than the 
+corresponding low-frequency driving for the dissociation to take place. We note that the isolated 
+
+(a)
+Ag231H2
+ty
+3
+I = 1.2 × 1013 W/cm²
+I = 1.6 × 1013 W/cm²
+A
+0
+H-
+6
+Hp
+(b)
+Au?
+Au231H,
+3
+I = 2.3 × 1013 W/cm²
+5
+0
+0
+100 150 2000
+100 150 200
+0
+50
+50
+Time [fs]20 fs
+50 fs
+100 fs
+200 fs 
+15 
+molecule dissociates at even higher intensities than we have presented. This confirms that 
+dissociation is enhanced by the presence of NPs and the enhancement factors strongly depend on 
+the NP size and driving frequencies. 
+ 
+ 
+Figure 6:  Threshold intensity (𝐼() as a function of particle size 𝜂 for 𝐴𝑔9𝐻! and 𝐴𝑢9𝐻! systems 
+with two different driving frequencies. 
+Table 1: Fitting parameters, 𝑎 and 𝑏, for threshold intensity as a function of particle size, 𝐼( =
+𝑎 × 𝜂=. 
+System          𝜔&	[eV]                𝑎                           𝑏 
+Ag9H!               2.7             6.4 × 10#!               -1.34 
+                           4               1.45 × 10#"              -0.34 
+Au9H!               2.3             1.5 × 10#"               -0.65 
+                           4               1 × 10#"                   -1.03 
+ 
+The above calculations are used to determine the threshold intensity for dissociation i.e., 
+the minimum intensity required for dissociation to take place. Interestingly, the threshold intensity 
+decreases with increasing particle size for both Ag and Au NPs but with different decay rates 
+depending on the driving frequency as shown in Fig.6. We fit the decay curves with the functional 
+
+14
+10
+13
+10
+Au
+= 2.3 eV
+12
+10
+=4 eV
+eV
+11
+10
+0
+5
+10
+15
+20
+Nanoparticle size n [nm] 
+16 
+form 𝐼( = 𝑎 × 𝜂=, where 𝐼(  is the threshold intensity, 𝜂 is the particle size, and 𝑎 and b are 
+parameters to be optimized. The fitting parameters 𝑎 and 𝑏 are listed in Table 1. All the decays are 
+approximated well with the same functional form 𝐼( = 𝑎 × 𝜂=.  The  Table shows that  b is negative 
+and hence the threshold intensity decreases with particle size.  The decay rate is maximum (𝑏 =
+−1.34) with minimum amplitude (a = 6.4 × 10#!) for Ag9H! systems with plasmon frequency 
+(𝜔& =	2.7 eV). This conclusion is in agreement with a recent experiment where Ag NPs are found 
+to be more efficient than Au NPs in generating hot electrons.24 On the other hand, for Au9H! 
+systems, the high frequency (𝜔& =	4 eV) driving provides a larger decay rate compared to the low-
+frequency driving as there is a significant inter-band transition amplitude, see Fig.2b, while the 
+plasmon amplitude is weak. For Ag9H! particles there are dominant plasmonic transitions for 𝑛 ≥ 
+85 and the plasmon resonance shows larger efficiency for dissociation compared to the inter-band 
+transition.  
+To provide insight about the comparison of calculated and experimental threshold 
+intensities, we extrapolated the plots to large particle sizes up to 20 nm where the intensities are 
+reduced significantly for large |𝑏| values. For  Ag9H!  at 2.7 eV (with the maximum dissociation 
+rate), the threshold intensity decreases from 10#" to 10## W/cm2 when we increase the particle 
+size from 0.5 nm to 20 nm. We note that the peak intensity can be further lowered to 10#& W/cm2 
+when we scale the pulse duration by a factor of 10 in order to compare with pulses of duration 200 
+fs used in the experiment.41 This is still 2 orders of magnitude larger than the experimental peak 
+amplitudes.  However, this can be compensated by including aggregated particles in the study, as 
+this would enable stronger hotspots, as known from surface-enhanced Raman spectroscopy 
+(SERS) studies and electrodynamics theory,60,61 and it would also provide a mechanism for 
+
+ 
+17 
+trapping of the dissociating molecule in the hot spot, which would enhance charge transfer 
+efficiency between nanoparticles and molecules.36  
+ 
+Figure 7: Trajectories for the molecular charge (𝑄>) for  𝐴𝑔!"#𝐻! (𝐴𝑢!"#𝐻!) with frequencies 
+2.7 (2.3) eV and 4 eV. Threshold intensities are used from Figs.3(c, g) for low frequencies and 
+Figs.4(c, d) for high frequencies. The same color code is used for respective trajectories compared 
+to Figs.3(c, g) and Figs.4(c, d) i.e., the color code from green to blue represents non-dissociative 
+to dissociative trajectories. 
+ 
+Figure 8: Orbital populations as a function of energy 	𝜀 − 𝜀? and time. The red (blue) color 
+represents population gain (de-population) with positive (negative) amplitudes. The same 
+(threshold) intensities are used as in Fig.7. (c, f) Density of states (DOS) where magenta lines 
+represent d band only and green shaded areas are for the complete systems (top) 𝐴𝑔!"#𝐻! and 
+(bottom) 𝐴𝑢!"#𝐻!. Red lines (in the DOS panels) represent the antibonding orbital of 𝐻!. 
+
+Ag231H2
+Au231H2
+0o = 2.7 eV
+0o = 2.3 eV
+e
+Q
+(b)
+(d)
+Wo = 4 eV
+Wo=4 eV
+0
+0
+50
+100 150 200 050
+100.150 200
+Time [fs]4
+(b)
+0o = 2.7eV
+Wo = 4 eV
+2
+0
+Tev
+2
+0o = 2.3 eV (d)
+(e)
+Wo = 4 eV
+2
+0
+-2
+25
+0
+50
+75
+1000
+25
+50
+75
+1000
+600
+DOS
+Time [fs]
+[states / eV] 
+18 
+Figure 7 displays the time evolution of molecular charge 𝑄> i.e., the combined charge of 
+the two hydrogen atoms as obtained through the Mulliken approximation for the Ag!"#H! and 
+Au!"#H! systems with the threshold intensities. Initially, the molecule is slightly positive  with 
+𝑄> ≈ 0.05	𝑒 when it is adsorbed on the surface of a NP. After the pulsed excitation, there is an 
+exchange of electrons between two sub-systems and  𝑄> follows the laser frequency oscillations.  
+We note that the total charge is conserved at all times i.e., 𝑄@A + 𝑄>= 0, where 𝑄@A is the NP 
+charge. There is an effective transfer of electrons to the molecule at a certain time after the pulse 
+is turned on (near the peak intensity of the pulses) and the average charge on the molecule becomes 
+negative, corresponding to the net transfer of electrons. Similar to inter-atomic distance 
+trajectories, the electron transfer process also depends on the initial vibrational phases. It is 
+interesting to notice that only dissociative trajectories exhibit substantial electron transfer (the 
+same color code is used as in Figs.3 and 4). This confirms that dissociation is driven by the electron 
+transfer process. After bond dissociation, one of the dissociated hydrogen atoms sticks to the 
+surface of the NPs (as shown in Fig.5), resulting in negative partial charges on the dissociated 
+atoms at longer times. Electron back transfer from the molecule to NP is also observed for some 
+trajectories when the molecule or dissociated atoms leave the surface of the NP and the final charge 
+at longer times becomes nearly zero.  In the earlier RT-TDDFT work,36 similar charge transfer 
+dynamics of H! in Au NPs were also observed. 
+Time-dependent orbital populations for the 𝑛 = 231 systems with plasmonic and inter-
+band transition frequencies are shown in Fig.8. Here we show population changes compared to the 
+initial populations (i.e., the populations prior to the pulse interaction).  Due to the interaction with 
+pulses, there is a transfer of populations across the Fermi level (with energy 𝜀?) from lower energy 
+to higher energy states covered by the bandwidth of the pulses in the frequency domain. As 
+
+ 
+19 
+expected plasmon excitations populate low energy states near the Fermi level while the inter-band 
+transitions are extended to high energy states with dominant d to sp transitions. In comparison to 
+Ag, Au NPs have a broader energy range for the d band that is closer to the Fermi level, which 
+allows significant inter-band transitions to populate states above the Fermi level. All the 
+populations show coherent oscillations following the laser frequencies during the pulse, and then 
+after the pulse, the coherence de-phases very quickly. These oscillations are known to appear upon 
+laser excitation of NPs which are referred to as “sloshing” and “inversion”.17,62,63 During the pulse 
+interaction (within 25 fs), adequate populations accumulate near the antibonding orbital of the H2 
+molecule (the red curves in Figs. 8c and 8f) which weakens the molecular bond and eventually the 
+molecule dissociates.  
+ 
+Conclusions 
+This work demonstrates the laser pulse driven dissociation of H! on octahedral Ag and Au 
+nanoparticle surfaces based on real-time density functional tight binding calculations within the 
+canonical ensemble trajectory approach. We have explored the effect of nanoparticle size and 
+driving frequencies on the dissociation intensity threshold, and we have also characterized the 
+influence of pulse width. Calculations are performed for nanoparticles with varying numbers of 
+atoms up to 489.  As expected, a plasmon peak appears for Au nanoparticles only for a large 
+number of atoms (more than 231) whereas Ag nanoparticles show a plasmon peak for much 
+smaller particle sizes. In order to investigate plasmon vs inter-band assisted dissociation of 
+molecules, two different driving frequencies are used in the real-time dynamics. Both particle size 
+and driving frequency are found to be crucial in driving dissociation dynamics and hence the 
+intensity threshold. Interestingly, we found that the intensity threshold decreases with particle size 
+
+ 
+20 
+following a  similar functional form irrespective of driving frequencies and systems. Using the 
+functional fittings, we scale the results to larger nanoparticle sizes up to 20 nm, for which the 
+largest decay rate is obtained for Ag nanoparticles excited at the plasmon frequency.  In this case, 
+the threshold power is about a factor of 100 larger than has been used in pulsed experiments, which 
+is a result that is consistent with known enhancements that arise when clusters of particles rather 
+than isolated particles are considered. For Au nanoparticles, inter-band transitions are more 
+effective than plasmon excitation, a result that has also been seen in experiments,64 and the 
+threshold power is similar to that for Ag. Additionally, we have calculated the charge on the 
+molecule as a function of time along each trajectory. After the pulsed excitation, we observe 
+electron transfer from the nanoparticle to the molecule which initiates the dissociation process. We 
+have also analyzed these results in terms of the orbital populations as a function of time. 
+ 
+ASSOCIATED CONTENT 
+The Supporting Information is available free of charge at 
+ 
+AUTHOR INFORMATION 
+Corresponding Author 
+George C. Schatz – Department of Chemistry, Northwestern University, 2145 Sheridan Road, 
+Evanston, Illinois 60208, United States; 
+Email: g-schatz@northwestern.edu 
+Author 
+Sajal Kumar Giri – Department of Chemistry, Northwestern University, 2145 Sheridan Road, 
+Evanston, Illinois 60208, United States 
+
+ 
+21 
+ 
+ACKNOWLEDGMENT 
+This research was supported by the Department of Energy, Office of Basic Energy Sciences 
+through grant DE-SC0004752. 
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+
+ 
+29 
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+nanoparticles and PCBM. J. Phys. Chem. C 2019, 123, 29908-29915. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+30 
+Supporting Information: 
+Photodissociation of H! on Ag and Au 
+Nanoparticles: Effect of Size, and Plasmon versus 
+Inter-Band Transitions on Threshold Intensities for 
+Dissociation  
+Sajal Kumar Giri and George C. Schatz* 
+Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, 
+Illinois 60208, United States 
+E-mail: g-schatz@northwestern.edu 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+31 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure S1: Comparison of absorption spectra obtained through RT-TDDFTB (time domain) and 
+TDDFTB (frequency domain) methods.  Vertical orange lines represent oscillator strengths. 
+ 
+ 
+ 
+Figure S2: (a) Time dependent field 𝐸B(𝑡) and (b) its frequency content (i.e., the Fourier transform 
+of 𝐸B(𝑡)) with polarization along z-axis. Here, 𝜔& = 2.7 eV.  
+ 
+
+[arb. units]
+units
+RT-TDDFTB
+RT-TDDFTB
+0.8
+0.8
+TDDFTB
+TDDFTB
+arb.
+0.6
+Absorbance L
+bsorbance
+0.2
+0.2
+0
+0
+1.0
+2.0
+3.0
+4.0
+5.0
+1.0
+2.0
+3.0
+4.0
+5.0
+Energy[e]
+Energy [eV](b)
+a
+[arb.units]
+E;(t) [V/A]
+0.5
+(m)
+E
+0
+0
+10
+20
+30
+1.7
+2.7
+3.7
+Time [fs]
+Frequency [eV] 
+32 
+ 
+Figure S3: (a) Nuclear trajectories from Fig.3(d) in the main text and (b) dissociation probability 
+as a function of time using a Gaussian weighting 𝐺(𝑑) = 𝑒)(C)C#)" !D"
+⁄
+  centered around 𝑑& =
+6	Å and with widths ranging from 𝜎 = 1 to 3.5 Å. Note that as soon as 𝑑 passes 6 Å, the probability 
+is kept at the peak value. 
+      
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+1.0
+5
+(a)
+0=1
+C = 2.5
+0.5
+(c)
+()
+3
+0.0
+1.0F
+o = 1.5
+0
+c=3
+0.5
+(d)
+Dissociation prob.
+0.7
+G
+(g)
+0.0
+1.0[
+0.35
+上
+3.5
+=2
+o = 3.5
+0.5
+(e)
+(h)
+(b)
+0.0
+0.0
+0
+50
+100
+150
+200
+0
+5
+10
+15
+200
+5
+10
+15
+20
+Time [fs]
+dH-H [A]
\ No newline at end of file
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf,len=1410
+page_content='1 Photodissociation of H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' on Ag and Au Nanoparticles: Effect of Size, and Plasmon versus Inter-Band Transitions on Threshold Intensities for Dissociation Sajal Kumar Giri and George C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Schatz* Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States E-mail: g-schatz@northwestern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='edu  This paper provides new insights concerning the simulation of plasmon driven chemical reactions using real-time TDDFT based on the tight-binding electronic structure code DFTB+, with applications to the dissociation of H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' on octahedral silver and gold nanoparticles with 19-489 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' A new component of these calculations involves sampling a 300 K canonical ensemble to determine the distribution of possible outcomes of the calculations, and with this approach we are able to determine the threshold for dissociation as a function of laser intensity, wavelength, and nanocluster size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We show that the threshold intensity varies as an inverse power of nanocluster size, which makes it possible to extrapolate the results to sizes that are more typical of experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The intensities obtained from this extrapolation are around a factor of 100 above powers used in the pulsed experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' This is a closer comparison of theory and experiment than has  2 been obtained in  previous real-time simulations, and the remaining discrepancy can be understood in terms of electromagnetic hot spots that are associated with cluster formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We also compare the influence of plasmon excitation versus inter-band excitation on reaction thresholds, revealing that for silver clusters plasmon excitation leads to lower thresholds, but for gold clusters, inter- band excitation is more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Our study also includes an analysis of charge transfer to and from the H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' molecule, and a determination of orbital populations during and after the pulse, showing the correlation between metal excitations and the location of the antibonding level of H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='.  Introduction Plasmon-driven chemistry (PDC) has recently gained substantial interest as it holds great promise for dramatically boosting the efficiency of sunlight absorption and solar energy conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='1-3 It is becoming a fast-growing field of surface photocatalysis where chemical reaction rates can be enhanced by several orders of magnitude, harvesting the electromagnetic field  14 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 nm H, dissociation 13 10 1 nm AunH, 2 0o = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' 12 AunH2 Wo = 4 eV 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7 nm 0o = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7 eV 11 10 5 10 15 20 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2 nm Nanoparticle size n [nm] 3 energy associated with localized surface plasmon resonance (LSPR) excitation of a metal nanoparticle (NP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4-6 Within a few femtoseconds after plasmonic excitation, dephasing leads to highly excited electron-hole pairs, that subsequently can undergo scattering processes to become hot carriers that provide an elevated electronic temperature in the nanoparticle compared to the lattice temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7,8 In addition, these hot carriers can be transferred to an electron (or hole)- accepting orbital in adsorbed molecules, leading to dissociation or other chemical reactions, either by directly populating antibonding electronic states, or by providing energy to overcome the barriers needed for dissociation, isomerization or other processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='9,10 The electronic subsystem can also equilibrate with the lattice subsystem on the picosecond timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Another process involves direct photoexcitation of electron on the metal to the lowest unoccupied molecular orbital (LUMO) of the adsorbed molecules, which can also lead to dissociation dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  This can circumvent the problems associated with hot-carrier generation and decay but efficiency is limited by the requirement that the metal atoms be very near the nanoparticle surface, and that the energy of the charge transfer matches the excitation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='10,11 LSPR-induced dissociation of small important molecules including H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' dissociation,12-15 H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='O splitting,16,17 CO!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' reduction,18,19 NH" decomposition,20 and others,21,22 has been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The efficiency of plasmon-enhanced dissociation is determined by several factors that influence hot-carrier generation and charge transfer dynamics, including the shape and size of NPs, and pulse properties like wavelength, pulse duration and intensity of the photoexcitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='23,24 The generation of hot carriers and charge transfer into surface molecules has been described using various theoretical models25,26 and ab-initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='27,28 Real-time (RT) dynamics of NPs provide an explicit time-dependent perspective and a natural choice when a time- dependent electric field interacts with the system to generate hot carriers, in contrast to the more  4 limited picture obtained from linear response frequency domain calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='29-31 The RT dynamics deals with the propagation of a single particle density matrix with an explicit time- dependent electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Some earlier studies have addressed RT-TDDFT simulations for NPs but mostly are restricted to small (few atom) systems or with serious approximations due to the systems’ complexity and computational expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='32-35 Schatz and co-workers36 have demonstrated plasmon-mediated H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' dissociation in a nanocavity performing RT-TDDFT calculations with a Gaussian pulse envelope but this required simplifications based on the jellium model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Dissociation was observed only for very high intensities (~10#$ W/cm2) and extremely short pulse durations (1fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Other theoretical works have reported the use of 10#$ − 10#% W/cm2  intensities and short pulse durations for dissociation to take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='37-39 Such powers are close to the limit where relativistic electrodynamics would be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='40 Pulsed laser-driven dissociation of molecules in presence of plasmonic NPs has been demonstrated experimentally with intensities and durations of 1-100 MW/cm2 and 100-250 fs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='41 It is of particular interest to explore the real-time dynamics of larger nanoparticles with sizes on the order of 10-20 nm and to establish the dependence of threshold intensity on particle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  Recently tight-binding density functional theory has been developed,42 enabling the study of the RT dynamics of relatively large particles of Ag and Au atoms with dimensions up to 2-3 nm, but this is still far below the sizes studied in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' In this work, we investigate the dissociation of  H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' on Ag and Au nanoparticles induced by laser pulses using the RT-TDDFTB method implemented in the DFTB+ code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' By introducing the canonical ensemble sampling of the initial vibrational coordinates and momenta of the molecules from an initial temperature of 300 K (but neglecting zero-point energies), we explore the effect of NPs size (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 nm to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2 nm), as well as driving frequencies, on the dissociation  5 intensity threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' This enables a determination of how the threshold intensity scales with particle size, providing a meaningful comparison with experimental intensity values for pulse durations in the range 100-250 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Two different driving frequencies are used to explore intra-band vs inter- band transition-assisted dissociations, and for each frequency, intensity, cluster size and element (Ag or Au) we perform trajectory simulations to determine dissociation probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Furthermore, we have calculated orbital populations and time-dependent charge transfer between NPs and molecules along the trajectories to demonstrate how electron transfer drives the dissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The pulse amplitude vanishes after 25 fs for the pulse parameters we have chosen, but we propagate the density matrix up to 200 fs to observe the subsequent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Larger pulse durations might be more effective as more interaction time enables additional energy transfer, however, pulses longer than 200 fs for fixed fluence are not beneficial as plasmon decay processes (that are not fully described in DFTB+) convert electronic excitation into vibrational excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' In Section II we provide details of dynamical equations, pulses, calculations of spectra and nuclear trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Then we discuss all the results in Section III and finally conclude in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  Theory and Computation Under the real-time dynamics used in DFTB+, the single-particle density matrix 𝜌(𝑡) is evolved in time using an initial condition defined by the ground state Hamiltonian 𝐻&, overlap matrix 𝑆, and density matrix 𝜌&.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=" The density matrix is propagated according to the Liouville-von Neumann equation within the Ehrenfest ansatz to include electron-nuclear dynamics43 using ' '( 𝜌(𝑡) = −i(𝑆)#𝐻𝜌 − 𝜌𝐻𝑆)#) − ( 𝑆)#𝐷𝜌 + 𝜌𝐷*𝑆)#),                                                              (1)  6 where 𝐻 is the Hamilton matrix, 𝑆)# is the inverse of the overlap matrix, and 𝐷+, = 〈𝜙+|𝜙̇,〉 are the non-adiabatic coupling matrix elements with localized atomic orbitals |𝜙⟩." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Note that the derivative coupling term 𝐷 arises from nuclear motion, so in contrast to past work with frequency domain calculations for frozen structures to determine optical spectra, the RT approach leads rather naturally to dissociation of metal cluster and adsorbed molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The first term in the above equation is only for electron dynamics whereas the second term is responsible for coupling between electron and nuclear motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' To describe photoexcitation within this framework, the Hamilton matrix 𝐻 includes a time-dependent electric field through the interaction 𝑉(𝑡) = −𝝁 ⋅ 𝑬(𝑡), with an external electric field 𝑬(𝑡) and transition dipole 𝝁, where the latter is obtained summing over partial charges on the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We note that both 𝐷 and 𝑉(𝑡) couple electronic states but in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' 𝐷 involves derivative couplings, while  𝑉(𝑡) includes the field-matter couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The force experienced by each atom is computed from the evolved non-equilibrium density matrix 𝜌(𝑡) and is determined by averaging over the distribution of electronic states:31,42 𝑀𝑹̈ = −𝛁-[Tr(𝜌(𝑡)𝐻(𝑡))/Tr(𝜌(𝑡))],                                                                                               (2) where 𝑹 denotes the atomic coordinates and 𝑀 specifies the masses of the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  Atomic charges are calculated using the Mulliken approximation31,42, 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' (𝑡) = Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' [𝜌(𝑡)𝑆], where Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' indicates the trace over the orbitals centered on atom A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  We consider the following initial perturbation (delta function in time) in all possible spatial directions to calculate the real-time absorption spectra: 𝑬(𝑡) = 𝐸&𝛿(𝑡)𝒆,                                                                                                                           (3) with the field amplitude 𝐸&, and a unit vector 𝒆 ∈ {𝑥, 𝑦, 𝑧}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' In the frequency domain (the Fourier transform of 𝑬(𝑡) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3), this perturbation corresponds to a constant value (𝐸&) independent of frequency, and therefore can include all possible frequency components in driving the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  7 From the Fourier transform of the resultant time-dependent dipole along each spatial direction, we compute absorption spectrum as44,45 𝜎(𝜔) = $/0 "1 ℑ[Tr[𝜶(𝜔)]],                                                                                                            (4)  where 𝑐 is the speed of light, and 𝜶 is the polarizability tensor,  𝜶(𝜔) = (𝝁U(𝜔) − 𝝁&)/𝐸&, with the frequency-dependent and initial (before the perturbation) dipoles 𝝁U(𝜔) and 𝝁& respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Alternatively, we compute spectra in the frequency domain using the linear response TDDFTB method through the calculation of oscillator strengths between the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We use equivalent damping factors of the Lorentzian lineshape in the time and frequency domain to obtain the final smoothed spectra in order to compare spectra from both domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  For laser pulse driven molecular dissociation dynamics, we have used an electric field of the following form 𝐸(𝑡) = 𝐸&e)(()(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' )"/5"sin (𝜔&(𝑡 − 𝑡1)),                                                                              (5)  with peak amplitude 𝐸&, driving frequency 𝜔&, peak position of the pulse 𝑡1, and pulse duration 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' During the calculation of nuclear trajectories, we consider different initial vibrational coordinates and momenta sampled from a canonical ensemble and calculate an ensemble of trajectories to generate a distribution of final outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' This type of calculation has not been considered in previous studies, but it is known to provide a more accurate treatment of the dynamics than the zero temperature approach involving a single trajectory, as has been used in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' From the trajectories, we compute dissociation probabilities as a function of time by using a Gaussian distribution along inter-atomic distances to weight the reactive probabilities associated with each trajectory i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=', identifying the fraction of dissociative trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' A Gaussian function centered around 6 Å with a width of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 Å is used, along with the assumption that as soon as the inter- atomic distance passes the center of the Gaussian (6 Å), the probability is kept at the peak value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  8 See the supporting information for the dependence of dissociation probability on Gaussian widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We note that the explicit nuclear motion of all atoms has been included during the trajectory calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  We have performed all the electronic structure calculations and solved the real-time dynamics using the DFTB+ code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='42 A step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='002 fs is used for the time evolution of the density matrix up to 200 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We have checked the propagation convergence using a smaller step size (smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='002 fs) and it turns out that a step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='002 fs is reasonable to keep a balance between the propagation error and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' For pulsed excitations, the laser pulse amplitude is chosen to vanish after 25 fs, but we further propagate the density matrix to observe the after-excitation dynamics, leading to charge transfer and dissociation of the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  Results and Discussion  In this work, we have studied octahedral geometries of Ag and Au nanoparticles (NPs) of various numbers of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We chose octahedrons for this work as this provides a simple motif for varying NP size over a wide range, and octahedron-shaped gold NPs have been studied experimentally,46 having plasmon resonance properties that are similar  (somewhat red-shifted) to what are found for spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Of course, the sharp points should provide hot spots with larger field enhancements than for spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Four different sizes have been used for each of the Ag and Au NPs with the number of atoms 19, 85, 231, and 489 corresponding to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2 nm dimensions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Also, these geometries have been considered in some previous experimental47-50 and theoretical51,52 works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We have optimized each NP geometry using the DFTB method with hyb-0-2 parameters set for Ag NPs and auorg-1-1 parameters set for Au NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='53 A H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' molecule is attached next to the tip of the optimized NP and then we further optimize the full system (NP with molecule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' All the optimized geometries are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='1 where the molecular  9 axis is perpendicular to the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The inter-atomic distance of the hydrogen molecule is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='85 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We note that the gas phase equilibrium bond length of the molecule is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='74 Å but due to the adsorption on the NP surfaces here the bond length is slightly larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The distance between the tip atom and molecule for systems involving Ag (Au) atoms is around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 Å (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='8 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' These optimized system parameters are comparable to previous DFT calculations for a tetrahedral Au6H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='15  Figure 1: 𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' molecule adsorbed on octahedral Ag and Au nanoparticle surfaces of different dimensions determined by the number of atoms 𝑛, (𝑛 = 19, 85, 231, and 489).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' All the geometries are optimized with the DFTB method using the parameters set hyb-0-2 and auorg-1-1 for Ag and Au NPs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' As a test of the RT code, we compute absorption spectra solving both the real-time dynamics (RT-TDDFTB) through the computation of time-dependent dipoles and linear response TDDFTB frequency domain calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' In the linear response limit (at low intensities involving a one photon process), both spectra should match each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' For the time domain spectra, we  AgnH2 00 n= 19 n = 85 n = 489 n = 231 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 nm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7 nm 1 nm AunH2 10 solve the Liouville-von Neumann equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='1) with a delta kick perturbation described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' From the resultant time-dependent dipole along each spatial direction, we calculate absorption spectra through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We set the peak amplitude of the perturbation (𝐸&) to a small value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='001 V/Å which corresponds to a peak intensity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 × 107 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We use the TDDFTB method in the Casida formulation in computing oscillator strengths between ground and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Spectra from both methods agree very well with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Throughout we have used a damping factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='16 eV in computing spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' See the supporting information for the comparison between the time and frequency domain spectra for Ag#8H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' and Au#8H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  Figure 2: Real-time absorption spectra for (a) 𝐴𝑔9𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' and (b) 𝐴𝑢9𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The two frequencies used for the dissociation dynamics are shown by vertical dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Real-time absorption spectra for all systems are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The absorption intensity increases gradually with the size of the NPs as the density of states (DOS) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' For Ag NPs, the LSPR peak appears between [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5, 3] eV and shows a prominent red shift with increasing NPs size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The peak shifts towards lower energy by around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='25 eV when we increase the number of  (a) n= 489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='8 n=231 n=85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='6 n=19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4 [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' units] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2 Absorbance [ - (b) - - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='8 - - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2 0 1 4 2 3 5 6 Energy [eV] 11 atoms from 19 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 nm) to 489 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' This trend was observed in previous studies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='0 nm clusters54 where it was attributed to quantum size effects, and is distinct from the red shift that is seen for much larger octahedral NPs49 which arises from electromagnetic depolarization effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='55 The high energy flat signals (above 3 eV) originate from inter-band transitions i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=', from d to sp band transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' For Au NPs, the plasmon peaks are relatively broader due to the short lifetime of the excited states compared to Ag NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' As a result, it becomes difficult to distinguish the plasmon peaks from high-energy inter-band transitions for the smaller particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The plasmon peak for the Au$:8H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' system arises at around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 eV, which is similar to the plasmon in larger gold NPs with roughly spherical shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='55,56 Here the plasmon peak is more pronounced and is distinct from the high-energy inter-band transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' This also agrees with the previous study by Berdakin and co-workers57 where plasmon peaks start to appear distinct from inter-band transitions for icosahedral Au NPs involving more than 300 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We note that the broad oscillating structures in the Au NP spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2 at energies above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 eV were also observed by Schatz and co-workers58,59 for tetrahedral Au NPs using the TDDFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  We use two different driving frequencies 𝜔& = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7 eV (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 eV) and 𝜔& = 4 eV for Ag (Au) NPs to explore the dissociation dynamics of H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' due to plasmon and inter-band transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' These two frequencies are labelled in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2 with vertical dashed lines and the width around each line is inversely proportional to the pulse duration 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The driving field described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4 is used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We vary the laser peak intensity through the parameter 𝐸&.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We consider the 𝑧-axis as the laser polarization direction for all systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Note that along this direction the tip atom of the cluster points towards the molecule and appears to be more suitable for bond dissociation compared to other possible directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We have included an example pulse and its frequency content (Fourier transform) in the supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  12  Figure 3: Trajectories showing inter-atomic distances and dissociation probabilities for (a-d) 𝐴𝑔9𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' and (e-h) 𝐴𝑢9𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' systems with driving frequencies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 eV, respectively, corresponding to plasmon-driven dissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' From top to bottom, the size of the particle increases, 𝑛 = 19, 85, 231, and 489 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Threshold intensities are given for all plots, denoting intensities where dissociation starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The red lines are the ensemble-averaged dissociation probability (obtained through the Gaussian binning along bond distances) as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Here we have calculated 20 trajectories for each plot and sorted the color code from green to blue for non-dissociative and dissociative trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Laser pulse-driven dissociative trajectories for Ag9H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' (Au9H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=') systems with 𝜔& = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7 eV (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 eV) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 where we have plotted inter-atomic distances 𝑑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  for an ensemble of trajectories and dissociation probabilities (averaging the trajectories) as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Note that these frequencies correspond to plasmon-driven dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We employ different pulse  AgnH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Au,H, 6 I = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 × 1013 W/cm² 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 3 I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='6 × 1013 0 6 Dissociation probability (b) 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 I = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 × 1012 W/cm A I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='6 × 1013 W/cm² 6 I = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 × 1012 W/cm2 I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='1 × 1013 W/cm2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 0 6 h 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 I = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='1 × 1012 W/cm² I = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 × 1012 W/cml 0 0 50 0 100 2000 50 100 150 200 150 Time [fs] 13 intensities for each plot, which are the lowest intensities necessary to initiate dissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We find that the dissociation probability increases linearly with the pulse intensity, from a threshold intensity up to a plateau where the probability is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The linear increase is because more hot carriers are generated with increasing photon flux and a one photon transition is sufficient to enable hot carrier generation leading to dissociation near threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The linear dependence was also observed in the previous experimental study of H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' dissociation on a Au NP surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='12,13 Most of the dissociations take place during interaction with the pulse i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=', within 25 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' At low intensities, dissociation strongly depends on the initial vibrational phases, and therefore only a fraction of trajectories dissociate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' For the remaining trajectories, the inter-atomic distance 𝑑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' shows coherent oscillation around the equilibrium bond length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We note that for most of these non- dissociative trajectories, molecules desorb from the surface very quickly within 20-30 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  At high intensities, the dissociation probability increases as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4(a, b) for the 𝑛 = 231 systems with intensities chosen as the threshold intensity for the 𝑛 = 19 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We found that beyond certain intensities, all the trajectories dissociate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The time-dependent dissociation probability calculated from an ensemble of trajectories through Gaussian binning along inter-atomic distances increases during the pulse interaction and saturates to a finite value within 40 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We note that there are interactions between dissociated atoms and the NPs as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Therefore, the dissociated outgoing trajectories are not completely smooth at the threshold intensities while at high intensities the dissociated atoms gain sufficient velocities to leave the surface as soon as the molecule dissociates, resulting in smoothed dissociative trajectories as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  14  Figure 4: Same as Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3(c, g) but for (a, b) high intensities, and (c, d) 𝜔& = 4 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  Figure 5: Geometries along the trajectories (a single trajectory for each system 𝐴𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='"#𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' or 𝐴𝑢#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='&𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=') at 20, 50, 100, and 200 fs from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3(c, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Here  𝑑<)< = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 Å (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='8 Å) for Ag (Au) NPs at 200 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Trajectories for the Ag!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='"#H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' and Au!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='"#H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' systems with 𝜔& = 4 eV are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4(c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' High-frequency driving shows similar dissociation but involves different intensities than the corresponding low-frequency driving for the dissociation to take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We note that the isolated  (a) Ag231H2 ty 3 I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2 × 1013 W/cm² I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='6 × 1013 W/cm² A 0 H- 6 Hp (b) Au?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Au231H, 3 I = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 × 1013 W/cm² 5 0 0 100 150 2000 100 150 200 0 50 50 Time [fs]20 fs 50 fs 100 fs 200 fs 15 molecule dissociates at even higher intensities than we have presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' This confirms that dissociation is enhanced by the presence of NPs and the enhancement factors strongly depend on the NP size and driving frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  Figure 6:  Threshold intensity (𝐼() as a function of particle size 𝜂 for 𝐴𝑔9𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' and 𝐴𝑢9𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' systems with two different driving frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Table 1: Fitting parameters, 𝑎 and 𝑏, for threshold intensity as a function of particle size, 𝐼( = 𝑎 × 𝜂=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' System          𝜔& [eV]                𝑎                           𝑏 Ag9H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7             6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4 × 10#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='34 4               1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='45 × 10#"              -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='34 Au9H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3             1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 × 10#"               -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='65 4               1 × 10#"                   -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='03  The above calculations are used to determine the threshold intensity for dissociation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=', the minimum intensity required for dissociation to take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Interestingly, the threshold intensity decreases with increasing particle size for both Ag and Au NPs but with different decay rates depending on the driving frequency as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We fit the decay curves with the functional  14 10 13 10 Au = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 eV 12 10 =4 eV eV 11 10 0 5 10 15 20 Nanoparticle size n [nm] 16 form 𝐼( = 𝑎 × 𝜂=, where 𝐼(  is the threshold intensity, 𝜂 is the particle size, and 𝑎 and b are parameters to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The fitting parameters 𝑎 and 𝑏 are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' All the decays are approximated well with the same functional form 𝐼( = 𝑎 × 𝜂=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The  Table shows that  b is negative and hence the threshold intensity decreases with particle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The decay rate is maximum (𝑏 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='34) with minimum amplitude (a = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4 × 10#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=') for Ag9H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' systems with plasmon frequency (𝜔& = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' This conclusion is in agreement with a recent experiment where Ag NPs are found to be more efficient than Au NPs in generating hot electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='24 On the other hand, for Au9H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' systems, the high frequency (𝜔& = 4 eV) driving provides a larger decay rate compared to the low- frequency driving as there is a significant inter-band transition amplitude, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='2b, while the plasmon amplitude is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' For Ag9H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' particles there are dominant plasmonic transitions for 𝑛 ≥ 85 and the plasmon resonance shows larger efficiency for dissociation compared to the inter-band transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' To provide insight about the comparison of calculated and experimental threshold intensities, we extrapolated the plots to large particle sizes up to 20 nm where the intensities are reduced significantly for large |𝑏| values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' For  Ag9H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7 eV (with the maximum dissociation rate), the threshold intensity decreases from 10#" to 10## W/cm2 when we increase the particle size from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5 nm to 20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We note that the peak intensity can be further lowered to 10#& W/cm2 when we scale the pulse duration by a factor of 10 in order to compare with pulses of duration 200 fs used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='41 This is still 2 orders of magnitude larger than the experimental peak amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' However, this can be compensated by including aggregated particles in the study, as this would enable stronger hotspots, as known from surface-enhanced Raman spectroscopy (SERS) studies and electrodynamics theory,60,61 and it would also provide a mechanism for  17 trapping of the dissociating molecule in the hot spot, which would enhance charge transfer efficiency between nanoparticles and molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='36  Figure 7: Trajectories for the molecular charge (𝑄>) for  𝐴𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='"#𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' (𝐴𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='"#𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=') with frequencies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3) eV and 4 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Threshold intensities are used from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3(c, g) for low frequencies and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4(c, d) for high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The same color code is used for respective trajectories compared to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3(c, g) and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='4(c, d) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=', the color code from green to blue represents non-dissociative to dissociative trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  Figure 8: Orbital populations as a function of energy  𝜀 − 𝜀?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The red (blue) color represents population gain (de-population) with positive (negative) amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' The same (threshold) intensities are used as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' (c, f) Density of states (DOS) where magenta lines represent d band only and green shaded areas are for the complete systems (top) 𝐴𝑔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='"#𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' and (bottom) 𝐴𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='"#𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='. Red lines (in the DOS panels) represent the antibonding orbital of 𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='.  Ag231H2 Au231H2 0o = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7 eV 0o = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 eV e Q (b) (d) Wo = 4 eV Wo=4 eV 0 0 50 100 150 200 050 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='150 200 Time [fs]4 (b) 0o = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='7eV Wo = 4 eV 2 0 Tev 2 0o = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 eV (d) (e) Wo = 4 eV 2 0 -2 25 0 50 75 1000 25 50 75 1000 600 DOS Time [fs] [states / eV] 18 Figure 7 displays the time evolution of molecular charge 𝑄> i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=', the combined charge of the two hydrogen atoms as obtained through the Mulliken approximation for the Ag!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='"#H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' and Au!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='"#H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' systems with the threshold intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Initially, the molecule is slightly positive  with 𝑄> ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='05 𝑒 when it is adsorbed on the surface of a NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' After the pulsed excitation, there is an exchange of electrons between two sub-systems and  𝑄> follows the laser frequency oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We note that the total charge is conserved at all times i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=', 𝑄@A + 𝑄>= 0, where 𝑄@A is the NP charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' There is an effective transfer of electrons to the molecule at a certain time after the pulse is turned on (near the peak intensity of the pulses) and the average charge on the molecule becomes negative, corresponding to the net transfer of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Similar to inter-atomic distance trajectories, the electron transfer process also depends on the initial vibrational phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' It is interesting to notice that only dissociative trajectories exhibit substantial electron transfer (the same color code is used as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' This confirms that dissociation is driven by the electron transfer process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  After bond dissociation, one of the dissociated hydrogen atoms sticks to the surface of the NPs (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='5), resulting in negative partial charges on the dissociated atoms at longer times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Electron back transfer from the molecule to NP is also observed for some trajectories when the molecule or dissociated atoms leave the surface of the NP and the final charge at longer times becomes nearly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' In the earlier RT-TDDFT work,36 similar charge transfer dynamics of H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' in Au NPs were also observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Time-dependent orbital populations for the 𝑛 = 231 systems with plasmonic and inter- band transition frequencies are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Here we show population changes compared to the initial populations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=', the populations prior to the pulse interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Due to the interaction with pulses, there is a transfer of populations across the Fermi level (with energy 𝜀?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=') from lower energy to higher energy states covered by the bandwidth of the pulses in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' As  19 expected plasmon excitations populate low energy states near the Fermi level while the inter-band transitions are extended to high energy states with dominant d to sp transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' In comparison to Ag, Au NPs have a broader energy range for the d band that is closer to the Fermi level, which allows significant inter-band transitions to populate states above the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' All the populations show coherent oscillations following the laser frequencies during the pulse, and then after the pulse, the coherence de-phases very quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' These oscillations are known to appear upon laser excitation of NPs which are referred to as “sloshing” and “inversion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='17,62,63 During the pulse interaction (within 25 fs), adequate populations accumulate near the antibonding orbital of the H2 molecule (the red curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' 8c and 8f) which weakens the molecular bond and eventually the molecule dissociates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  Conclusions This work demonstrates the laser pulse driven dissociation of H!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' on octahedral Ag and Au nanoparticle surfaces based on real-time density functional tight binding calculations within the canonical ensemble trajectory approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We have explored the effect of nanoparticle size and driving frequencies on the dissociation intensity threshold, and we have also characterized the influence of pulse width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Calculations are performed for nanoparticles with varying numbers of atoms up to 489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' As expected, a plasmon peak appears for Au nanoparticles only for a large number of atoms (more than 231) whereas Ag nanoparticles show a plasmon peak for much smaller particle sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' In order to investigate plasmon vs inter-band assisted dissociation of molecules, two different driving frequencies are used in the real-time dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Both particle size and driving frequency are found to be crucial in driving dissociation dynamics and hence the intensity threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Interestingly, we found that the intensity threshold decreases with particle size  20 following a  similar functional form irrespective of driving frequencies and systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Using the functional fittings, we scale the results to larger nanoparticle sizes up to 20 nm, for which the largest decay rate is obtained for Ag nanoparticles excited at the plasmon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' In this case, the threshold power is about a factor of 100 larger than has been used in pulsed experiments, which is a result that is consistent with known enhancements that arise when clusters of particles rather than isolated particles are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' For Au nanoparticles, inter-band transitions are more effective than plasmon excitation, a result that has also been seen in experiments,64 and the threshold power is similar to that for Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Additionally, we have calculated the charge on the molecule as a function of time along each trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' After the pulsed excitation, we observe electron transfer from the nanoparticle to the molecule which initiates the dissociation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' We have also analyzed these results in terms of the orbital populations as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='  ASSOCIATED CONTENT The Supporting Information is available free of charge at  AUTHOR INFORMATION Corresponding Author George C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Schatz – Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Email: g-schatz@northwestern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content='edu Author Sajal Kumar Giri – Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States  21  ACKNOWLEDGMENT This research was supported by the Department of Energy, Office of Basic Energy Sciences through grant DE-SC0004752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' 2015, 10, 25-34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' ACS Photonics 2017, 4, 2759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Mechanistic studies of plasmon chemistry on metal catalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' 2019, 58, 4800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' ACS Nano 2016, 10, 6108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Liao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Gu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Yan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Devasia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Heo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Jain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Watching visible light-driven CO2 reduction on a plasmonic nanoparticle catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' ACS Nano 2018, 12, 8330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Swearer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Quantifying hot carrier and thermal contributions in plasmonic photocatalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Science 2018, 362, 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Science 2013, 339, 1590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content='  Santiago, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' 2014, 5, 5788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Pandeya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Guo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Kuda-Singappulige, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Andrade, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Lorenzen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Marques, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content='  29 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Townsend, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Bryant, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Which resonances in small metallic nanoparticles are plasmonic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' 2014, 16, 114022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Interplay between plasmon and single-particle excitations in a metal nanocluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' 2015, 6, 10107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Mosquera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Schatz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' SERS study of the mechanism of plasmon-driven hot electron transfer between gold nanoparticles and PCBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+page_content=' on Ag and Au Nanoparticles: Effect of Size, and Plasmon versus Inter-Band Transitions on Threshold Intensities for Dissociation Sajal Kumar Giri and George C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
+page_content=' Schatz* Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States E-mail: g-schatz@northwestern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtAyT4oBgHgl3EQfmfgA/content/2301.00469v1.pdf'}
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+Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+Neural Network Approximation of Optimal Controls for Stochastic
+Reaction-Diffusion Equations
+W. Stannat,1 A. Vogler,1 and L. Wessels1
+Department of Mathematics, Technical University of Berlin.
+(*Electronic mail: stannat@math.tu-berlin.de, vogler@math.tu-berlin.de, wessels@math.tu-berlin.de.)
+(Dated: 25 January 2023)
+We present a numerical algorithm that allows the approximation of optimal controls for stochastic reaction-diffusion
+equations with additive noise by first reducing the problem to controls of feedback form and then approximating the
+feedback function using finitely based approximations. Using structural assumptions on the finitely based approxima-
+tions, rates for the approximation error of the cost can be obtained. Our algorithm significantly reduces the computa-
+tional complexity of finding controls with asymptotically optimal cost. Numerical experiments using artificial neural
+networks as well as radial basis function networks illustrate the performance of our algorithm. Our approach can
+also be applied to stochastic control problems for high dimensional stochastic differential equations and more general
+stochastic partial differential equations.
+There is a huge body of literature on optimal control prob-
+lems with partial differential equation (PDE) constraints
+and their numerical treatment. Recent years have shown
+a rising interest in the optimal control of stochastic par-
+tial differential equations (SPDEs). However, the numer-
+ical approximation of optimal controls, let alone its prac-
+tical implementation in the stochastic case, faces serious
+obstacles due to the computational complexity of classical
+algorithms. In this work, we present a new numerical al-
+gorithm that approximates feedback controls for SPDEs
+with asymptotically optimal cost. The algorithm is based
+on adjoint calculus applied to gradient descent. For the
+approximation of feedback controls, we use finitely based
+approximations such as artificial neural networks or ra-
+dial basis function networks. The restriction to additive
+noise and our approach for the approximation of feedback
+functions enables us to significantly reduce the algorithmic
+complexity of our approach in comparison with classical
+algorithms.
+I.
+INTRODUCTION
+For a fixed finite time horizon T > 0, we consider the ran-
+domly forced reaction-diffusion equation
+�
+∂tu(t,x) = ∆u(t,x)+ f(u(t,x))+ξ(t,x), (t,x) ∈ [0,T]×Λ
+u(0,x) = u(x),
+x ∈ Λ
+(1)
+on a bounded domain Λ ⊂ Rd with Neumann boundary con-
+ditions. Here, ∆ := ∑d
+k=1
+∂ 2
+∂x2
+k denotes the Laplace operator,
+f : R → R models a local reaction term, and ξ(t,x) are ran-
+dom fluctuations. Deterministic reaction-diffusion equations
+are ubiquitous in the natural sciences and, in many situations,
+taking into account random fluctuations leads to the more re-
+alistic model (1).
+Typically, these random fluctuations are
+highly irregular, and therefore, in order to treat this equation
+rigorously, we reformulate it as the following L2(Λ)-valued
+SPDE:
+�
+dut = [∆ut +F(ut)]dt +σdWt,
+t ∈ [0,T]
+u0 = u ∈ L2(Λ),
+(2)
+where ut(x) = u(t,x) for fixed t ∈ [0,T] is considered as an
+element in L2(Λ). The random fluctuations ξ(t,x) are mod-
+elled by a cylindrical Wiener process (Wt)t∈[0,T] on L2(Λ),
+defined on some underlying probability space (Ω,F,P), and
+σ : L2(Λ) → L2(Λ) is a Hilbert-Schmidt operator. For more
+details on the mathematical theory of SPDEs, see1. Further-
+more, F : L2(Λ) → L2(Λ) denotes the Nemytskii operator as-
+sociated with f, i.e.,
+F(u)(x) := f(u(x)),
+u ∈ L2(Λ), x ∈ Λ.
+The objective of control theory is to achieve a desired out-
+come for a dynamical system by applying an external in-
+put which can be chosen freely among a set of admissible
+inputs.
+In order to set up a mathematical formulation for
+the control of the SPDE (2), we introduce a control process
+g : [0,T] × Ω → U ⊂ L2(Λ), adapted to the filtration gen-
+erated by ut, as a forcing term on the right-hand side of the
+equation
+�
+dug
+t = [∆ug
+t +F(ug
+t )+gt]dt +σdWt,
+t ∈ [0,T]
+ug
+0 = u ∈ L2(Λ),
+(3)
+and define the cost functional
+J(g) := E
+�� T
+0
+�
+Λ l(t,x,ug
+t (x))dx+ ν
+2 ∥gt∥2
+L2(Λ)dt
++
+�
+Λ m(x,ug
+T(x))dx
+�
+.
+(4)
+Here, the running cost l(t,x,u) : [0,T] × Λ × R → R and the
+terminal cost m(x,u) : Λ×R → R are assumed to be differen-
+tiable and Lipschitz continuous in the state variable u. Now,
+the objective is to find a control process g that minimizes the
+cost functional (4) over some set of admissible controls sub-
+ject to (3).
+arXiv:2301.11926v1  [math.OC]  25 Jan 2023
+
+Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+2
+In the deterministic case, there is a huge body of litera-
+ture on necessary and sufficient optimality conditions2–4, as
+well as numerical algorithms which efficiently approximate
+optimal controls, see e.g.5–8. In recent years, extensions to
+the stochastic case have seen a rising interest in the math-
+ematical literature leading to necessary and sufficient opti-
+mality conditions in great generality9–21. However, classical
+algorithms for the numerical approximation of optimal con-
+trols in the stochastic case either require the approximation
+of backward SPDEs or the approximation of infinite dimen-
+sional Hamilton-Jacobi-Bellman (HJB) equations. Due to the
+curse of dimensionality, both of these alternatives are compu-
+tationally very expensive, leading to an increased interest in
+the development of new, more efficient algorithms22–31.
+A considerable reduction of the problem can be achieved in
+the case when an optimal control g∗ is of feedback-type, i.e.,
+g∗
+t = G∗(t,uG∗
+t ),
+(5)
+for some G∗ : [0,T] × L2(Λ) → L2(Λ). In this work, we con-
+sider a mathematical setting in which the optimal control is
+indeed of the above type. In a first step we then use finitely
+based approximations for G∗ in order to approximate the op-
+timal control g∗. This approach together with the restriction
+to additive noise enables us to replace the backward SPDE
+arising in adjoint calculus by a random backward PDE which
+significantly reduces its computational complexity. A similar
+idea was already used in32.
+The reduction to finitely based feedback controls then al-
+lows in a second step the local uniform approximation of
+finitely based feedback controls in appropriate ansatz spaces
+of variable, but finite, dimension. Solving the corresponding
+finite dimensional optimal control problem, e.g. with artifi-
+cial neural networks, leads to an efficient computation of fi-
+nite dimensional controls, whose costs approximate the opti-
+mal cost with increasing dimension, see Theorem 4. Impos-
+ing additional smoothness assumptions on the optimal control
+g∗
+t = G∗(t,uG∗
+t ), in particular G∗ globally Lipschitz, also en-
+ables us to derive rates for the approximation error in ansatz
+spaces consisting of Lipschitz continuous feedback functions,
+see Theorem 5. We discuss the example of neural network
+approximation in detail; see Theorem 8 for the universal ap-
+proximation of finitely based feedback controls with 1-layer
+artificial neural networks and Proposition 9 for rates on the
+approximation error.
+The practical implementation of the gradient descent algo-
+rithm for the approximation of finite dimensional feedback
+controls also requires the numerical discretization of the con-
+trolled state equation. In Proposition 11, we derive the com-
+bined approximation error for the optimal cost.
+Detailed
+proofs of our results, that also hold for more general ansatz
+spaces beyond artificial neural networks, can be found in33.
+The performance of our gradient descent algorithm is illus-
+trated with three examples. The first example deals with the
+validation of our algorithm in the case of linear quadratic con-
+trol. The other two examples consider the problem of stabi-
+lizing a bump solution of the stochastic Nagumo equation, in
+one case with general feedback controls, in the other case with
+feedback controls of Nemytskii-type.
+The remainder of the paper is organized as follows: First,
+in Section II, we show that it is sufficient to consider feed-
+back controls. In Section III, we explain how to approximate
+the optimal control by introducing ansatz spaces of controls
+that are suited for the numerical implementation. Section IV
+contains our main approximation results. In Section V, we
+describe our gradient descent algorithm. Next, in Section VI,
+we discuss the explicit example of artificial neural networks
+for the finite dimensional ansatz spaces. In Section VII, we
+investigate the error resulting from the numerical discretiza-
+tion. Finally, in Section VIII, we present numerical experi-
+ments using artificial neural networks and radial basis func-
+tion networks.
+II.
+OPTIMAL CONTROLS OF FEEDBACK-TYPE
+In order to reduce the complexity of the problem, we as-
+sume that there exists an optimal control g∗ in feedback form
+(5), for some continuous G∗ : [0,T]×L2(Λ) → L2(Λ) that sat-
+isfies a linear growth condition. This can be achieved, using
+the solution of the associated HJB equation
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂tV + 1
+2tr(σ∗D2Vσ)+⟨DV,∆u+F(u)⟩L2(Λ)
++
+�
+Λ l(t,x,u(x))dx+infG∈U {⟨DV,G⟩+ ν
+2 ∥G∥2
+L2(Λ)} = 0,
+(t,u) ∈ [0,T]×L2(Λ)
+V(T,u) =
+�
+Λ m(x,u(x))dx,
+u ∈ L2(Λ).
+(6)
+For solution theories regarding equations of this type, see11.
+In particular, if (6) has a unique mild solution satisfying cer-
+tain regularity assumptions, and
+γ(p) := arginfG∈U
+�
+⟨p,G⟩+ ν
+2 ∥G∥2
+L2(Λ)
+�
+is continuous, then
+g∗
+t := G∗(t,uG∗
+t ) := γ(DV(t,uG∗
+t )).
+is an optimal control in feedback form, provided that uG∗ is a
+unique strong solution of the closed loop equation
+�
+duG
+t = [∆uG
+t +F(uG
+t )+G(t,uG
+t )]dt +σdWt, t ∈ [0,T]
+uG
+0 = u ∈ L2(Λ)
+(7)
+with G = G∗. A sufficient condition for (7) to have a unique
+strong solution is that G∗ is Lipschitz continuous in u, see
+e.g.1. This is in particular the case if the solution V of the
+HJB equation (6) has a bounded second derivative in u, see
+Theorem 4.155 and Remark 4.202 in11. However, directly
+tackling the optimal control problem by approximating the so-
+lution of the HJB equation (6) numerically is very challenging
+due to the infinite dimensionality of the domain of V. Instead,
+our approach is to approximate the feedback function directly.
+Therefore we consider the following feedback control prob-
+
+Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+3
+lem: Minimize
+J(G) := E
+�� T
+0
+�
+Λ l(t,x,uG
+t (x))dx+ ν
+2 ∥G(t,uG
+t )∥2
+L2(Λ)dt
++
+�
+Λ m(x,uG
+T (x))dx
+�
+subject to equation (7). We seek to minimize J over the set of
+admissible controls
+Uad := {G : [0,T]×L2(Λ) → U |G(·,uG
+· ) ∈ A},
+where A = L2([0,T]×Ω;L2(Λ)).
+Example 1 (Linear Quadratic Control) Let us consider the
+linear quadratic control problem
+�
+dug
+t = [∆ug
+t +gt]dt +σdWt,
+t ∈ [0,T]
+ug
+0 = u ∈ L2(Λ),
+where g : [0,T]×Ω → L2(Λ) is an adapted process, and
+J(g) := 1
+2E
+�� T
+0
+�
+Λ(ug
+t (x))2 +g2
+t (x)dxdt +
+�
+Λ(ug
+T(x))2dx
+�
+.
+In this case, the optimal control g∗ is indeed of feedback form,
+given by
+g∗
+t = P(t)ug∗
+t
+where P : [0,T] → L(L2(Λ)) is the solution of the Riccati
+equation34
+�
+∂tP(t)+P(t)∆+∆P(t)−Id+P2(t) = 0, t ∈ [0,T]
+P(T) = −Id.
+(8)
+III.
+CONSTRUCTION OF ANSATZ SPACES
+For the efficient numerical implementation we need to re-
+strict ourselves to a subset of controls U ⊂Uad that are suitable
+for the implementation in a gradient descent algorithm. How-
+ever, we need to ensure that when minimizing over U, we do
+not end up much worse than in the original control problem.
+In this section we will provide a method to construct suitable
+ansatz spaces U when we do not have any particular control
+constraints, i.e. U = L2(Λ).
+In order to construct a suitable space U, we introduce
+the so-called finitely based approximation of a function G :
+[0,T] × L2(Λ) → L2(Λ). To this end, we consider finite di-
+mensional subspaces
+Sn ⊂ L2(Λ)
+with orthonormal basis e0,...,en and orthogonal projections
+Pn : L2(Λ) → Sn, such that
+∥Pnu−u∥L2(Λ) → 0.
+The finitely based approximations of G with respect to Sn are
+then defined by
+Gn(t,u) := PnG(t,Pnu)
+(9)
+=
+n
+∑
+k=0
+⟨G(t,
+n
+∑
+j=0
+ujej),ek⟩L2(Λ)ek,
+where uj := ⟨u,ej⟩L2(Λ). One possible choice for the finite
+dimensional subspaces in the case of Λ = (0,1) is
+Sn := span
+�
+1,
+√
+2cos(kπ ·)
+���k = 1,...,n
+�
+.
+In this case the finitely based approximation of G is given by
+Gn(t,u)(x)
+:=
+n
+∑
+k=0
+2⟨G(t,
+n
+∑
+j=0
+uj
+√
+2cos( jπ ·)),cos(kπ ·)⟩L2(0,1) cos(kπx),
+for u = ∑∞
+j=0 uj cos( jπ ·) ∈ L2(Λ).
+Our main results in Section IV provide approximation re-
+sults for ansatz spaces U ⊂ Uad. The main assumption on the
+ansatz spaces U ⊂ Uad is the following approximation prop-
+erty with respect to the optimal feedback G∗:
+Definition 2 Let G : [0,T] × L2(Λ) → L2(Λ). We say that a
+subset U ⊂ Uad satisfies the universal approximation property
+with respect to G, if there exists a sequence (Gn,m)n,m∈N ⊂ U
+that satisfies the following linear growth condition uniformly
+in m
+∥Gn,m(t,u)∥L2(Λ) ≤ Cn(1+∥u∥L2(Λ)),
+for some constant Cn > 0, such that for any R > 0
+lim
+m→∞
+sup
+(t,u)∈[0,T]×BL2(Λ)(0,R)
+∥Gn(t,u)−Gn,m(t,u)∥2
+L2(Λ) = 0,
+where
+BL2(Λ)(0,R) := {u ∈ L2(Λ)|∥u∥L2(Λ) ≤ R}
+and Gn is given as in (9).
+We would like to stress that in the above definition we only
+require that the finitely based approximations of G can be ap-
+proximated uniformly on bounded sets, and not G itself.
+In the first part of this section we will explain how to con-
+struct ansatz spaces U that satisfies the universal approxima-
+tion property with respect to the optimal feedback G∗. These
+ansatz spaces consider finitely based controls of arbitrary di-
+mension n ∈ N, which are of the type
+G(t,u) =
+n
+∑
+k=0
+ψk(t,Pnu)ek,
+for some functions ψk : [0,T] × Sn → R. However, in prac-
+tice the dimension n ∈ N of the ansatz space need to be fixed
+
+Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+4
+a priori and one is interested in how close one can get to the
+optimal cost. Our second main result, Theorem 5, provides
+explicit convergence rates, however, we need to strengthen
+the assumption on our ansatz spaces. In the second part of
+this section we will explain how to construct for any n ∈ N a
+sequence of ansatz spaces (Un,m)m∈N ⊂ Uad that satisfies the
+following approximation property with respect to the optimal
+feedback G∗:
+Definition 3 Let G : [0,T] × L2(Λ) → L2(Λ) and n ∈ N. We
+say that a sequence of subsets (Un,m)m∈N ⊂ Uad satisfies the
+Lipschitz approximation property with respect to G in dimen-
+sion n, if there exists a sequence of Lipschitz continuous con-
+trols (Gn,m)m∈N with Lipschitz constants independent of m,
+such that Gn,m ∈ Un,m, and a sequence of radii (Rn
+m)m∈N with
+limm→∞ Rn
+m = ∞, such that
+εn
+m :=
+sup
+(t,u)∈[0,T]×BL2(Λ)(0,Rnm)
+∥Pn(Gn(t,u)−Gn,m(t,u))∥2
+L2(Λ)
+→ 0,
+(10)
+as m → ∞.
+For this type of ansatz spaces, Theorem 5 provides error esti-
+mates for
+| inf
+g∈Un,m J(g)− inf
+g∈AJ(g)|
+in terms of εn
+m and the projection error γn (11).
+A.
+Universal Ansatz Spaces
+We start by constructing an ansatz space that satisfies a uni-
+versal approximation property with respect to G∗. For n ∈ N,
+we consider the function gn : [0,T]×Rn → Rn given by
+gn
+k(t,u1,...,un) = ⟨G∗(t,
+n
+∑
+j=1
+u je j),ek⟩L2(Λ),
+k = 1,...,n.
+Since G∗ is continuous, the functions (gn)n∈N are also contin-
+uous. In particular, it is possible to approximate these func-
+tions by simpler functions that can be treated numerically, e.g.,
+artificial neural networks. In the following we consider for
+any n ∈ N a set N n of Lipschitz continuous approximations,
+such that for all R > 0 there exists a sequence (ψn
+m)m∈N ∈ N n
+with
+sup
+(t,x)∈[0,T]×BRn(0,R)
+|ψn
+m(t,x)−gn(t,x)|2 → 0,
+m → ∞,
+where
+BRn(0,R) := {x ∈ Rn||x| ≤ R}.
+For a particular choice of N n we refer to our examples in
+Section VI. Then we define the ansatz space
+U :=
+�
+G(t,
+∞
+∑
+j=0
+ujej) :=
+n
+∑
+k=0
+ψk(t,ηl(u1,...,un))ek
+���ψ ∈ N n,n,l ∈ N
+�
+,
+where ηl : Rn → Rn
+ηl(x) =
+�
+x
+|x| ≤ l
+l x
+|x|
+|x| > l
+is a smooth cutoff function. It is not difficult to see that for
+any n ∈ N and R > 0 there exists a sequence (Gn,m)m∈N ⊂ U
+that satisfies a linear growth condition of the type
+∥Gn,m(t,u)∥L2(Λ) ≤ Cn(1+∥u∥L2(Λ)),
+such that for any n ∈ N
+lim
+m→∞
+sup
+(t,u)∈[0,T]×BL2(Λ)(0,R)
+∥Gn(t,u)−Gn,m(t,u)∥2
+L2(Λ) = 0,
+where
+BL2(Λ)(0,R) := {u ∈ L2(Λ)|∥u∥L2(Λ) ≤ R}.
+In particular U satisfies a universal approximation property
+with respect to G∗. We will give a short proof for this in Sec-
+tion VI where we consider artificial neural networks, mapping
+from Rn+1 to Rn, as an explicit example for Lipschitz contin-
+uous approximations N n.
+B.
+Finite Dimensional Ansatz Spaces
+Our second main result, Theorem 5, requires that the con-
+trols of our ansatz space take values in the Sobolev space
+H1(Λ). Therefore we strengthen our assumption on the finite
+dimensional subspaces and assume that
+Sn ⊂ H1(Λ).
+In order to obtain convergence rates, we also need to specify
+the rate of convergence of the orthogonal projections. There-
+fore, we assume that
+∥Pnu−u∥L2(Λ) ≤ γn∥u∥H1(Λ),
+(11)
+for some γn → 0, as n → ∞. Furthermore, we assume that there
+exists an optimal control g∗ in feedback form with a Lipschitz
+continuous feedback function G∗. A simple example for such
+a situation is the linear quadratic case discussed in Example 1.
+Since G∗ is Lipschitz continuous, the functions gn are also
+Lipschitz continuous. As we will see in the examples in Sec-
+tion VI, it is therefore possible to approximate gn by artificial
+neural networks that have uniformly bounded Lipschitz con-
+stants. For any n ∈ N, let (N n,m)m∈N be a sequence of sets of
+Lipschitz continuous approximations, such that for all m ∈ N
+there exists a ψn,m ∈ N n,m with Lipschitz constant indepen-
+dent of m and
+sup
+(t,x)∈[0,T]×BRn(0,Rnm)
+|ψn,m(t,x)−gn(t,x)|2 =: εn
+m → 0,
+as m → ∞, for some sequence (Rn
+m)m∈N of radii with Rn
+m → ∞,
+as m → ∞. Then we define the sequence of ansatz spaces of
+
+Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+5
+dimension n ∈ N by
+Un,m :=
+�
+G(t,
+∞
+∑
+j=0
+ujej) :=
+n
+∑
+k=0
+ψk(t,u1,...,un)ek
+���ψ ∈ N n,m�
+.
+It is not difficult to observe that for any m ∈ N, there exists a
+Lipschitz continuous control Gn,m ∈ Un,m with Lipschitz con-
+stant independent of m, such that
+sup
+(t,u)∈[0,T]×BL2(Λ)(0,Rnm)
+∥Pn(Gn,m(t,u)−Gn(t,u))∥2
+L2(Λ) → 0.
+In particular, for any n ∈ N, the sequence (Un,m)m∈N satisfies
+the Lipschitz approximation property with respect to G∗ in
+dimension n.
+IV.
+MAIN APPROXIMATION RESULTS
+Our first main result, Theorem 4, provides an approxima-
+tion result for universal ansatz spaces of the type constructed
+in Section III. In the second main result, Theorem 5, we pro-
+vide error estimates for ansatz spaces of fixed dimension.
+A.
+Universal Approximation
+The following result shows that we can reach the optimal
+cost of the control problem when we consider ansatz spaces
+of the type constructed in Section III.
+Theorem 4 Assume that U ⊂ Uad satisfies the universal ap-
+proximation property with respect to G∗. Then
+inf
+g∈AJ(g) = inf
+G∈UJ(G).
+B.
+Approximation with Ansatz Spaces of Fixed Dimensions
+In Section VI, we construct as a particular example our
+ansatz space using artificial neural networks. By Theorem 4
+we can get arbitrarily close to the optimal cost of the original
+control problem. However, at this point we do not know how
+close we can get to the optimal cost using ansatz spaces of a
+fixed size. Our second main result, Theorem 5, provides error
+estimates for the convergence of the cost of our approximation
+and the optimal cost. Similar to Section III B, we consider fi-
+nite dimensional subspaces
+Sn ⊂ H1(Λ).
+and assume that for any u ∈ H1(Λ) the orthogonal projections
+satisfy
+∥Pnu−u∥L2(Λ) ≤ γn∥u∥H1(Λ),
+for some γn → 0, as n → ∞. Furthermore we assume that
+there exists an optimal control g∗ in feedback form with a Lip-
+schitz continuous feedback function G∗, as already mentioned
+in Section III B.
+Theorem 5 Let n ∈ N. Let (Un,m)m∈N be a sequence of sub-
+sets of Uad that satisfies the Lipschitz approximation property
+with respect to G∗ in dimension n. Then it holds
+inf
+G∈Un,m J(G)− inf
+g∈AJ(g)
+≤ C
+�
+1+
+�
+E
+�� T
+0 ∥g∗t ∥2
+H1(Λ)dt
+��
+γn +Cn
+�
+εnm,
+for some universal constant C independent of n and m and
+some constant Cn which is independent of m. Here εn
+m is given
+by (10).
+V.
+GRADIENT DESCENT ALGORITHM
+In this section, we describe our gradient descent algorithm.
+Theorem 4 enables us to consider the approximating optimal
+control problem of minimizing optimal costs on a finite di-
+mensional ansatz space Un,m. We assume that
+Un,m =
+�
+Φ(·,·,α)
+���α ∈ Rdm
+�
+for a parametrization Φ : [0,T]×L2(Λ)×Rdm → L2(Λ). Re-
+placing
+G(t,u) = Φ(t,u,α)
+for some given α ∈ Rdm leads to the state equation
+�
+duα
+t = [∆uα
+t +F(uα
+t )+Φ(t,uα
+t ,α)]dt +σdWt
+uα
+0 = u ∈ L2(Λ),
+(12)
+and the cost functional J : Rdm → R,
+J(α) = E
+�� T
+0
+�
+Λ l(t,x,uα
+t (x))dx+ ν
+2 ∥Φ(t,uα
+t ,α)∥2
+L2(Λ)dt
++
+�
+Λ m(x,uα
+T (x))dx
+�
+.
+Using this parametrization, we obtain under suitable regular-
+ity assumptions on Φ the following representation of the gra-
+dient of the cost functional:
+∇J(α)
+= E
+�� T
+0 νΦ∗
+α(t,uα
+t ,α)Φ(t,uα
+t ,α)+Φ∗
+α(t,uα
+t ,α)ptdt
+�
+,
+(13)
+where p is the solution of the so-called adjoint equation
+�
+�
+�
+�
+�
+dpt = −[(∆+F ′(uα
+t )+Φ∗
+u(t,uα
+t ,α))pt +L ′(t,uα
+t )
++νΦ∗
+u(t,uα
+t ,α)Φ(t,uα
+t ,α)]dt
+pT = M ′(uα
+T ).
+(14)
+
+Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+6
+Here
+L (t,u)(x) := l(t,x,u(x)),
+u ∈ L2(Λ),(t,x) ∈ [0,T]×Λ
+denotes the Nemytskii operator associated with l, and
+M (u)(x) := m(x,u(x)),
+u ∈ L2(Λ),x ∈ Λ
+denotes the Nemytskii operator associated with m. Further-
+more, Φα(t,u,α) : Rdm → L2(Λ) (resp. Φu(t,u,α) : L2(Λ) →
+L2(Λ)) denotes the derivative of Φ with respect to α (resp.
+u), and Φ∗
+α(t,u,α) (resp. Φ∗
+u(t,u,α)) denotes the adjoint of
+Φα(t,u,α) (resp. Φu(t,u,α)). For a derivation of the gradi-
+ent, see the supplemental material.
+Note that equation (14) is a linear backward PDE with ran-
+dom coefficients which are given by the state equation (12).
+For details concerning the numerical implementation, see the
+supplemental material.
+Example 6 (Artificial Neural Network) In the case of an ar-
+tificial neural network with activator function θ, the controls
+in the ansatz space Un,m can be parametrized as
+Φ(t,u,α) =
+n
+∑
+i=1
+�
+Cθ
+�
+A
+�
+t
+πnu
+�
++a
+�
+,ei
+�
+ei,
+where α = (A,a,C) consists of A ∈ Rk×(n+1), a ∈ Rk, and C ∈
+Rn×k, for respective dimensions n = nm and k = km, and πnu =
+(⟨u,e1⟩,...,⟨u,en⟩)⊤.
+Based on this representation, we implement the following
+algorithm:
+Algorithm 7 Fix an initial control α0, a stopping criterion
+ρ > 0, and a step size s > 0.
+1. Solve the state equation (12) for one realization of the
+noise.
+2. Solve the adjoint equation (14) with the data given by
+the sample calculated in Step 1.
+3. Repeat Step 1 and Step 2 to approximate the gradient
+(13) using Monte Carlo approximation.
+4. Compute new control via αn+1 = αn −s∇J(αn).
+5. Stop if ∥∇J(αn)∥ < ρ.
+VI.
+ARTIFICIAL NEURAL NETWORKS AS ANSATZ
+SPACES
+In this section, we discuss explicit examples for ansatz
+spaces U ⊂ Uad that satisfy the assumptions of our main re-
+sults, Theorem 4 and Theorem 5, and are suitable for the
+numerical implementation of our gradient descent algorithm.
+In the first part of this section, we focus on universal spaces
+and Theorem 4. The second part is devoted to ansatz spaces
+of fixed size and Theorem 5 with corresponding convergence
+rates. In the whole section, we consider the finite dimensional
+subspaces
+Sn := span
+�
+1,
+√
+2cos(kπ ·)
+���k = 1,...,n
+�
+⊂ H1(0,1),
+with orthonormal basis in L2(Λ)
+e0 = 1,
+ek =
+√
+2cos(kπ ·),
+k = 1,...,n.
+In particular we have for any u ∈ H1(0,1)
+∥Pnu−u∥2
+L2(0,1) =
+∞
+∑
+k=n+1
+|⟨u,
+√
+2cos(kπ ·)⟩L2(0,1)|2
+≤
+1
+n2π2
+∞
+∑
+k=n+1
+|⟨∆1/2u,
+√
+2cos(kπ ·)⟩L2(0,1)|2
+≤
+1
+n2π2 ∥u∥2
+H1(0,1) =: γn∥u∥2
+H1(0,1),
+where we used that the k-th eigenvalue of the Neumann
+Laplace operator with respect to the eigenfunction ek is given
+by −
+1
+k2π2 .
+A.
+Neural Network Approximation
+Regarding Theorem 4, we will show in our first example
+that it is indeed sufficient to consider the type of ansatz space
+constructed in the first part of Section III, using 1-layer arti-
+ficial neural networks for the approximating sets, to get arbi-
+trarily close to the optimal cost. More precisely, we show that
+the set
+U :=
+�
+G(t,
+∞
+∑
+i=1
+uiei)(x) =
+n
+∑
+i=1
+ψi(t,ηl(u1,...,un))ei(x)
+����ψ ∈ N n,n,l ∈ N
+�
+satisfies the universal approximation property with respect to
+G∗, where
+N n :=
+∞
+�
+k=1
+N n
+k
+and
+N n
+k :=
+�
+ψ(x) = Cθ(Ax+a)
+��A ∈ Rk×n,a ∈ Rk,C ∈ Rn×k�
+denotes the set of all 1-layer artificial neural networks from Rn
+to Rn with k neurons, for a given non-polynomial, Lipschitz
+continuous activator function θ.
+To this end, we recall the following classical universal ap-
+proximation result by35:
+
+Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+7
+Theorem 8 Let
+θ ∈ C (R),
+then
+we
+define
+for
+u =
+(u1,...,ud) ∈ Rd
+θ(u)i := θ(ui).
+If θ is not polynomial, then for any n,m ∈ N, compact set K ⊂
+Rn, h ∈ C (K,Rm) and ε > 0, there exists k ∈ N,A ∈ Rk×n,a ∈
+Rk,C ∈ Rm×k, such that
+sup
+u∈K
+|h(u)−ψ(u)| < ε,
+where ψ is the 1-layer artificial neural network
+ψ(u) := Cθ(Au+a).
+Let n ∈ N and recall from Section III the finitely based ap-
+proximation of G∗
+gn : [0,T]×Rn → Rn
+gn
+i (t,u) := ⟨G∗(t,
+n
+∑
+j=1
+ujej),ei⟩,
+i = 1,...,n.
+Since gn is continuous, there exists for any m ∈ N a 1-layer
+artificial neural network ψn,m : [0,T]×Rn → Rn, such that
+sup
+(t,u)∈[0,T]×BRn(0,m)
+|gn(t,u)−ψn,m(t,u)| < 1
+m.
+Since G∗ satisfies a linear growth condition, i.e.,
+∥G∗(t,u)∥L2(Λ) ≤ C(1+∥u∥L2(Λ)),
+for some C > 0, we have
+|gn(t,u)| = ∥PnG∗(t,
+n
+∑
+i=1
+uiei)∥L2(Λ)
+≤ ∥G∗(t,
+n
+∑
+i=1
+uiei)∥L2(Λ)
+≤ C(1+|u|),
+where u = (u1,...,un). Hence, if we consider the continuous
+function ηm : Rn → Rn
+ηm(x) =
+�
+x
+|x| ≤ m
+m x
+|x|
+|x| > m
+and define
+˜ψn,m(t,u) := ψn,m(t,η(u)),
+then clearly ˜ψn,m = ψn,m on [0,T] × BRn(0,m) and for any
+(t,u) ∈ [0,T]×BRn(0,m)
+|ψn,m(t,u)| ≤ |gn(t,u)|+|ψn,m(t,u)−gn(t,u)|
+≤ C(1+|u|)+1.
+Furthermore, on [0,T]×BRn(0,m)c we have
+| ˜ψn,m(t,u)| = |ψn,m(t,m u
+|u|)|
+≤ |gn(t,m u
+|u|)|+|ψn,m(t,m u
+|u|)−gn(t,m u
+|u|)|
+≤ C(1+m)+1
+≤ C(1+|u|)+1.
+Therefore ˜ψn,m satisfies a linear growth condition with some
+constant C > 0 independent of m. Now, we define
+Gn,m(t,u) :=
+n
+∑
+i=1
+˜ψn,m
+i
+(t,⟨u,e1⟩L2(Λ),...,⟨u,en⟩L2(Λ)))ei.
+One can easily check that all the assumptions of Theorem 5
+are satisfied for U. Indeed, due to the Lipschitz continuity of
+the elements in N n
+k , any control in U is indeed admissible,
+i.e., U ⊂ Uad. Furthermore, (Gn,m)m∈N is a sequence in U that
+satisfies a linear growth condition with some constant C >
+0 independent of m and approximates any finitely based Gn.
+Indeed, for any R > 0 and any ε > 0 there exists an M ∈ N,
+such that BL2(Λ)(0,R) ⊂ BL2(Λ)(0,m) and 1
+m < ε for every
+m ≥ M. Therefore, we have for any m ≥ M
+sup
+(t,u)∈[0,T]×BL2(Λ)(0,R)
+∥Gn,m(t,u)−Gn(t,u)∥2
+≤
+sup
+(t,u)∈[0,T]×BL2(Λ)(0,m)
+∥Gn,m(t,u)−Gn(t,u)∥2
+≤
+sup
+(t,u)∈[0,T]×BRn(0,m)
+| ˜ψn,m(t,u)−gn(t,u)|2
+≤
+sup
+(t,u)∈[0,T]×BRn(0,m)
+|ψn,m(t,u)−gn(t,u)|2
+< 1
+m < ε.
+In the case of bounded controls, for example if U =
+BL2(Λ)(0,R), we could consider the ansatz space
+U :=
+�
+G(t,
+∞
+∑
+i=1
+uiei)(x) =
+n
+∑
+i=1
+ψi(t,ηl(u1,...,un))ei(x)
+����∥G(t,u)∥L2(Λ) ≤ R where ψ ∈ N n,n,l ∈ N
+�
+and the sequence
+Gn,m(t,u)
+:= (1−
+1
+m(R+1))
+n
+∑
+i=1
+˜ψn,m
+i
+(t,⟨u,e1⟩L2(Λ),...,⟨u,en⟩L2(Λ)))ei.
+
+Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+8
+B.
+Convergence Rates
+Next, we provide explicit convergence rates for the se-
+quence of approximating spaces
+Un,m :=
+�
+G(t,
+∞
+∑
+i=1
+uiei)(x) =
+n
+∑
+i=1
+ψi(t,(u1,...,un))ei(x)
+���ψ ∈ N n
+m
+�
+using Theorem 5. We will mainly follow the ideas of36. In
+the following we consider a 2π-periodic activator function
+θ : R → R that satisfies ˆθ :=
+� π
+−π θ(x)e−ixdx ̸= 0. Recall the
+following result from36:
+Proposition 9 Let
+K,L1,L2
+> 0
+and
+h : [0,T] ×
+BRn+1(0,R) → Rn be Lipschitz continuous in (t,x) with
+∥h∥C 0([0,T]×BRn+1(0,R)) ≤ K and Lipschitz constant bounded
+by L1. Furthermore we assume that h is twice differentiable
+in x and ∂xih is Lipschitz continuous with Lipschitz constant
+bounded by L2. Then there exists a constant C > 0 depending
+only on the above constants, on n,T and on the activator
+function θ through ˆθ, ∥θ ′∥C 0, ∥θ ′′∥C 0 and ∥θ ′′′∥C 0, and
+there exists a constant m0 depending only on n with the
+following property.
+For every R > 0 and every min > m0,
+there exists a one-hidden layer artificial neural network
+ψh ∈ N n
+min such that
+∥h−ψh∥C 0([0,T]×BRn(0,R);Rn) ≤ C(1+R)m−1/(2(n+1))
+in
+and such that the Lipschitz constants of ψh,∂xψh are at most
+C(1+Rm−1/(2(n+1))
+in
+).
+If we now set Rn(m) := m1/(3(n+1)), then by Proposition 9
+there exists for any sufficiently large m ∈ N a network ψn,m ∈
+N n
+m , such that for
+Gn,m(t,
+∞
+∑
+i=1
+uiei)(x) =
+n
+∑
+i=1
+ψn,m
+i
+(t,(u1,...,un))ei(x),
+it holds
+sup
+(t,x)∈[0,T]×BL2(Λ)(0,Rn(m))
+∥Pn(Gn,m(t,x)−Gn(t,x))∥2
+L2(Λ)
+≤ m−1/(3(n+1)) → 0.
+Furthermore Gn,m ∈ Un,m is Lipschitz continuous with some
+Lipschitz constant independent of n. Therefore, applying The-
+orem 5 yields
+inf
+G∈Un,m J(G)− inf
+g∈AJ(g)
+≤C
+�
+1+
+�
+E
+�� T
+0 ∥g∗t ∥2
+H1(Λ)dt
+��
+1
+n2π2 +Cn
+�
+m−1/(3(n+1)).
+VII.
+NUMERICAL DISCRETIZATION OF THE CONTROL
+PROBLEM
+In order to implement a numerical algorithm, the control
+problem needs to be discretized. In this section, we introduce
+the discretized version of our control problem and provide a
+bound for the error resulting from the numerical discretiza-
+tion.
+Problem 10 (Discretized Control Problem) Let
+n ∈ N.
+Minimize
+Jn(G) := E
+�� T
+0
+�
+Λ l(t,x,uG,n
+t
+(x))dx+ ν
+2 ∥PnG(t,uG,n
+t
+)∥2
+L2(Λ)dt
+�
++E
+��
+Λ m(x,uG,n
+T (x))dx
+�
+over the set Uad subject to the discretized SPDE
+�
+duG,n
+t
+= [−∆nuG,n
+t
++PnF(uG,n
+t
+)+PnG(t,uG,n
+t
+)]dt +σPndWt
+uG,n
+0
+= Pnu ∈ Sn,
+where ∆nxn is defined as the unique element in Sn with
+⟨∇xn,∇yn⟩L2(Λ) = ⟨∆nxn,yn⟩L2(Λ)
+∀yn ∈ Sn.
+Proposition 11 Under the same assumptions as in Theorem
+5, it holds
+inf
+G∈Un,m Jn(G)− inf
+g∈AJ(g)
+≤ C
+�
+1+
+�
+E
+�� T
+0 ∥g∗t ∥2
+H1(Λ)dt
+��
+γn +Cn
+�
+εnm,
+for some constant Cn > 0 that only depends on n ∈ N and on
+the Lipschitz constant of G∗.
+Remark 12 Under additional assumptions, in particular con-
+vexity of the Hamiltonian, one can show that for any U ⊂ Uad,
+it holds
+inf
+g∈AJ(g)− inf
+G∈UJn(G) ≤ Cγn,
+for some constant C > 0 which is independent of n.
+VIII.
+SIMULATIONS
+In this whole section, we consider equation (2) on some
+interval Λ = (0,L) of length L. For the approximation we
+consider the Galerkin finite dimensional subspace
+Sn := span
+�
+1,
+�
+2
+L cos
+� k
+Lπ ·
+������k = 1,...,n
+�
+⊂ H1(0,L),
+with orthonormal basis
+e0 =
+�
+1
+L,
+ek =
+�
+2
+L cos
+� k
+Lπ ·
+�
+,
+k = 1,...,n,
+
+Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+9
+or the finite element subspace
+Sn := span{ek|k = 1,...,n} ⊂ H1(0,L),
+with basis
+ek(x) =
+�
+�
+�
+�
+�
+n
+�
+x− k−1
+n L
+�
+if x ∈
+� k−1
+n L, k
+nL
+�
+n
+� k+1
+n L−x
+�
+if x ∈
+� k
+nL, k+1
+n L
+�
+0
+otherwise
+for k = 1,...,n.
+A.
+Heat Equation
+In this subsection, we consider the controlled stochastic
+heat equation in order to validate our algorithm by compar-
+ing with the optimal feedback control obtained from the asso-
+ciated Riccati equation, see Example 1. The controlled state
+equation is given by
+�
+dug
+t = [∆ug
+t +gt]dt +0.05dWt,
+t ∈ [0,20]
+ug
+0 = u ∈ L2(0,20),
+(15)
+where u = 1[20/3,40/3]. We consider the problem of steering
+the solution of the stochastic heat equation into the constant
+zero profile. To this end, we introduce the cost functional
+J(g) = 1
+2E
+�� 20
+0
+∥ug
+t ∥2
+L2(0,20) +∥gt∥2
+L2(0,20)dt
+�
+.
+Note that the second term is a regularization, which is neces-
+sary in linear quadratic control theory. We approximate the
+Riccati equation (8) numerically based on 400 Fourier coeffi-
+cients to obtain the approximated optimal feedback control
+gRic
+t
+= Pn(t)ugRic
+t
+and use this approximation as a benchmark for our gradient
+descent algorithm.
+For the approximation of the optimal control, we use the
+ansatz space
+Un,k1+k2 :=
+�
+G(t,
+∞
+∑
+i=1
+uiei)(x) =
+n
+∑
+i=1
+ψi(t,(u1,...,un))ei(x)
+����ψ ∈ N n
+k1,k2
+�
+,
+(16)
+for k = k1 = k2 = 100, where
+N n
+k,k :=
+�
+ψ(t,u) = Cθ
+�
+Bθ
+�
+A
+�
+t
+u
+�
++a
+�
++b
+�
+����A ∈ Rk×(n+1),B ∈ Rk×k,C ∈ R(n+1)×k,a,b ∈ Rk
+�
+denotes the set of all 2-layer neural networks with k neurons
+in the first and second layer, and ReLU activator function
+θ. After about 20,000 iterations of our stochastic gradient
+descent algorithm, we end up with an approximated cost of
+J ≈ 5.3, and approximated L2-distance of our approximation
+gapprox
+t
+= Gapprox(t,uGapprox
+t
+) to the optimal control gRic given
+by
+E
+�� 20
+0
+∥gRic
+t
+−gapprox
+t
+∥2
+L2(Λ)dt
+�
+≈ 0.05.
+Below, we display our simulation results. Figure 1 displays
+one realization of the uncontrolled stochastic heat equation.
+Figure 2 displays our neural network approximation of the
+optimal feedback control, and Figure 3 shows its impact when
+applied to the system (15). Figure 4 displays the optimal con-
+trol obtained using the Riccati equation, and shows that our
+approximated feedback control is indeed qualitatively close.
+FIG. 1. Sample of uncontrolled solution
+FIG. 2. Sample of approximated optimal control gapprox
+FIG. 3. Sample of controlled solution ugapprox
+
+1.0
+1.0
+0.8
+0.8
+0.6
+0.4
+0.6
+0.2
+0.4
+0.0
+0.2
+20
+0.0
+15
+10
+5
+5
+x
+10
+t
+15
+20
+00.2
+0.2
+0.0
+0.2
+0.0
+0.4
+0.2
+0.6
+0.4
+0.8
+1.0
+0.6
+20
+0.8
+15
+0
+10
+5
+5
+x
+10
+t
+15
+20
+01.0
+0.8
+0.8
+0.6
+0.4
+0.6
+0.2
+0.4
+0.0
+0.2
+20
+0.0
+15
+10
+5
+x
+10
+5
+t
+15
+20
+0Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+10
+FIG. 4. Sample of optimal control gRic
+B.
+L2-Feedback Control of the Nagumo Equation
+In this example, we apply our algorithm to the controlled
+stochastic Nagumo equation
+�
+duG
+t =
+�
+∆uG
+t −uG
+t (uG
+t − 1
+2)(uG
+t −1)+G(t,uG
+t )
+�
+dt +0.05dWt
+uG
+0 = u ∈ L2(0,20),
+(17)
+where u = 1[5,15]. We consider the problem of stabilizing a
+bump profile given by the solution to the uncontrolled deter-
+ministic Nagumo equation, i.e., equation (17) with G ≡ 0 and
+without noise, see Figure 5. To this end, we introduce the cost
+functional
+J(G) = E
+�� 100
+0
+∥uG
+t −u0
+t ∥2
+L2(0,20) + 1
+2∥G(t,uG
+t )∥2
+L2(0,20)dt
+�
++E
+�
+∥uG
+100 −u0
+100∥2
+L2(0,20)
+�
+.
+(18)
+We consider again the ansatz space (16) with the same pa-
+rameters as in the previous example. In this case, our optimal
+control achieves an approximated cost of J ≈ 8.1.
+FIG. 5. Reference profile u0
+Figures 6 and 7 display two realizations of the uncontrolled
+stochastic Nagumo equation (17), i.e., G ≡ 0. Without control,
+the bump is unstable and the noise pushes the solution to one
+of the stable steady states u ≡ 0 or u ≡ 1. Figure 8 displays
+one realization of the approximated optimal control and shows
+that the control mostly acts on the interface, but also reacts
+to the noise in the system. Figure 9 illustrates the impact of
+the approximated optimal control when applied to the system
+(17); it shows that our feedback control indeed stabilizes the
+bump.
+FIG. 6. Sample of uncontrolled state
+FIG. 7. Sample of uncontrolled state
+FIG. 8. Sample of approximated optimal control
+
+0.2
+0.0
+0.0
+0.2
+-0.2
+0.4
+0.4
+0.6
+0.8
+0.6
+1.0
+0.8
+20
+15
+0
+10
+5
+5
+x
+10
+t
+15
+20
+01.0
+1.0
+0.8
+0.8
+0.6
+0.4
+0.6
+0.2
+0.4
+0.0
+0.2
+20
+15
+0.0
+0
+10
+20
+40
+x
+60
+5
+t
+80
+100
+01.0
+1.00
+0.8
+0.6
+0.75
+0.4
+0.50
+0.2
+0.0
+0.25
+20
+0.00
+15
+10
+20
+40
+x
+60
+5
+t
+80
+100
+01.0
+1.0
+0.8
+0.8
+0.6
+0.4
+0.6
+0.2
+0.4
+0.0
+0.2
+20
+0.0
+15
+10
+20
+40
+5
+x
+60
+t
+80
+100
+00.15
+0.10
+0.10
+0.05
+0.05
+0.00
+0.05
+-0.10
+0.00
+0.15
+0.20
+0.05
+20
+15
+-0.10
+0
+10
+20
+40
+x
+60
+5
+t
+80
+100
+0Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+11
+FIG. 9. Sample of controlled state
+C.
+Nemytskii Feedback Control of the Nagumo Equation
+In our final example, we consider again the controlled
+stochastic Nagumo equation (17) with the same cost func-
+tional (18). However, now we only consider feedback controls
+of Nemytskii-type, i.e., the control G is a Nemytskii operator
+G (t,u)(x) = g(t,x,u(x))
+for some function g : [0,100] × [0,20] × R → R. This means
+that the control at point x ∈ Λ depends on u(x), the value of the
+solution at point x, but not on the whole function u ∈ L2(Λ)
+as in the previous example. This restriction to feedback con-
+trols of Nemytskii-type significantly reduces the computa-
+tional complexity and therefore leads to a more efficient ap-
+proximation.
+For the approximation of the optimal control, we consider
+the following ansatz space of Gaussian radial basis function
+neural networks with m = 40 neurons:
+Un,m
+:=
+�
+G(t,u)(x) =
+n
+∑
+i=1
+m
+∑
+j=1
+r
+∑
+k=1
+αi jk1[tk−1,tk](t)e−κ|u(x)−u j|2ei(x)
+����αi jk ∈ R,uj ∈ R
+�
+,
+where {0 = t0 < t1 < ... < tr = 100} and κ = 6.
+In this
+case, our optimal control achieves an approximated cost of
+J ≈ 1.25.
+Similar to the approximated control of Section
+VIII B, Figure 10 shows that the control again mostly acts
+on the interface and also reacts to the noise in the system
+(compare with Figure 8). Figure 11 shows that the Nemytskii
+feedback control achieves a better result than the L2-feedback
+control from Subsection VIII B (compare with Figure 9). Fig-
+ure 12 displays the approximation of the feedback function
+(x,u) �→ g(t,x,u) at time t = 40. Observe that the feedback
+function is only trained for profiles arising in simulations. In
+particular, the feedback function is not trained in the middle
+of the bump (x ≈ 10) for u ≈ 0.
+FIG. 10. Sample of approximated optimal control
+FIG. 11. Sample of controlled state
+FIG. 12. Approximation of feedback function (x,u) �→ g(t,x,u) at
+time t = 40
+ACKNOWLEDGMENTS
+This work was funded by the Deutsche Forschungsgemein-
+schaft (DFG) via grant CRC 910, “Control of self-organizing
+nonlinear systems: Theoretical methods and concepts of ap-
+plication,” project A10, “Control of stochastic mean-field
+equations with applications to brain networks.”
+
+0.10
+0.05
+0.050
+0.00
+0.05
+0.025
+0.10
+0.000
+0.15
+0.20
+0.025
+20
+0.050
+15
+10
+20
+40
+5
+x
+60
+t
+80
+100
+01.0
+1.0
+0.8
+0.8
+0.6
+- 0.6
+0.4
+0.2
+0.4
+0.0
+0.2
+20
+0.0
+15
+0
+10
+20
+40
+x
+60
+5
+t
+80
+100
+00.2
+0.2
+0.1
+0.1
+0.0
+0.1
+0.0
+0.2
+0.1
+20
+0.2
+15
+0.0
+10
+0.2
+0.4
+x
+0.6
+5
+0.8
+u
+1.0
+1.2
+01.0
+1.00
+0.8
+0.6
+0.75
+0.4
+0.2
+0.50
+0.0
+0.25
+20
+0.00
+15
+10
+20
+40
+x
+60
+5
+t
+80
+100
+0Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations
+12
+AUTHOR DECLARATIONS
+Conflict of Interest
+The authors have no conflicts to disclose.
+Author Contributions
+Wilhelm Stannat: Mathematical theory (equal). Alexander
+Vogler: Mathematical theory (equal); Software. Lukas Wes-
+sels: Mathematical theory (equal).
+DATA AVAILABILITY
+Software and simulation output freely available with pub-
+lication on GitHub at https://github.com/AVoglerTu/
+SFB910Feedback.
+Appendix: Derivation of the Gradient of the Cost Functional
+As discussed in Section V, we now consider the following
+approximating optimal control problem: Minimize
+J(α) = E
+�� T
+0
+�
+Λ l(t,x,uα
+t (x))dx+ ν
+2 ∥Φ(t,uα
+t ,α)∥2
+L2(Λ)dt
++
+�
+Λ m(x,uα
+T (x))dx
+�
+.
+subject to
+�
+duα
+t = [∆uα
+t +F(uα
+t )+Φ(t,uα
+t ,α)]dt +σdWt
+uα
+0 = u ∈ L2(Λ).
+(A.1)
+In the following derivation of the gradient of the cost func-
+tional, we assume that l and m are differentiable with respect
+to u, F is differentiable, and Φ is differentiable with respect to
+u and α. The derivation of the gradient of the cost functional
+follows along the same lines as the derivation in32, however, in
+the present setting, the control enters the state equation (A.1)
+in a nonlinear fashion, which requires slight modifications of
+the arguments. We begin by differentiating the cost functional
+in some direction β ∈ Rdm which yields
+∂J(α)
+∂β
+(A.2)
+= E
+�� T
+0 ⟨L ′(t,uα
+t ),yβ
+t ⟩+ν⟨Φ(t,uα
+t ,α),Φα(t,uα
+t ,α)β⟩
++ν⟨Φ(t,uα
+t ,α),Φu(t,uα
+t ,α)yβ
+t ⟩dt +⟨M ′(uα
+t ),yβ
+T⟩
+�
+,
+where yβ is the solution of the linearized state equation
+�
+�
+�
+�
+�
+dyβ
+t = [∆yβ
+t +F ′(uα
+t )yβ
+t +Φu(t,uα,α)yβ
+t
++Φα(t,uα
+t ,α)β]dt
+yβ
+0 = 0.
+Next, we introduce the adjoint state p as the solution of the
+adjoint equation
+�
+�
+�
+�
+�
+dpt = −[(∆+F ′(uα
+t )+Φ∗
+u(t,uα
+t ,α))pt +L ′(t,uα
+t )
++νΦ∗
+u(t,uα
+t ,α)Φ(t,uα
+t ,α)]dt
+pT = M ′(uα
+T ).
+A straightforward computation leads to the following adjoint
+state property:
+d⟨yβ
+t , pt⟩ = ⟨yβ
+t ,dpt⟩+⟨pt,dyβ
+t ⟩
+=
+�
+−ν⟨Φ∗
+u(t,uα
+t ,α)Φ(t,uα
+t ,α),yβ
+t ⟩−⟨L ′(t,uα
+t ),yβ
+t ⟩
++⟨Φα(t,uα
+t ,α)β, pt⟩
+�
+dt.
+Integrating over [0,T] and taking expectations, we obtain from
+equation (A.2) the desired representation of the gradient of the
+cost functional:
+∇J(α)
+= E
+�� T
+0 νΦ∗
+α(t,uα
+t ,α)Φ(t,uα
+t ,α)+Φ∗
+α(t,uα
+t ,α)ptdt
+�
+.
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+
diff --git a/t9FKT4oBgHgl3EQf3C5S/content/tmp_files/load_file.txt b/t9FKT4oBgHgl3EQf3C5S/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..54224a1683d0d98e34cd58eaab7dafe922270f65
--- /dev/null
+++ b/t9FKT4oBgHgl3EQf3C5S/content/tmp_files/load_file.txt
@@ -0,0 +1,722 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf,len=721
+page_content='Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  Neural Network Approximation of Optimal Controls for Stochastic  Reaction-Diffusion Equations  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Stannat,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Vogler,1 and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Wessels1  Department of Mathematics, Technical University of Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  (*Electronic mail: stannat@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='de, vogler@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='de, wessels@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=')  (Dated: 25 January 2023)  We present a numerical algorithm that allows the approximation of optimal controls for stochastic reaction-diffusion  equations with additive noise by first reducing the problem to controls of feedback form and then approximating the  feedback function using finitely based approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Using structural assumptions on the finitely based approxima-  tions, rates for the approximation error of the cost can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Our algorithm significantly reduces the computa-  tional complexity of finding controls with asymptotically optimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Numerical experiments using artificial neural  networks as well as radial basis function networks illustrate the performance of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Our approach can  also be applied to stochastic control problems for high dimensional stochastic differential equations and more general  stochastic partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  There is a huge body of literature on optimal control prob-  lems with partial differential equation (PDE) constraints  and their numerical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Recent years have shown  a rising interest in the optimal control of stochastic par-  tial differential equations (SPDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' However, the numer-  ical approximation of optimal controls, let alone its prac-  tical implementation in the stochastic case, faces serious  obstacles due to the computational complexity of classical  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In this work, we present a new numerical al-  gorithm that approximates feedback controls for SPDEs  with asymptotically optimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' The algorithm is based  on adjoint calculus applied to gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' For the  approximation of feedback controls, we use finitely based  approximations such as artificial neural networks or ra-  dial basis function networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' The restriction to additive  noise and our approach for the approximation of feedback  functions enables us to significantly reduce the algorithmic  complexity of our approach in comparison with classical  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  INTRODUCTION  For a fixed finite time horizon T > 0, we consider the ran-  domly forced reaction-diffusion equation  �  ∂tu(t,x) = ∆u(t,x)+ f(u(t,x))+ξ(t,x), (t,x) ∈ [0,T]×Λ  u(0,x) = u(x),  x ∈ Λ  (1)  on a bounded domain Λ ⊂ Rd with Neumann boundary con-  ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Here, ∆ := ∑d  k=1  ∂ 2  ∂x2  k denotes the Laplace operator,  f : R → R models a local reaction term, and ξ(t,x) are ran-  dom fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Deterministic reaction-diffusion equations  are ubiquitous in the natural sciences and, in many situations,  taking into account random fluctuations leads to the more re-  alistic model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Typically, these random fluctuations are  highly irregular, and therefore, in order to treat this equation  rigorously, we reformulate it as the following L2(Λ)-valued  SPDE:  �  dut = [∆ut +F(ut)]dt +σdWt,  t ∈ [0,T]  u0 = u ∈ L2(Λ),  (2)  where ut(x) = u(t,x) for fixed t ∈ [0,T] is considered as an  element in L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' The random fluctuations ξ(t,x) are mod-  elled by a cylindrical Wiener process (Wt)t∈[0,T] on L2(Λ),  defined on some underlying probability space (Ω,F,P), and  σ : L2(Λ) → L2(Λ) is a Hilbert-Schmidt operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' For more  details on the mathematical theory of SPDEs, see1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Further-  more, F : L2(Λ) → L2(Λ) denotes the Nemytskii operator as-  sociated with f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',  F(u)(x) := f(u(x)),  u ∈ L2(Λ), x ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  The objective of control theory is to achieve a desired out-  come for a dynamical system by applying an external in-  put which can be chosen freely among a set of admissible  inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In order to set up a mathematical formulation for  the control of the SPDE (2), we introduce a control process  g : [0,T] × Ω → U ⊂ L2(Λ), adapted to the filtration gen-  erated by ut, as a forcing term on the right-hand side of the  equation  �  dug  t = [∆ug  t +F(ug  t )+gt]dt +σdWt,  t ∈ [0,T]  ug  0 = u ∈ L2(Λ),  (3)  and define the cost functional  J(g) := E  �� T  0  �  Λ l(t,x,ug  t (x))dx+ ν  2 ∥gt∥2  L2(Λ)dt  +  �  Λ m(x,ug  T(x))dx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  (4)  Here, the running cost l(t,x,u) : [0,T] × Λ × R → R and the  terminal cost m(x,u) : Λ×R → R are assumed to be differen-  tiable and Lipschitz continuous in the state variable u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Now,  the objective is to find a control process g that minimizes the  cost functional (4) over some set of admissible controls sub-  ject to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='11926v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='OC]  25 Jan 2023  Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  2  In the deterministic case, there is a huge body of litera-  ture on necessary and sufficient optimality conditions2–4, as  well as numerical algorithms which efficiently approximate  optimal controls, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='5–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In recent years, extensions to  the stochastic case have seen a rising interest in the math-  ematical literature leading to necessary and sufficient opti-  mality conditions in great generality9–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' However, classical  algorithms for the numerical approximation of optimal con-  trols in the stochastic case either require the approximation  of backward SPDEs or the approximation of infinite dimen-  sional Hamilton-Jacobi-Bellman (HJB) equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Due to the  curse of dimensionality, both of these alternatives are compu-  tationally very expensive, leading to an increased interest in  the development of new, more efficient algorithms22–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  A considerable reduction of the problem can be achieved in  the case when an optimal control g∗ is of feedback-type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',  g∗  t = G∗(t,uG∗  t ),  (5)  for some G∗ : [0,T] × L2(Λ) → L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In this work, we con-  sider a mathematical setting in which the optimal control is  indeed of the above type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In a first step we then use finitely  based approximations for G∗ in order to approximate the op-  timal control g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' This approach together with the restriction  to additive noise enables us to replace the backward SPDE  arising in adjoint calculus by a random backward PDE which  significantly reduces its computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' A similar  idea was already used in32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  The reduction to finitely based feedback controls then al-  lows in a second step the local uniform approximation of  finitely based feedback controls in appropriate ansatz spaces  of variable, but finite, dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Solving the corresponding  finite dimensional optimal control problem, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' with artifi-  cial neural networks, leads to an efficient computation of fi-  nite dimensional controls, whose costs approximate the opti-  mal cost with increasing dimension, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Impos-  ing additional smoothness assumptions on the optimal control  g∗  t = G∗(t,uG∗  t ), in particular G∗ globally Lipschitz, also en-  ables us to derive rates for the approximation error in ansatz  spaces consisting of Lipschitz continuous feedback functions,  see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We discuss the example of neural network  approximation in detail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' see Theorem 8 for the universal ap-  proximation of finitely based feedback controls with 1-layer  artificial neural networks and Proposition 9 for rates on the  approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  The practical implementation of the gradient descent algo-  rithm for the approximation of finite dimensional feedback  controls also requires the numerical discretization of the con-  trolled state equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In Proposition 11, we derive the com-  bined approximation error for the optimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Detailed  proofs of our results, that also hold for more general ansatz  spaces beyond artificial neural networks, can be found in33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  The performance of our gradient descent algorithm is illus-  trated with three examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' The first example deals with the  validation of our algorithm in the case of linear quadratic con-  trol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' The other two examples consider the problem of stabi-  lizing a bump solution of the stochastic Nagumo equation, in  one case with general feedback controls, in the other case with  feedback controls of Nemytskii-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  The remainder of the paper is organized as follows: First,  in Section II, we show that it is sufficient to consider feed-  back controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In Section III, we explain how to approximate  the optimal control by introducing ansatz spaces of controls  that are suited for the numerical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Section IV  contains our main approximation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In Section V, we  describe our gradient descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Next, in Section VI,  we discuss the explicit example of artificial neural networks  for the finite dimensional ansatz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In Section VII, we  investigate the error resulting from the numerical discretiza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Finally, in Section VIII, we present numerical experi-  ments using artificial neural networks and radial basis func-  tion networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  OPTIMAL CONTROLS OF FEEDBACK-TYPE  In order to reduce the complexity of the problem, we as-  sume that there exists an optimal control g∗ in feedback form  (5), for some continuous G∗ : [0,T]×L2(Λ) → L2(Λ) that sat-  isfies a linear growth condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' This can be achieved, using  the solution of the associated HJB equation  �  �  �  �  �  �  �  �  �  ∂tV + 1  2tr(σ∗D2Vσ)+⟨DV,∆u+F(u)⟩L2(Λ)  +  �  Λ l(t,x,u(x))dx+infG∈U {⟨DV,G⟩+ ν  2 ∥G∥2  L2(Λ)} = 0,  (t,u) ∈ [0,T]×L2(Λ)  V(T,u) =  �  Λ m(x,u(x))dx,  u ∈ L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  (6)  For solution theories regarding equations of this type, see11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In particular, if (6) has a unique mild solution satisfying cer-  tain regularity assumptions, and  γ(p) := arginfG∈U  �  ⟨p,G⟩+ ν  2 ∥G∥2  L2(Λ)  �  is continuous, then  g∗  t := G∗(t,uG∗  t ) := γ(DV(t,uG∗  t )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  is an optimal control in feedback form, provided that uG∗ is a  unique strong solution of the closed loop equation  �  duG  t = [∆uG  t +F(uG  t )+G(t,uG  t )]dt +σdWt, t ∈ [0,T]  uG  0 = u ∈ L2(Λ)  (7)  with G = G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' A sufficient condition for (7) to have a unique  strong solution is that G∗ is Lipschitz continuous in u, see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' This is in particular the case if the solution V of the  HJB equation (6) has a bounded second derivative in u, see  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='155 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='202 in11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' However, directly  tackling the optimal control problem by approximating the so-  lution of the HJB equation (6) numerically is very challenging  due to the infinite dimensionality of the domain of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Instead,  our approach is to approximate the feedback function directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Therefore we consider the following feedback control prob-  Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  3  lem: Minimize  J(G) := E  �� T  0  �  Λ l(t,x,uG  t (x))dx+ ν  2 ∥G(t,uG  t )∥2  L2(Λ)dt  +  �  Λ m(x,uG  T (x))dx  �  subject to equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We seek to minimize J over the set of  admissible controls  Uad := {G : [0,T]×L2(Λ) → U |G(·,uG  ) ∈ A},  where A = L2([0,T]×Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='L2(Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Example 1 (Linear Quadratic Control) Let us consider the  linear quadratic control problem  �  dug  t = [∆ug  t +gt]dt +σdWt,  t ∈ [0,T]  ug  0 = u ∈ L2(Λ),  where g : [0,T]×Ω → L2(Λ) is an adapted process, and  J(g) := 1  2E  �� T  0  �  Λ(ug  t (x))2 +g2  t (x)dxdt +  �  Λ(ug  T(x))2dx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In this case, the optimal control g∗ is indeed of feedback form,  given by  g∗  t = P(t)ug∗  t  where P : [0,T] → L(L2(Λ)) is the solution of the Riccati  equation34  �  ∂tP(t)+P(t)∆+∆P(t)−Id+P2(t) = 0, t ∈ [0,T]  P(T) = −Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  (8)  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  CONSTRUCTION OF ANSATZ SPACES  For the efficient numerical implementation we need to re-  strict ourselves to a subset of controls U ⊂Uad that are suitable  for the implementation in a gradient descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' How-  ever, we need to ensure that when minimizing over U, we do  not end up much worse than in the original control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In this section we will provide a method to construct suitable  ansatz spaces U when we do not have any particular control  constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' U = L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In order to construct a suitable space U, we introduce  the so-called finitely based approximation of a function G :  [0,T] × L2(Λ) → L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' To this end, we consider finite di-  mensional subspaces  Sn ⊂ L2(Λ)  with orthonormal basis e0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',en and orthogonal projections  Pn : L2(Λ) → Sn, such that  ∥Pnu−u∥L2(Λ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  The finitely based approximations of G with respect to Sn are  then defined by  Gn(t,u) := PnG(t,Pnu)  (9)  =  n  ∑  k=0  ⟨G(t,  n  ∑  j=0  ujej),ek⟩L2(Λ)ek,  where uj := ⟨u,ej⟩L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' One possible choice for the finite  dimensional subspaces in the case of Λ = (0,1) is  Sn := span  �  1,  √  2cos(kπ ·)  ���k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In this case the finitely based approximation of G is given by  Gn(t,u)(x)  :=  n  ∑  k=0  2⟨G(t,  n  ∑  j=0  uj  √  2cos( jπ ·)),cos(kπ ·)⟩L2(0,1) cos(kπx),  for u = ∑∞  j=0 uj cos( jπ ·) ∈ L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Our main results in Section IV provide approximation re-  sults for ansatz spaces U ⊂ Uad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' The main assumption on the  ansatz spaces U ⊂ Uad is the following approximation prop-  erty with respect to the optimal feedback G∗:  Definition 2 Let G : [0,T] × L2(Λ) → L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We say that a  subset U ⊂ Uad satisfies the universal approximation property  with respect to G, if there exists a sequence (Gn,m)n,m∈N ⊂ U  that satisfies the following linear growth condition uniformly  in m  ∥Gn,m(t,u)∥L2(Λ) ≤ Cn(1+∥u∥L2(Λ)),  for some constant Cn > 0, such that for any R > 0  lim  m→∞  sup  (t,u)∈[0,T]×BL2(Λ)(0,R)  ∥Gn(t,u)−Gn,m(t,u)∥2  L2(Λ) = 0,  where  BL2(Λ)(0,R) := {u ∈ L2(Λ)|∥u∥L2(Λ) ≤ R}  and Gn is given as in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  We would like to stress that in the above definition we only  require that the finitely based approximations of G can be ap-  proximated uniformly on bounded sets, and not G itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In the first part of this section we will explain how to con-  struct ansatz spaces U that satisfies the universal approxima-  tion property with respect to the optimal feedback G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' These  ansatz spaces consider finitely based controls of arbitrary di-  mension n ∈ N, which are of the type  G(t,u) =  n  ∑  k=0  ψk(t,Pnu)ek,  for some functions ψk : [0,T] × Sn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' However, in prac-  tice the dimension n ∈ N of the ansatz space need to be fixed  Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  4  a priori and one is interested in how close one can get to the  optimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Our second main result, Theorem 5, provides  explicit convergence rates, however, we need to strengthen  the assumption on our ansatz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In the second part of  this section we will explain how to construct for any n ∈ N a  sequence of ansatz spaces (Un,m)m∈N ⊂ Uad that satisfies the  following approximation property with respect to the optimal  feedback G∗:  Definition 3 Let G : [0,T] × L2(Λ) → L2(Λ) and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We  say that a sequence of subsets (Un,m)m∈N ⊂ Uad satisfies the  Lipschitz approximation property with respect to G in dimen-  sion n, if there exists a sequence of Lipschitz continuous con-  trols (Gn,m)m∈N with Lipschitz constants independent of m,  such that Gn,m ∈ Un,m, and a sequence of radii (Rn  m)m∈N with  limm→∞ Rn  m = ∞, such that  εn  m :=  sup  (t,u)∈[0,T]×BL2(Λ)(0,Rnm)  ∥Pn(Gn(t,u)−Gn,m(t,u))∥2  L2(Λ)  → 0,  (10)  as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  For this type of ansatz spaces, Theorem 5 provides error esti-  mates for  | inf  g∈Un,m J(g)− inf  g∈AJ(g)|  in terms of εn  m and the projection error γn (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Universal Ansatz Spaces  We start by constructing an ansatz space that satisfies a uni-  versal approximation property with respect to G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' For n ∈ N,  we consider the function gn : [0,T]×Rn → Rn given by  gn  k(t,u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',un) = ⟨G∗(t,  n  ∑  j=1  u je j),ek⟩L2(Λ),  k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Since G∗ is continuous, the functions (gn)n∈N are also contin-  uous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In particular, it is possible to approximate these func-  tions by simpler functions that can be treated numerically, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',  artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In the following we consider for  any n ∈ N a set N n of Lipschitz continuous approximations,  such that for all R > 0 there exists a sequence (ψn  m)m∈N ∈ N n  with  sup  (t,x)∈[0,T]×BRn(0,R)  |ψn  m(t,x)−gn(t,x)|2 → 0,  m → ∞,  where  BRn(0,R) := {x ∈ Rn||x| ≤ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  For a particular choice of N n we refer to our examples in  Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Then we define the ansatz space  U :=  �  G(t,  ∞  ∑  j=0  ujej) :=  n  ∑  k=0  ψk(t,ηl(u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',un))ek  ���ψ ∈ N n,n,l ∈ N  �  ,  where ηl : Rn → Rn  ηl(x) =  �  x  |x| ≤ l  l x  |x|  |x| > l  is a smooth cutoff function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' It is not difficult to see that for  any n ∈ N and R > 0 there exists a sequence (Gn,m)m∈N ⊂ U  that satisfies a linear growth condition of the type  ∥Gn,m(t,u)∥L2(Λ) ≤ Cn(1+∥u∥L2(Λ)),  such that for any n ∈ N  lim  m→∞  sup  (t,u)∈[0,T]×BL2(Λ)(0,R)  ∥Gn(t,u)−Gn,m(t,u)∥2  L2(Λ) = 0,  where  BL2(Λ)(0,R) := {u ∈ L2(Λ)|∥u∥L2(Λ) ≤ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In particular U satisfies a universal approximation property  with respect to G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We will give a short proof for this in Sec-  tion VI where we consider artificial neural networks, mapping  from Rn+1 to Rn, as an explicit example for Lipschitz contin-  uous approximations N n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Finite Dimensional Ansatz Spaces  Our second main result, Theorem 5, requires that the con-  trols of our ansatz space take values in the Sobolev space  H1(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Therefore we strengthen our assumption on the finite  dimensional subspaces and assume that  Sn ⊂ H1(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In order to obtain convergence rates, we also need to specify  the rate of convergence of the orthogonal projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' There-  fore, we assume that  ∥Pnu−u∥L2(Λ) ≤ γn∥u∥H1(Λ),  (11)  for some γn → 0, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Furthermore, we assume that there  exists an optimal control g∗ in feedback form with a Lipschitz  continuous feedback function G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' A simple example for such  a situation is the linear quadratic case discussed in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Since G∗ is Lipschitz continuous, the functions gn are also  Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' As we will see in the examples in Sec-  tion VI, it is therefore possible to approximate gn by artificial  neural networks that have uniformly bounded Lipschitz con-  stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' For any n ∈ N, let (N n,m)m∈N be a sequence of sets of  Lipschitz continuous approximations, such that for all m ∈ N  there exists a ψn,m ∈ N n,m with Lipschitz constant indepen-  dent of m and  sup  (t,x)∈[0,T]×BRn(0,Rnm)  |ψn,m(t,x)−gn(t,x)|2 =: εn  m → 0,  as m → ∞, for some sequence (Rn  m)m∈N of radii with Rn  m → ∞,  as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Then we define the sequence of ansatz spaces of  Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  5  dimension n ∈ N by  Un,m :=  �  G(t,  ∞  ∑  j=0  ujej) :=  n  ∑  k=0  ψk(t,u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',un)ek  ���ψ ∈ N n,m�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  It is not difficult to observe that for any m ∈ N, there exists a  Lipschitz continuous control Gn,m ∈ Un,m with Lipschitz con-  stant independent of m, such that  sup  (t,u)∈[0,T]×BL2(Λ)(0,Rnm)  ∥Pn(Gn,m(t,u)−Gn(t,u))∥2  L2(Λ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In particular, for any n ∈ N, the sequence (Un,m)m∈N satisfies  the Lipschitz approximation property with respect to G∗ in  dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  MAIN APPROXIMATION RESULTS  Our first main result, Theorem 4, provides an approxima-  tion result for universal ansatz spaces of the type constructed  in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In the second main result, Theorem 5, we pro-  vide error estimates for ansatz spaces of fixed dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Universal Approximation  The following result shows that we can reach the optimal  cost of the control problem when we consider ansatz spaces  of the type constructed in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Theorem 4 Assume that U ⊂ Uad satisfies the universal ap-  proximation property with respect to G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Then  inf  g∈AJ(g) = inf  G∈UJ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Approximation with Ansatz Spaces of Fixed Dimensions  In Section VI, we construct as a particular example our  ansatz space using artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' By Theorem 4  we can get arbitrarily close to the optimal cost of the original  control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' However, at this point we do not know how  close we can get to the optimal cost using ansatz spaces of a  fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Our second main result, Theorem 5, provides error  estimates for the convergence of the cost of our approximation  and the optimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Similar to Section III B, we consider fi-  nite dimensional subspaces  Sn ⊂ H1(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  and assume that for any u ∈ H1(Λ) the orthogonal projections  satisfy  ∥Pnu−u∥L2(Λ) ≤ γn∥u∥H1(Λ),  for some γn → 0, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Furthermore we assume that  there exists an optimal control g∗ in feedback form with a Lip-  schitz continuous feedback function G∗, as already mentioned  in Section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Theorem 5 Let n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Let (Un,m)m∈N be a sequence of sub-  sets of Uad that satisfies the Lipschitz approximation property  with respect to G∗ in dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Then it holds  inf  G∈Un,m J(G)− inf  g∈AJ(g)  ≤ C  �  1+  �  E  �� T  0 ∥g∗t ∥2  H1(Λ)dt  ��  γn +Cn  �  εnm,  for some universal constant C independent of n and m and  some constant Cn which is independent of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Here εn  m is given  by (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  GRADIENT DESCENT ALGORITHM  In this section, we describe our gradient descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Theorem 4 enables us to consider the approximating optimal  control problem of minimizing optimal costs on a finite di-  mensional ansatz space Un,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We assume that  Un,m =  �  Φ(·,·,α)  ���α ∈ Rdm  �  for a parametrization Φ : [0,T]×L2(Λ)×Rdm → L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Re-  placing  G(t,u) = Φ(t,u,α)  for some given α ∈ Rdm leads to the state equation  �  duα  t = [∆uα  t +F(uα  t )+Φ(t,uα  t ,α)]dt +σdWt  uα  0 = u ∈ L2(Λ),  (12)  and the cost functional J : Rdm → R,  J(α) = E  �� T  0  �  Λ l(t,x,uα  t (x))dx+ ν  2 ∥Φ(t,uα  t ,α)∥2  L2(Λ)dt  +  �  Λ m(x,uα  T (x))dx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Using this parametrization, we obtain under suitable regular-  ity assumptions on Φ the following representation of the gra-  dient of the cost functional:  ∇J(α)  = E  �� T  0 νΦ∗  α(t,uα  t ,α)Φ(t,uα  t ,α)+Φ∗  α(t,uα  t ,α)ptdt  �  ,  (13)  where p is the solution of the so-called adjoint equation  �  �  �  �  �  dpt = −[(∆+F ′(uα  t )+Φ∗  u(t,uα  t ,α))pt +L ′(t,uα  t )  +νΦ∗  u(t,uα  t ,α)Φ(t,uα  t ,α)]dt  pT = M ′(uα  T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  (14)  Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  6  Here  L (t,u)(x) := l(t,x,u(x)),  u ∈ L2(Λ),(t,x) ∈ [0,T]×Λ  denotes the Nemytskii operator associated with l, and  M (u)(x) := m(x,u(x)),  u ∈ L2(Λ),x ∈ Λ  denotes the Nemytskii operator associated with m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Further-  more, Φα(t,u,α) : Rdm → L2(Λ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Φu(t,u,α) : L2(Λ) →  L2(Λ)) denotes the derivative of Φ with respect to α (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  u), and Φ∗  α(t,u,α) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Φ∗  u(t,u,α)) denotes the adjoint of  Φα(t,u,α) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Φu(t,u,α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' For a derivation of the gradi-  ent, see the supplemental material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Note that equation (14) is a linear backward PDE with ran-  dom coefficients which are given by the state equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  For details concerning the numerical implementation, see the  supplemental material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Example 6 (Artificial Neural Network) In the case of an ar-  tificial neural network with activator function θ, the controls  in the ansatz space Un,m can be parametrized as  Φ(t,u,α) =  n  ∑  i=1  �  Cθ  �  A  �  t  πnu  �  +a  �  ,ei  �  ei,  where α = (A,a,C) consists of A ∈ Rk×(n+1), a ∈ Rk, and C ∈  Rn×k, for respective dimensions n = nm and k = km, and πnu =  (⟨u,e1⟩,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',⟨u,en⟩)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Based on this representation, we implement the following  algorithm:  Algorithm 7 Fix an initial control α0, a stopping criterion  ρ > 0, and a step size s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Solve the state equation (12) for one realization of the  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Solve the adjoint equation (14) with the data given by  the sample calculated in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Repeat Step 1 and Step 2 to approximate the gradient  (13) using Monte Carlo approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Compute new control via αn+1 = αn −s∇J(αn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Stop if ∥∇J(αn)∥ < ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  ARTIFICIAL NEURAL NETWORKS AS ANSATZ  SPACES  In this section, we discuss explicit examples for ansatz  spaces U ⊂ Uad that satisfy the assumptions of our main re-  sults, Theorem 4 and Theorem 5, and are suitable for the  numerical implementation of our gradient descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In the first part of this section, we focus on universal spaces  and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' The second part is devoted to ansatz spaces  of fixed size and Theorem 5 with corresponding convergence  rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In the whole section, we consider the finite dimensional  subspaces  Sn := span  �  1,  √  2cos(kπ ·)  ���k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',n  �  ⊂ H1(0,1),  with orthonormal basis in L2(Λ)  e0 = 1,  ek =  √  2cos(kπ ·),  k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In particular we have for any u ∈ H1(0,1)  ∥Pnu−u∥2  L2(0,1) =  ∞  ∑  k=n+1  |⟨u,  √  2cos(kπ ·)⟩L2(0,1)|2  ≤  1  n2π2  ∞  ∑  k=n+1  |⟨∆1/2u,  √  2cos(kπ ·)⟩L2(0,1)|2  ≤  1  n2π2 ∥u∥2  H1(0,1) =: γn∥u∥2  H1(0,1),  where we used that the k-th eigenvalue of the Neumann  Laplace operator with respect to the eigenfunction ek is given  by −  1  k2π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Neural Network Approximation  Regarding Theorem 4, we will show in our first example  that it is indeed sufficient to consider the type of ansatz space  constructed in the first part of Section III, using 1-layer arti-  ficial neural networks for the approximating sets, to get arbi-  trarily close to the optimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' More precisely, we show that  the set  U :=  �  G(t,  ∞  ∑  i=1  uiei)(x) =  n  ∑  i=1  ψi(t,ηl(u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',un))ei(x)  ����ψ ∈ N n,n,l ∈ N  �  satisfies the universal approximation property with respect to  G∗, where  N n :=  ∞  �  k=1  N n  k  and  N n  k :=  �  ψ(x) = Cθ(Ax+a)  ��A ∈ Rk×n,a ∈ Rk,C ∈ Rn×k�  denotes the set of all 1-layer artificial neural networks from Rn  to Rn with k neurons, for a given non-polynomial, Lipschitz  continuous activator function θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  To this end, we recall the following classical universal ap-  proximation result by35:  Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  7  Theorem 8 Let  θ ∈ C (R),  then  we  define  for  u =  (u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',ud) ∈ Rd  θ(u)i := θ(ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  If θ is not polynomial, then for any n,m ∈ N, compact set K ⊂  Rn, h ∈ C (K,Rm) and ε > 0, there exists k ∈ N,A ∈ Rk×n,a ∈  Rk,C ∈ Rm×k, such that  sup  u∈K  |h(u)−ψ(u)| < ε,  where ψ is the 1-layer artificial neural network  ψ(u) := Cθ(Au+a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Let n ∈ N and recall from Section III the finitely based ap-  proximation of G∗  gn : [0,T]×Rn → Rn  gn  i (t,u) := ⟨G∗(t,  n  ∑  j=1  ujej),ei⟩,  i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Since gn is continuous, there exists for any m ∈ N a 1-layer  artificial neural network ψn,m : [0,T]×Rn → Rn, such that  sup  (t,u)∈[0,T]×BRn(0,m)  |gn(t,u)−ψn,m(t,u)| < 1  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Since G∗ satisfies a linear growth condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',  ∥G∗(t,u)∥L2(Λ) ≤ C(1+∥u∥L2(Λ)),  for some C > 0, we have  |gn(t,u)| = ∥PnG∗(t,  n  ∑  i=1  uiei)∥L2(Λ)  ≤ ∥G∗(t,  n  ∑  i=1  uiei)∥L2(Λ)  ≤ C(1+|u|),  where u = (u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Hence, if we consider the continuous  function ηm : Rn → Rn  ηm(x) =  �  x  |x| ≤ m  m x  |x|  |x| > m  and define  ˜ψn,m(t,u) := ψn,m(t,η(u)),  then clearly ˜ψn,m = ψn,m on [0,T] × BRn(0,m) and for any  (t,u) ∈ [0,T]×BRn(0,m)  |ψn,m(t,u)| ≤ |gn(t,u)|+|ψn,m(t,u)−gn(t,u)|  ≤ C(1+|u|)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Furthermore, on [0,T]×BRn(0,m)c we have  | ˜ψn,m(t,u)| = |ψn,m(t,m u  |u|)|  ≤ |gn(t,m u  |u|)|+|ψn,m(t,m u  |u|)−gn(t,m u  |u|)|  ≤ C(1+m)+1  ≤ C(1+|u|)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Therefore ˜ψn,m satisfies a linear growth condition with some  constant C > 0 independent of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Now, we define  Gn,m(t,u) :=  n  ∑  i=1  ˜ψn,m  i  (t,⟨u,e1⟩L2(Λ),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',⟨u,en⟩L2(Λ)))ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  One can easily check that all the assumptions of Theorem 5  are satisfied for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Indeed, due to the Lipschitz continuity of  the elements in N n  k , any control in U is indeed admissible,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=', U ⊂ Uad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Furthermore, (Gn,m)m∈N is a sequence in U that  satisfies a linear growth condition with some constant C >  0 independent of m and approximates any finitely based Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Indeed, for any R > 0 and any ε > 0 there exists an M ∈ N,  such that BL2(Λ)(0,R) ⊂ BL2(Λ)(0,m) and 1  m < ε for every  m ≥ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Therefore, we have for any m ≥ M  sup  (t,u)∈[0,T]×BL2(Λ)(0,R)  ∥Gn,m(t,u)−Gn(t,u)∥2  ≤  sup  (t,u)∈[0,T]×BL2(Λ)(0,m)  ∥Gn,m(t,u)−Gn(t,u)∥2  ≤  sup  (t,u)∈[0,T]×BRn(0,m)  | ˜ψn,m(t,u)−gn(t,u)|2  ≤  sup  (t,u)∈[0,T]×BRn(0,m)  |ψn,m(t,u)−gn(t,u)|2  < 1  m < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In the case of bounded controls, for example if U =  BL2(Λ)(0,R), we could consider the ansatz space  U :=  �  G(t,  ∞  ∑  i=1  uiei)(x) =  n  ∑  i=1  ψi(t,ηl(u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',un))ei(x)  ����∥G(t,u)∥L2(Λ) ≤ R where ψ ∈ N n,n,l ∈ N  �  and the sequence  Gn,m(t,u)  := (1−  1  m(R+1))  n  ∑  i=1  ˜ψn,m  i  (t,⟨u,e1⟩L2(Λ),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',⟨u,en⟩L2(Λ)))ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  8  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Convergence Rates  Next, we provide explicit convergence rates for the se-  quence of approximating spaces  Un,m :=  �  G(t,  ∞  ∑  i=1  uiei)(x) =  n  ∑  i=1  ψi(t,(u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',un))ei(x)  ���ψ ∈ N n  m  �  using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We will mainly follow the ideas of36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In  the following we consider a 2π-periodic activator function  θ : R → R that satisfies ˆθ :=  � π  −π θ(x)e−ixdx ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Recall the  following result from36:  Proposition 9 Let  K,L1,L2  > 0  and  h : [0,T] ×  BRn+1(0,R) → Rn be Lipschitz continuous in (t,x) with  ∥h∥C 0([0,T]×BRn+1(0,R)) ≤ K and Lipschitz constant bounded  by L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Furthermore we assume that h is twice differentiable  in x and ∂xih is Lipschitz continuous with Lipschitz constant  bounded by L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Then there exists a constant C > 0 depending  only on the above constants, on n,T and on the activator  function θ through ˆθ, ∥θ ′∥C 0, ∥θ ′′∥C 0 and ∥θ ′′′∥C 0, and  there exists a constant m0 depending only on n with the  following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  For every R > 0 and every min > m0,  there exists a one-hidden layer artificial neural network  ψh ∈ N n  min such that  ∥h−ψh∥C 0([0,T]×BRn(0,R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='Rn) ≤ C(1+R)m−1/(2(n+1))  in  and such that the Lipschitz constants of ψh,∂xψh are at most  C(1+Rm−1/(2(n+1))  in  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  If we now set Rn(m) := m1/(3(n+1)), then by Proposition 9  there exists for any sufficiently large m ∈ N a network ψn,m ∈  N n  m , such that for  Gn,m(t,  ∞  ∑  i=1  uiei)(x) =  n  ∑  i=1  ψn,m  i  (t,(u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',un))ei(x),  it holds  sup  (t,x)∈[0,T]×BL2(Λ)(0,Rn(m))  ∥Pn(Gn,m(t,x)−Gn(t,x))∥2  L2(Λ)  ≤ m−1/(3(n+1)) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Furthermore Gn,m ∈ Un,m is Lipschitz continuous with some  Lipschitz constant independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Therefore, applying The-  orem 5 yields  inf  G∈Un,m J(G)− inf  g∈AJ(g)  ≤C  �  1+  �  E  �� T  0 ∥g∗t ∥2  H1(Λ)dt  ��  1  n2π2 +Cn  �  m−1/(3(n+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  NUMERICAL DISCRETIZATION OF THE CONTROL  PROBLEM  In order to implement a numerical algorithm, the control  problem needs to be discretized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In this section, we introduce  the discretized version of our control problem and provide a  bound for the error resulting from the numerical discretiza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Problem 10 (Discretized Control Problem) Let  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Minimize  Jn(G) := E  �� T  0  �  Λ l(t,x,uG,n  t  (x))dx+ ν  2 ∥PnG(t,uG,n  t  )∥2  L2(Λ)dt  �  +E  ��  Λ m(x,uG,n  T (x))dx  �  over the set Uad subject to the discretized SPDE  �  duG,n  t  = [−∆nuG,n  t  +PnF(uG,n  t  )+PnG(t,uG,n  t  )]dt +σPndWt  uG,n  0  = Pnu ∈ Sn,  where ∆nxn is defined as the unique element in Sn with  ⟨∇xn,∇yn⟩L2(Λ) = ⟨∆nxn,yn⟩L2(Λ)  ∀yn ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Proposition 11 Under the same assumptions as in Theorem  5, it holds  inf  G∈Un,m Jn(G)− inf  g∈AJ(g)  ≤ C  �  1+  �  E  �� T  0 ∥g∗t ∥2  H1(Λ)dt  ��  γn +Cn  �  εnm,  for some constant Cn > 0 that only depends on n ∈ N and on  the Lipschitz constant of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Remark 12 Under additional assumptions, in particular con-  vexity of the Hamiltonian, one can show that for any U ⊂ Uad,  it holds  inf  g∈AJ(g)− inf  G∈UJn(G) ≤ Cγn,  for some constant C > 0 which is independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  SIMULATIONS  In this whole section, we consider equation (2) on some  interval Λ = (0,L) of length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' For the approximation we  consider the Galerkin finite dimensional subspace  Sn := span  �  1,  �  2  L cos  � k  Lπ ·  ������k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',n  �  ⊂ H1(0,L),  with orthonormal basis  e0 =  �  1  L,  ek =  �  2  L cos  � k  Lπ ·  �  ,  k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',n,  Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  9  or the finite element subspace  Sn := span{ek|k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',n} ⊂ H1(0,L),  with basis  ek(x) =  �  �  �  �  �  n  �  x− k−1  n L  �  if x ∈  � k−1  n L, k  nL  �  n  � k+1  n L−x  �  if x ∈  � k  nL, k+1  n L  �  0  otherwise  for k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Heat Equation  In this subsection, we consider the controlled stochastic  heat equation in order to validate our algorithm by compar-  ing with the optimal feedback control obtained from the asso-  ciated Riccati equation, see Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' The controlled state  equation is given by  �  dug  t = [∆ug  t +gt]dt +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='05dWt,  t ∈ [0,20]  ug  0 = u ∈ L2(0,20),  (15)  where u = 1[20/3,40/3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We consider the problem of steering  the solution of the stochastic heat equation into the constant  zero profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' To this end, we introduce the cost functional  J(g) = 1  2E  �� 20  0  ∥ug  t ∥2  L2(0,20) +∥gt∥2  L2(0,20)dt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Note that the second term is a regularization, which is neces-  sary in linear quadratic control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We approximate the  Riccati equation (8) numerically based on 400 Fourier coeffi-  cients to obtain the approximated optimal feedback control  gRic  t  = Pn(t)ugRic  t  and use this approximation as a benchmark for our gradient  descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  For the approximation of the optimal control, we use the  ansatz space  Un,k1+k2 :=  �  G(t,  ∞  ∑  i=1  uiei)(x) =  n  ∑  i=1  ψi(t,(u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=',un))ei(x)  ����ψ ∈ N n  k1,k2  �  ,  (16)  for k = k1 = k2 = 100, where  N n  k,k :=  �  ψ(t,u) = Cθ  �  Bθ  �  A  �  t  u  �  +a  �  +b  �  ����A ∈ Rk×(n+1),B ∈ Rk×k,C ∈ R(n+1)×k,a,b ∈ Rk  �  denotes the set of all 2-layer neural networks with k neurons  in the first and second layer, and ReLU activator function  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' After about 20,000 iterations of our stochastic gradient  descent algorithm, we end up with an approximated cost of  J ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='3, and approximated L2-distance of our approximation  gapprox  t  = Gapprox(t,uGapprox  t  ) to the optimal control gRic given  by  E  �� 20  0  ∥gRic  t  −gapprox  t  ∥2  L2(Λ)dt  �  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Below, we display our simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Figure 1 displays  one realization of the uncontrolled stochastic heat equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Figure 2 displays our neural network approximation of the  optimal feedback control, and Figure 3 shows its impact when  applied to the system (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Figure 4 displays the optimal con-  trol obtained using the Riccati equation, and shows that our  approximated feedback control is indeed qualitatively close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Sample of uncontrolled solution  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Sample of approximated optimal control gapprox  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Sample of controlled solution ugapprox  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='2  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='0  15  10  5  5  x  10  t  15  20  00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='6  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='8  15  0  10  5  5  x  10  t  15  20  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='2  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='0  15  10  5  x  10  5  t  15  20  0Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Sample of optimal control gRic  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  L2-Feedback Control of the Nagumo Equation  In this example, we apply our algorithm to the controlled  stochastic Nagumo equation  �  duG  t =  �  ∆uG  t −uG  t (uG  t − 1  2)(uG  t −1)+G(t,uG  t )  �  dt +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='05dWt  uG  0 = u ∈ L2(0,20),  (17)  where u = 1[5,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We consider the problem of stabilizing a  bump profile given by the solution to the uncontrolled deter-  ministic Nagumo equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=', equation (17) with G ≡ 0 and  without noise, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' To this end, we introduce the cost  functional  J(G) = E  �� 100  0  ∥uG  t −u0  t ∥2  L2(0,20) + 1  2∥G(t,uG  t )∥2  L2(0,20)dt  �  +E  �  ∥uG  100 −u0  100∥2  L2(0,20)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  (18)  We consider again the ansatz space (16) with the same pa-  rameters as in the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In this case, our optimal  control achieves an approximated cost of J ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Reference profile u0  Figures 6 and 7 display two realizations of the uncontrolled  stochastic Nagumo equation (17), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=', G ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Without control,  the bump is unstable and the noise pushes the solution to one  of the stable steady states u ≡ 0 or u ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Figure 8 displays  one realization of the approximated optimal control and shows  that the control mostly acts on the interface, but also reacts  to the noise in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Figure 9 illustrates the impact of  the approximated optimal control when applied to the system  (17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' it shows that our feedback control indeed stabilizes the  bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content=' Sample of uncontrolled state  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='10  0  10  20  40  x  60  5  t  80  100  0Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  11  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Sample of controlled state  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Nemytskii Feedback Control of the Nagumo Equation  In our final example, we consider again the controlled  stochastic Nagumo equation (17) with the same cost func-  tional (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' However, now we only consider feedback controls  of Nemytskii-type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=', the control G is a Nemytskii operator  G (t,u)(x) = g(t,x,u(x))  for some function g : [0,100] × [0,20] × R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' This means  that the control at point x ∈ Λ depends on u(x), the value of the  solution at point x, but not on the whole function u ∈ L2(Λ)  as in the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' This restriction to feedback con-  trols of Nemytskii-type significantly reduces the computa-  tional complexity and therefore leads to a more efficient ap-  proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  For the approximation of the optimal control, we consider  the following ansatz space of Gaussian radial basis function  neural networks with m = 40 neurons:  Un,m  :=  �  G(t,u)(x) =  n  ∑  i=1  m  ∑  j=1  r  ∑  k=1  αi jk1[tk−1,tk](t)e−κ|u(x)−u j|2ei(x)  ����αi jk ∈ R,uj ∈ R  �  ,  where {0 = t0 < t1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' < tr = 100} and κ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  In this  case, our optimal control achieves an approximated cost of  J ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Similar to the approximated control of Section  VIII B, Figure 10 shows that the control again mostly acts  on the interface and also reacts to the noise in the system  (compare with Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Figure 11 shows that the Nemytskii  feedback control achieves a better result than the L2-feedback  control from Subsection VIII B (compare with Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Fig-  ure 12 displays the approximation of the feedback function  (x,u) �→ g(t,x,u) at time t = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Observe that the feedback  function is only trained for profiles arising in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' In  particular, the feedback function is not trained in the middle  of the bump (x ≈ 10) for u ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Sample of approximated optimal control  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Sample of controlled state  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Approximation of feedback function (x,u) �→ g(t,x,u) at  time t = 40  ACKNOWLEDGMENTS  This work was funded by the Deutsche Forschungsgemein-  schaft (DFG) via grant CRC 910, “Control of self-organizing  nonlinear systems: Theoretical methods and concepts of ap-  plication,” project A10, “Control of stochastic mean-field  equations with applications to brain networks.”  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='00  15  10  20  40  x  60  5  t  80  100  0Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  12  AUTHOR DECLARATIONS  Conflict of Interest  The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Author Contributions  Wilhelm Stannat: Mathematical theory (equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Alexander  Vogler: Mathematical theory (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Lukas Wes-  sels: Mathematical theory (equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  DATA AVAILABILITY  Software and simulation output freely available with pub-  lication on GitHub at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='com/AVoglerTu/  SFB910Feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Appendix: Derivation of the Gradient of the Cost Functional  As discussed in Section V, we now consider the following  approximating optimal control problem: Minimize  J(α) = E  �� T  0  �  Λ l(t,x,uα  t (x))dx+ ν  2 ∥Φ(t,uα  t ,α)∥2  L2(Λ)dt  +  �  Λ m(x,uα  T (x))dx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  subject to  �  duα  t = [∆uα  t +F(uα  t )+Φ(t,uα  t ,α)]dt +σdWt  uα  0 = u ∈ L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='1)  In the following derivation of the gradient of the cost func-  tional, we assume that l and m are differentiable with respect  to u, F is differentiable, and Φ is differentiable with respect to  u and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' The derivation of the gradient of the cost functional  follows along the same lines as the derivation in32, however, in  the present setting, the control enters the state equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='1)  in a nonlinear fashion, which requires slight modifications of  the arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' We begin by differentiating the cost functional  in some direction β ∈ Rdm which yields  ∂J(α)  ∂β  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='2)  = E  �� T  0 ⟨L ′(t,uα  t ),yβ  t ⟩+ν⟨Φ(t,uα  t ,α),Φα(t,uα  t ,α)β⟩  +ν⟨Φ(t,uα  t ,α),Φu(t,uα  t ,α)yβ  t ⟩dt +⟨M ′(uα  t ),yβ  T⟩  �  ,  where yβ is the solution of the linearized state equation  �  �  �  �  �  dyβ  t = [∆yβ  t +F ′(uα  t )yβ  t +Φu(t,uα,α)yβ  t  +Φα(t,uα  t ,α)β]dt  yβ  0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Next, we introduce the adjoint state p as the solution of the  adjoint equation  �  �  �  �  �  dpt = −[(∆+F ′(uα  t )+Φ∗  u(t,uα  t ,α))pt +L ′(t,uα  t )  +νΦ∗  u(t,uα  t ,α)Φ(t,uα  t ,α)]dt  pT = M ′(uα  T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  A straightforward computation leads to the following adjoint  state property:  d⟨yβ  t , pt⟩ = ⟨yβ  t ,dpt⟩+⟨pt,dyβ  t ⟩  =  �  −ν⟨Φ∗  u(t,uα  t ,α)Φ(t,uα  t ,α),yβ  t ⟩−⟨L ′(t,uα  t ),yβ  t ⟩  +⟨Φα(t,uα  t ,α)β, pt⟩  �  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Integrating over [0,T] and taking expectations, we obtain from  equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='2) the desired representation of the gradient of the  cost functional:  ∇J(α)  = E  �� T  0 νΦ∗  α(t,uα  t ,α)Φ(t,uα  t ,α)+Φ∗  α(t,uα  t ,α)ptdt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='  7C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='  8C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='  9F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Cordoni and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Di Persio, “Optimal control for the stochastic FitzHugh-  Nagumo model with recovery variable,” Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Control Theory 7, 571–  585 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Du and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='  11G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Fabbri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Gozzi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' ´Swi˛ech, Stochastic Optimal Control in Infinite  Dimension (Springer, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  12H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Frankowska and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Zhang, “Necessary conditions for stochastic optimal  control problems in infinite dimensions,” Stochastic Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 130,  4081–4103 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  13M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Fuhrman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Hu, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Tessitore, “Stochastic maximum principle for  optimal control of SPDEs,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 68, 181–217 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  14M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Fuhrman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Hu,  and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Tessitore, “Stochastic maximum principle  for optimal control of partial differential equations driven by white noise,”  Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Partial Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 6, 255–285 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  15M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Fuhrman and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Orrieri, “Stochastic maximum principle for optimal  control of a class of nonlinear SPDEs with dissipative drift,” SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Con-  trol Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' 54, 341–371 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations  13  16Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Lü and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Zhang, General Pontryagin-Type Stochastic Maximum Prin-  ciple and Backward Stochastic Evolution Equations in Infinite Dimensions  (Springer, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  17Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Lü and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Zhang, “Transposition method for backward stochastic evo-  lution equations revisited, and its application,” Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Control Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Fields  5, 529–555 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  18Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Lü and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Zhang, “Operator-valued backward stochastic Lyapunov  equations in infinite dimensions, and its application,” Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Control Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  Fields 8, 337–381 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  19W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Stannat and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content=' 59, 3552–3573 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  20W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Stannat and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Wessels, “Necessary and sufficient conditions for opti-  mal control of semilinear stochastic partial differential equations,” preprint,  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='org/abs/2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='09639 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content='  21L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='  22C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Beck, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' E, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='  23S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Dolgov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Kalise, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='  24T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content=' Vallet, “On stochastic optimal con-  trol in ferromagnetism,” Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content=' Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='  29N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Nüsken and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+page_content='  30M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9FKT4oBgHgl3EQf3C5S/content/2301.11926v1.pdf'}
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+InsPro: Propagating Instance Query and Proposal
+for Online Video Instance Segmentation
+Fei He1,2, Haoyang Zhang4, Naiyu Gao4, Jian Jia1,2, Yanhu Shan4,
+Xin Zhao1,2∗, Kaiqi Huang1,2,3
+1 CRISE, Institute of Automation, Chinese Academy of Sciences
+2 School of Artificial Intelligence, University of Chinese Academy of Sciences
+3 CAS Center for Excellence in Brain Science and Intelligence Technology 4 Horizon Robotics
+{hefei2018,jiajian2018}@ia.ac.cn, {haoyang.zhang,naiyu01.gao,yanhu.shan}@horizon.ai
+{xzhao,kaiqi.huang}@nlpr.ia.ac.cn
+Abstract
+Video instance segmentation (VIS) aims at segmenting and tracking objects in
+videos. Prior methods typically generate frame-level or clip-level object instances
+first and then associate them by either additional tracking heads or complex instance
+matching algorithms. This explicit instance association approach increases system
+complexity and fails to fully exploit temporal cues in videos. In this paper, we
+design a simple, fast and yet effective query-based framework for online VIS.
+Relying on an instance query and proposal propagation mechanism with several
+specially developed components, this framework can perform accurate instance
+association implicitly. Specifically, we generate frame-level object instances based
+on a set of instance query-proposal pairs propagated from previous frames. This
+instance query-proposal pair is learned to bind with one specific object across
+frames through conscientiously developed strategies. When using such a pair to
+predict an object instance on the current frame, not only the generated instance
+is automatically associated with its precursors on previous frames, but the model
+gets a good prior for predicting the same object. In this way, we naturally achieve
+implicit instance association in parallel with segmentation and elegantly take
+advantage of temporal clues in videos. To show the effectiveness of our method
+InsPro, we evaluate it on two popular VIS benchmarks, i.e., YouTube-VIS 2019
+and YouTube-VIS 2021. Without bells-and-whistles, our InsPro with ResNet-50
+backbone achieves 43.2 AP and 37.6 AP on these two benchmarks respectively,
+outperforming all other online VIS methods.
+1
+Introduction
+Video instance segmentation (VIS) [1] is a challenging but important computer vision task. It requires
+not only segmenting object instances on each video frame but also associating them across all frames.
+Due to its fine-grained object representation form, it has got a wide range of applications in various
+areas such as autonomous driving and video editing.
+Existing VIS methods can be categorized into two groups: frame-level methods and clip-level
+methods. Frame-level methods [1, 2, 3, 4] generally follow a ‘tracking-by-detection’ paradigm,
+which first generate per-frame object instances by existing instance segmentation models [5, 6],
+and then associate them across frames via additional tracking heads (as shown in Figure 1 (a)). In
+comparison, clip-level methods [7, 8, 9, 10] take a ‘clip-matching’ paradigm, which divide a video
+∗Corresponding author
+36th Conference on Neural Information Processing Systems (NeurIPS 2022).
+arXiv:2301.01882v1  [cs.CV]  5 Jan 2023
+
+(a) Previous VIS paradigm 
+(b) Our InsPro
+Instance 
+association
+Ins 1 Ins 2
+Ins 1 Ins 2
+Ins 𝑦
+Ins 𝑥
+assign ID
+Previous
+Current
+SegNet
+SegNet
+temporal propagation
+Previous
+Current
+Ins 1
+Ins 2
+Ins 1
+Ins 2
+Our
+SegNet
+Our
+SegNet
+query-proposal pair
+query-proposal pair
+Figure 1: (a) Previous methods take a two-step approach to VIS. They first generate object instances
+and then perform explicit instance association to link them across frames. (b) Our InsPro implements
+implicit instance association through a temporal propagation mechanism, achieving object instance
+segmentation and tracking in one shot. It generates frame-level object instances based on a set of
+instance query and proposal pairs propagated from previous frames. Since the instance query-proposal
+pair is learned to represent one specific object across frames, the instance association is naturally
+achieved in parallel with segmentation and video temporal clues are elegantly exploited meanwhile.
+into multiple overlapped clips, generate instance predictions for each clip, and then associate these
+clip-level predictions by some hand-crafted instance matching algorithms. Whether frame-level or
+clip-level methods, both of them inevitably need an explicit instance association step to fulfill object
+tracking. This generally requires to design a complicated association strategy to achieve good tracking
+performance, which is not trivial. More importantly, the explicit association step increases model
+complexity and slows inference speed. Furthermore, this extra step also indicates that the temporal
+clues intrinsic in videos are not well utilized, as the instance prediction is performed separately on
+each frame or each clip.
+In this work, inspired by the recent success of query-based object detectors [11, 12], we propose a
+simple, fast and yet effective query-based framework for online VIS. Our system, dubbed as InsPro,
+segments and tracks objects in one shot through an instance query and proposal propagation strategy
+with carefully designed modules (Figure 1 (b)) , which eliminates the explicit instance association
+step. Specifically, our approach generates frame-level object instances based on a set of instance
+query-proposal pairs propagated from previous frames. In the learning process, we develop several
+techniques to make sure that the generated instance query-proposal pair corresponds to one specific
+object across frames. Thus, when an object instance is generated using such a query-proposal pair
+on the current frame, it is automatically associated with its precursors on all previous frames. In
+this way, we achieve implicit object association without a linking step. Meanwhile, this instance
+query-proposal propagation mechanism also enables our VIS system to achieve a better prediction
+accuracy (see Table 1). This benefits from the instance query-proposal pair’s encoding one object’s
+temporal and spatial information across all previous frames, which provides a very good prior for
+the model to infer the same object on the current frame. In this sense, our query-based VIS method
+actually implements an efficient way to exploit the intrinsic temporal clues in videos.
+To fulfill the advantages of our VIS system, learning exclusive and expressive instance query-proposal
+pairs is the key. In this work, we develop several strategies to ensure the learning effectiveness. First,
+we design a temporally consistent matching mechanism to enforce the one-to-one correspondence
+between the instance query-proposal pair and a specific ground truth object across frames during
+training. Second, we propose a box deduplication loss to enlarge the distance between instance
+proposals. This helps suppress duplicate proposals on the same object and increase the exclusivity of
+the generated instance query-proposal pair. At the same time, the sparsely distributed unoccupied
+query-proposal pairs can serve as candidates in the next frame to detect new objects, allowing our
+system to achieve new object detection and tracking effortlessly. Third, we propose an intra-query
+attention module that enhances instance query with its predecessors encoding the same object. This
+explicitly aggregates long-range object information into the query, augmenting its representation
+capacity, which helps handle occlusion and motion blur.
+To validate the effectiveness and efficiency of our InsPro, we conduct extensive experiments on two
+popular VIS benchmarks [1], i.e., YouTube-VIS 2019 and YouTube-VIS 2021. Without bells-and-
+whistles, our InsPro with ResNet-50 [13] backbone achieves 43.2 AP on YouTube-VIS 2019 and 37.6
+AP on YouTube-VIS 2021 respectively, outperforming all other online VIS models. Moreover, our
+2
+
+lite variant, InsPro-lite, reaches 38.7 AP on YouTube-VIS 2019 at impressive 45.7 FPS on a Nvidia
+RTX2080Ti GPU.
+In summary, we make the following contributions in this paper. 1) We propose a simple, fast and
+yet effective query-based framework for online VIS. 2) We develop several techniques to make the
+query-proposal pair propagation mechanism work smoothly. These techniques distinguish our work
+from other query propagation-based object association methods [14, 15, 16], and make our work
+simpler, more elegant and more effective than them. 3) Our VIS system achieves the state-of-the-art
+performances on two popular VIS benchmarks.
+2
+Related Work
+Frame-level VIS Methods
+mainly adopt a ‘tracking-by-detection’ paradigm and can run in an
+online fashion. They first generate instance predictions frame by frame and then perform explicit
+instance association. MaskTrack R-CNN [1] first proposes the VIS task and simply adds an additional
+tracking head to Mask R-CNN [5] for instance association. Follow-up works [2, 3, 4, 17] improve
+either the segmentation or the tracking algorithm to achieve better performance. On the other
+hand, some works [8, 18, 19, 20, 21, 22] attempt to perform temporal feature fusion to improve
+instance segmentation and association. For example, PCAN [22] proposes frame- and instance-level
+prototypical cross-attention modules to leverage rich spatio-temporal information to facilitate better
+segmentation. All these methods require additional modules to achieve explicit instance association,
+which expands model complexity and reduces inference speed. By contrast, our method performs
+instance association implicitly through an instance query and proposal propagation mechanism, which
+is simpler and naturally exploits the temporal and spatial consistency in videos.
+Clip-level VIS Methods
+take a ‘clip-matching’ paradigm, which process multiple frames within
+a clip simultaneously and then perform instance matching between clips to complete VIS. While
+some methods [7, 23, 9] propagate instance information within a clip with well-designed propagation
+modules to model temporal context, recent works [8, 10] utilize transformer [24] to model temporal
+context in an end-to-end manner. These methods normally need hand-crafted matching algorithms to
+complete instance association between clips. Although they usually achieve high performance, they
+can only run in an offline mode, which restricts their application to limited areas. In contrast, our
+method achieves comparable performance but can run online.
+Query-based Methods
+have attracted increasing attention in recent years due to their flexibility
+and simplicity. DETR [11] first uses a set of learned queries interacting with image features to
+encode objects, and then directly outputs detections by decoding the transformed queries. Following
+works [25, 26, 27, 28, 29] improve DETR in terms of either training efficiency or detection perfor-
+mance. Sparse R-CNN [12] builds a query-based detector on top of R-CNN architecture [30, 31]. Its
+follow-up works [3, 32] extend it to instance segmentation and video object detection. Besides, Max-
+DeepLab [33] proposes a box-free panoptic segmentation method with external query. DAFL [34]
+uses a set of queries to encode pedestrian information for pedestrian attribute recognition.
+The success of DETR has also inspired query-based VIS methods. VisTR [8] adapts DETR to the
+VIS task. It takes a video clip as input and directly outputs the sequence of masks for each instance
+orderly. IFC [10] proposes inter-frame communication transformers to reduce the heavy computation
+and memory usage of VisTR-like VIS methods. Similar to VisTR, Mask2Former [35] applies masked
+attention to a video clip and directly predicts a 3D instance volume. To learn a powerful video-level
+instance query, SeqFormer [36] aggregates temporal information from each frame to the instance
+query. These methods work on clips rather than frames, and achieve object association through
+sharing of the queries within a clip rather than query propagation. Thus, they still need instance
+matching between clips. Instead, our method applies to frames, and can propagate query-proposal
+pairs through the entire video and thereby can associate object instances over any frame length.
+Query Propagation-based Object Association Methods
+have been recently explored in several
+works, such as TransTrack [14], TrackFormer [15], MOTR [16] and EfficientVIS [37], which are also
+inspired by query-based methods [11, 12]. This shows the effectiveness and potential of such a new
+object linking approach. The differences between our InsPro and them are as follows.
+3
+
+SegHead
+SegHead
+SegHead
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+𝑝𝑒𝑟𝑠𝑜𝑛
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+∅
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+(a) Overview of InsPro
+(b) Details of SegHead
+Weight
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+DIIM
+Figure 2: (a) Overview of our InsPro. It performs VIS by propagating instance query-proposal pairs
+across frames. qinit ∈ RN×C and pinit ∈ RN×4 are initial instance queries and proposals on the first
+video frame, respectively. They are used in SegHead to predict instance results r0 on frame I0, and
+to produce updated q0 and p0 which are propagated to the next frame. By repeating this process, we
+complete the VIS task. (b) Details of SegHead. It is a multi-stage network, consisting of a dynamic
+instance interaction module (DIIM) and an instance segmentation module. The former transforms
+instance queries with RoI features of corresponding proposals and produces object features, while the
+latter predicts object instances based on the object features and conditional convolution [39].
+First, our InsPro is different from them in the way of either tracking seen objects or detecting new
+objects. TransTrack is basically a ‘tracking-by-detection’ method, because it still needs to explicitly
+match detection boxes to tracked boxes in each frame, while our InsPro performs implicit association.
+More importantly, TransTrack, TrackFormer and MOTR adopt a track query subset to track seen
+objects and an extra object query subset to detect new objects. This requires additional heuristic rules
+to combine two type queries, and may miss occluded or blurred objects with low scores, which can
+result in object trajectory break [38]. Our InsPro simply propagates all object queries produced in
+the previous frame to the current frame, and keeps using this set to track seen objects and detect
+new objects, which is much simpler and more elegant. As for EfficientVIS, our concurrent work, it
+does not consider this new object detection problem, and its performance will probably be impacted
+greatly if there are new objects in the next clip.
+Furthermore, we design a more intelligent strategy to suppress duplicates. TransTrack and Track-
+Former employ score filtering or NMS to reduce duplicate predictions. MOTR builds a temporal
+aggregation network to learn more discriminative features to address this problem, while EfficientVIS
+does not discuss this problem. By contrast, we design a Box Deduplication Loss to suppress dupli-
+cates and an Inter-query attention module to enhance queries with their predecessors. Our solution
+avoids heuristic rules and post-processing steps, and is more effective according to the experimental
+results (see Table 2 (e)).
+3
+InsPro
+We aim to design a simple and fast VIS system that performs instance association implicitly and
+exploits video temporal clues elegantly. To this end, we take a query-based VIS approach that
+predicts object instances on each frame based on a set of instance query-proposal pairs propagated
+from previous frames. In this section, we introduce our VIS system, InsPro, including an instance
+query and proposal propagation mechanism and an instance segmentation head. Meanwhile, we also
+describe those proposed techniques that make our propagation mechanism work well.
+3.1
+Instance Query and Proposal Propagation
+The instance query and proposal propagation mechanism enables our VIS system to perform object
+instance association implicitly in parallel with instance segmentation. Since it is inspired by the
+recent query-based object detector Sparse R-CNN [12], we first briefly review Sparse R-CNN.
+4
+
+Sparse R-CNN [12] formulates object detection as a set prediction problem and achieves state-of-the-
+art performance. It simplifies the detection pipeline and removes heuristic components like NMS.
+Specifically, it first initializes a fixed set of learnable instance queries (N × C, N denotes the number
+of queries and C the query dimension) paired with learnable instance proposals (N × 4) to describe
+objects in an image. As illustrated in Figure 2 (b), each instance query is convolved with the RoI
+feature of the corresponding proposal to output a more discriminate feature ot [12]. After multi-stage
+iterative updating, the instance query encodes more accurate object appearance information while the
+proposal captures more precise location. Finally, decoding the object feature ot produced using the
+instance query-proposal pairs, we get the detection results.
+Inspired by this instance query-proposal representation of an object, we design a query-proposal
+temporal propagation mechanism (as shown in Figure 2 (a)) to achieve implicit object instance associ-
+ation and temporal cue utilization in VIS. Our key insight is that there is a one-to-one correspondence
+between the learned instance query-proposal pair and a specific object. If we manage to preserve this
+correspondence from the first frame to the one where the object finally disappears, then we realize
+object tracking and object information propagation spontaneously.
+To this end, we first initialize a set of instance queries qinit ∈ RN×C and proposals pinit ∈ RN×4 on
+the first video frame I0, where qinit and pinit are learnable parameters and arranged in pairs. After
+learning, they are able to encode objects on the first frame. Decoding them with the first frame image
+feature inside the SegHead, we obtain instance results r0 as well as a new set of updated pairs (q0,
+p0). Then we propagate this pair set (q0, p0) to the next frame as input to the SegHead. Similarly,
+we get the instance results r1 and another new set of (q1, p1) on this frame. Among them, the object
+instance produced on this frame shares the same ID with the one on the previous frame if they are
+both decoded by the same slice of the instance queries. In this way, we automatically link object
+instances belonging to an identical object across frames and elegantly make use of object priors from
+the past. Repeating the above process until the last video frame, we then accomplish the VIS task on
+this video.
+Please note that our InsPro simply propagates all object queries produced in the previous frame to
+the current frame, and keeps using this set to track seen objects and detect new objects. Instead,
+recent works [14, 15, 16] that take a similar query-propagation mechanism use a track query set
+to track seen objects and a new object query set to detect new objects respectively, which requires
+additional heuristic rules to combine these two type queries. Moreover, they rely on hand-crafted rules
+like a score threshold to select a subset of track queries, and occluded objects with low prediction
+scores are probably filtered out, which results in non-negligible true object missing and fragmented
+trajectories [38]. In comparison, our method is obviously simpler, more elegant and more effective
+(see Table 2 (e)).
+Intra-query Attention
+Since frame-by-frame temporal propagation encodes only short-range
+temporal cues, the instance query from just the last frame shows limitations in dealing with tough
+scenarios, e.g., occlusion and motion blur. To boost the representation capacity of instance query,
+we augment it in practice with instance features from previous T frames [40, 41]. Specifically, we
+build a feature bank that caches instance features from previous T frames and perform intra-query
+attention inside this bank to aggregate long-range temporal cues into the current instance query, as
+shown in the upper part of Figure 2 (b). Formally, at frame It, instance features o from previous T
+frames are put together to form a feature bank fb = {ot−T +1, . . . , ot}. Then, the enhanced instance
+query is computed as:
+qi
+t =
+�T −1
+n=0 oi
+t−n exp(ε(oi
+t−n))
+�T −1
+m=0 exp(ε(oi
+t−m))
++ oi
+t,
+(1)
+where i denotes the i-th query and ε(·) is a linear transformation function. The enhanced qt is
+basically a weighted sum of instance features inside the feature bank, and the weights are learned
+upon the quality of the queries. Experiments (Table 6 (c)) show that this augmentation improves the
+query representation capacity greatly.
+3.2
+Segmentation Head
+The segmentation head transforms the instance query and performs instance segmentation on each
+frame. As illustrated in Figure 2 (b), it is a multi-stage network and has two main parts: a dynamic
+instance interaction module and an instance segmentation module.
+5
+
+Dynamic Instance Interaction Module
+transforms the instance query with the proposal RoI
+feature and yields object features. It has M stages and forms an iterative structure. At the first stage,
+given a pair of instance queries qt−1 ∈ RN×C and proposals pt−1 ∈ RN×4 that propagated from
+the previous frame It−1, it first augments the instance queries by a self-attention module [24], which
+models the inter-query relations. At the same time, it extracts the RoI feature of each proposal on the
+feature map by RoIAlign [5]. Then, each enhanced instance query convolves with its corresponding
+RoI feature through dynamic convolution [42] to get the object feature ot ∈ RN×C. Since the object
+feature absorbs temporal cues encoded in instance queries, it has better representation ability than the
+single-frame RoI feature (see Table 2 (d)). The object feature is used to predict object instances in the
+following instance segmentation module, and the newly generated object boxes together with the
+object features proceed as input to the next stage in the iterative process. At the final stage, the object
+feature is processed together with its precursors from previous frames through the aforementioned
+intra-query attention module, to produce instance queries for the next frame.
+Instance Segmentation Module
+decodes the object feature ot and produces VIS predictions. It has
+three main heads. While the classification head predicts object classes, the regression one generates
+object boxes. Another mask head is responsible for producing instance masks through a conditional
+convolution [39] approach. Specifically, it first uses ot to generate conditional convolution weights in
+weight generator. As shown in Figure 2 (b), it inputs oi
+t that represents the i-th object instance feature
+to the weight generator and outputs a set of convolution parameters ωi. Meanwhile, it produces
+mask feature maps by transforming FPN [43] feature maps through a mask branch. Note that the
+output mask feature maps have 8 channels and a 1
+8 resolution of the input image, and are boosted by
+concatenating a 2-channel relative coordinates map to it. This relative coordinates map is computed
+using the center of predicted object boxes and provides strong location cues for predicting instance
+masks. Finally, we feed the combined feature maps f mask
+t
+∈ R10× H
+8 × W
+8 and convolution parameters
+ωi to a mask FCN head, predicting the i-th object instance mask via a conditional convolution:
+mi
+t = CondConv(f mask
+t
+, ωi).
+(2)
+For more details about the instance segmentation module, please refer to [39].
+3.3
+Temporally Consistent Matching
+The key to the success of our InsPro is to make sure that the evolving instance query-proposal pair
+corresponds to the same object across frames in a video. To ensure this, one technique we propose is
+temporally consistent matching. This technique matches predictions and ground truth during training,
+assigns each ground truth object a proper prediction, and propagates the matching made on previous
+frames to subsequent frames.
+Specifically, given a training batch consisting of multiple consecutive frames, we first compute the
+matching cost Lmatch between predictions and ground truth objects on the first frame:
+Lmatch = λcls · Lcls + λL1 · LL1 + λgiou · Lgiou,
+(3)
+where Lcls is the focal loss [44] between predicted classifications and ground-truth labels, LL1 and
+Lgiou are L1 loss and the generalized IoU loss [45] between predicted boxes and ground-truth boxes,
+respectively. λcls, λL1 and λgiou are loss weights and set as 2, 5, and 2, respectively.
+We search for the best bipartite matching that minimizes the matching cost Lmatch with the Hungarian
+algorithm [46]. After finding the best matching on the first frame, we propagate this matching to other
+frames. Concretely, if one ground truth object still exists on subsequent frames, it will be matched to
+the prediction that is generated by the same instance query on the first frame. If there are new objects
+emerging, new matching will be made between the new objects and yet unmatched predictions. If a
+ground truth object disappears, its corresponding predictions will not participate in a new matching
+process. Through this temporally consistent matching mechanism, we bind one ground truth object
+to a single instance query during training.
+3.4
+Loss Function
+Box Deduplication Loss
+Although the self-attention mechanism between queries has driven the
+model to generate fewer duplication predictions [11], we still observe multiple overlapped proposal
+6
+
+time
+(a) w/o BDL
+(b) w/ BDL
+Figure 3: (a) Multiple duplicate boxes exist on the same object across frames. (b) After applying
+the proposed box deduplication loss (BDL) in training, the duplicate predictions are significantly
+suppressed along with temporal propagation.
+boxes on the same object across many frames, as displayed in Figure 3 (a). We conjecture this is
+caused by those unmatched queries which cannot be pushed away from those matched queries due
+to lack of supervision. To address this problem, we propose a box deduplication loss to push away
+prediction boxes in terms of the center-to-center distance between them. As a result, not only the
+duplication problem is alleviated, but the sparsely distributed unmatched query-proposal pairs can
+serve as candidates in the next frame to detect and track new objects (see Figure 7 in Appendix). The
+loss is defined as:
+Ldedup = 1
+k
+k
+�
+i=1
+max(β − C2(b, ˆb
+i
+neg)
+D2(b)
+, 0),
+(4)
+where b is a ground truth box, ˆb
+i
+neg is a negative box that has the i-th largest IoU with b among those
+unmatched predicted boxes, C(·) is the center distance calculation function, and D(·) is the diagonal
+length calculation function. β is set as 0.1 and k as 5. This loss penalizes the short distance between
+b and ˆb
+i
+neg, and drags all other duplicate boxes away from b [47]. With this new loss, our final box
+loss function is formed as:
+Lbox = λL1 · LL1 + λgiou · Lgiou + λdedup · Ldedup,
+(5)
+where λL1 and λgiou have the same values as in Equation 3, and λdedup is set as 1.
+Overall Loss
+Given the one-to-one matching results, the final loss on each training frame is a sum
+of classification, box and mask losses:
+L = λcls · Lcls + λbox · Lbox + λdice · Ldice + λfocal · Lfocal,
+(6)
+where Ldice and Lfocal are dice loss [48] and focal loss [44] for foreground mask prediction,
+respectively. We set λbox = 1, λdice = 5 and λfocal = 5. Finally, the losses of all training frames
+inside a batch are summed together and normalized by the number of frames.
+4
+Experiments
+4.1
+Datasets and Evaluation Metrics
+We evaluate our method on YouTube-VIS 2019 and 2021 benchmarks [1]. YouTube-VIS 2019
+consists of 2,238 training videos and 302 validation videos, and labels 40 object categories. YouTube-
+VIS 2021 is an extended version, which comprises 2,985 training videos and 421 validation videos,
+and labels improved 40 categories. All videos in these two datasets are annotated every 5 frames with
+object bounding box, object category, instance mask and instance ID. Following [1], we report the
+video-level average precision (AP) and average recall (AR) on the validation sets as the evaluation
+metrics, where both accurate instance segmentation and instance association are necessary to achieve
+high performance.
+7
+
+Table 1: Comparison of our InsPro to state-of-the-art methods. All methods use ResNet-50 as
+backbone. C: additionally using COCO train2017 images that contain YouTube-VIS categories for
+training. The inference speed is tested on a Nvidia RTX2080Ti GPU. ∗ indicates using deformable
+convolution [52] in backbone. ‡ indicates that the FPS is measured by parallel processing of images
+in one clip rather than sequential processing.
+YouTube-VIS 2019 Val.
+YouTube-VIS 2021 Val.
+Method
+Online
+AP
+AP50 AP75 AR1 AR10
+AP
+AP50 AP75 AR1 AR10
+FPS
+STEm-Seg [7] (C)
+
+30.6
+50.7
+33.5
+31.6
+37.1
+-
+-
+-
+-
+-
+4.4
+VisTR [8]
+
+35.6
+56.8
+37.0
+35.2
+40.2
+-
+-
+-
+-
+-
+30.0‡
+Propose-Reduce [9] (C)
+
+40.4
+63.0
+43.8
+41.1
+49.7
+-
+-
+-
+-
+-
+< 20
+MaskProp∗ [23]
+
+40.0
+-
+42.9
+-
+-
+-
+-
+-
+-
+-
+< 10
+IFC [10]
+
+39.0
+60.4
+42.7
+41.7
+51.6
+35.2
+57.2
+37.5
+-
+-
+46.5‡
+EfficientVIS [37]
+
+37.9
+59.7
+43.0
+40.3
+46.6
+34.0
+57.5
+37.3
+33.8
+42.5
+36‡
+MaskTrack R-CNN [1]
+
+30.3
+51.1
+32.6
+31.0
+35.5
+28.6
+48.9
+29.6
+26.5
+33.8
+26.1
+SipMask [4]
+
+33.7
+54.1
+35.8
+35.4
+40.1
+31.7
+52.5
+34.0
+30.8
+37.8
+30
+STMask∗ [20]
+
+33.5
+52.1
+36.9
+31.1
+39.2
+-
+-
+-
+-
+-
+28.6
+SG-Net [2]
+
+34.8
+56.1
+36.8
+35.8
+40.8
+-
+-
+-
+-
+-
+23.0
+PCAN [22]
+
+36.1
+54.9
+39.4
+36.3
+41.6
+-
+-
+-
+-
+-
+-
+CrossVIS [18]
+
+36.3
+56.8
+38.9
+35.6
+40.7
+34.2
+54.4
+37.9
+30.4
+38.2
+25.6
+HybridVIS [21] (C)
+
+41.3
+61.5
+43.5
+42.7
+47.8
+35.8
+56.3
+39.1
+33.6
+40.3
+< 20
+InsPro-lite
+
+38.7
+60.9
+41.7
+36.9
+43.6
+-
+-
+-
+-
+-
+45.7
+InsPro
+
+40.2
+62.9
+43.1
+37.6
+44.5
+36.1
+57.6
+39.6
+30.9
+40.4
+26.3
+InsPro (C)
+
+43.2
+65.3
+48.0
+38.8
+49.0
+37.6
+58.7
+40.9
+32.7
+41.4
+26.3
+4.2
+Implementation Details
+We implement our InsPro with Detectron2 [49], and most hyperparameters are set following Sparse
+R-CNN [12] and CondInst [39] unless otherwise specified. More implementation details can be found
+in Appendix A.1.
+Training Details
+We employ AdamW [50] with an initial learning rate of 2.5 × 10−5 and weight
+decay 0.0001 as our model optimizer. We initialize our model with parameters pre-trained on
+COCO [51], and train it for 32k iterations where the learning rate is divided by 10 at iterations 24k
+and 28k, respectively. The training is performed end-to-end on 8 Nvidia RTX2080Ti GPUs and each
+GPU holds one mini-batch which contains three frame images randomly sampled from the same
+video. Data augmentation includes only random horizontal flip and multi-scale training where the
+training image is resized so that the length of its shortest side is at least 288 and at most 512. Unless
+otherwise noted, our InsPro adopts ResNet-50 [13] as backbone and uses 100 instance queries in our
+experiments.
+Inference Details
+In inference, we resize the frame image size to 640 × 360, which follows
+MaskTrack R-CNN [1]. The size of the feature bank is set to 18 by default. If the generated proposal
+box exceeds the frame’s boundaries, it will be clipped to corresponding boundaries. No multi-scale
+testing is adopted in our experiments.
+InsPro-lite
+We also build a lite version of our method, named InsPro-lite. In this variant, inspired
+by [32], we divide video frames into key frames and non-key frames, i.e., we select one key frame
+per K frames in a video and treat other frames as non-key ones. K is 10 by default. On key frames,
+we conduct the dynamic instance interaction 6 times while only once on non-key frames. This takes
+advantage of the redundancy of videos and helps reduce inference computation time. Our InsPro-lite
+reaches a high inference speed of 45.7 FPS at a small accuracy loss (Table 1).
+4.3
+Main Results
+We perform a thorough comparison of our InsPro to state-of-the-art VIS methods on YouTube-VIS
+2019 and 2021. Existing VIS methods can be divided into two categories according to whether they
+run online or offline [53]. Since some methods [7, 9] use 80k transformed COCO images [51] as
+extra training data to prevent overfitting to YouTube-VIS, for a fair comparison, we also report our
+8
+
+Table 2: Ablation studies on YouTube-VIS 2019.
+(a) Effectiveness of instance query and pro-
+posal propagation, and temporally consis-
+tent matching (TCM).
+query
+proposal
+TCM
+AP
+AP50
+AP75
+(A)
+24.0
+41.3
+24.2
+(B)
+
+
+36.3
+56.3
+38.9
+(C)
+
+
+
+37.4
+57.6
+41.1
+(D)
+
+
+36.7
+57.3
+39.9
+(E)
+
+
+36.6
+55.5
+40.3
+(b) Effectiveness of the proposed
+box deduplication loss (BDL).
+AP
+AP50
+AP75
+FPS
+w/o BDL
+37.4
+57.6
+41.1
+26.3
+w/ BDL
+38.4
+57.7
+41.6
+26.3
+(c) Intra-query attention. T is the
+length of the feature bank.
+AP
+AP50
+AP75
+FPS
+T=1
+38.4
+57.7
+41.6
+26.3
+T=9
+39.7
+61.6
+42.1
+26.3
+T=18
+40.2
+62.9
+43.1
+26.3
+T=27
+40.1
+62.6
+42.2
+26.3
+T=36
+40.1
+62.5
+42.2
+26.3
+(d) Comparison between our temporal propagation paradigm
+and ‘tracking-by-detection’ paradigm.
+AP
+AP50
+AP75
+Param (M)
+FLOPs (G)
+FPS
+Tracking-by-detection
+31.5
+49.3
+34.1
+119.9
+48.3
+25.4
+Ours
+37.4
+57.6
+41.1
+106.1
+45.5
+26.3
+(e) Comparison between our united query
+and ‘track-and-detect query’.
+AP
+AP50
+AP75
+Track-and-detect query
+37.4
+56.9
+40.3
+Ours
+38.4
+57.7
+41.6
+results with and without extra COCO training data. Table 1 presents all the results obtained with a
+ResNet-50 backbone on a Nvidia RTX2080Ti GPU.
+YouTube-VIS 2019
+Table 1 (left) shows the comparison between our InsPro and other state-of-the-
+art methods on YouTube-VIS 2019 validation set. We can see that, in the online group, our InsPro
+outperforms all other popular methods under the same data setting. Specifically, our InsPro achieves
+40.2 AP without COCO data and 43.2 AP with COCO data respectively, surpassing other online VIS
+methods by a large margin. Even our lite version, InsPro-lite, performs better than all other online
+methods trained without COCO data, reaching 38.7 AP at an impressive speed of 45.7 FPS.
+YouTube-VIS 2021
+Table 1 (right) displays results on YouTube-VIS 2021 validation set. It shows
+a similar comparison pattern to YouTube-VIS 2019 and our InsPro achieves the state-of-the-art
+performance once again.
+4.4
+Ablation Study
+We conduct extensive experiments on YouTube-VIS 2019 to study the effectiveness and individual
+performance contribution of our proposed modules.
+Temporal Propagation and Matching
+Our InsPro is built on the proposed instance query and
+proposal temporal propagation mechanism. Table 2 (a) shows how this mechanism contributes to
+our high performance. In this table, method A represents the video instance segmentation baseline,
+where each frame is processed individually without any temporal propagation, and object instances
+generated on each frame are linked if they are produced from the same instance query slice. Since
+this method lacks the mechanism to ensure the instance query-proposal pair corresponds to the
+same object across frames, it only achieves 24.0 AP due to inaccurate instance association. By
+contrast, when we add the temporal propagation (method B), we can easily improve the performance
+significantly to 36.3 AP. This evidences the importance and effectiveness of the proposed temporal
+propagation technique in a query-based VIS framework. If we further adopt the temporally consistent
+matching strategy during training (method C), we achieve an even better performance of 37.4 AP.
+We also analyze the separate performance of propagating only instance query (method D) or instance
+proposal (method E). The results show that these two settings achieve a similar performance boost
+(36.7 AP vs 36.6 AP). Applying them together yields 37.4 AP, bringing further performance gain.
+Box Deduplication Loss
+We propose a box deduplication loss to suppress the duplicate proposal
+boxes on the same object across frames. With the qualitative results shown in Figure 3, we show the
+quantitative comparison in Table 2 (b). We can see that supervising the learning with this loss during
+training can lead 1.0 AP improvement (38.4 AP vs 37.4 AP). This performance gain is brought by
+fewer duplicate boxes and fewer missed detections.
+9
+
+Intra-query Attention
+We perform intra-query attention inside a feature bank to augment the
+instance query so that it can capture long-range temporal cues. As we can see in Table 2 (c), this
+simple method works well and improves the performance considerably. In particular, T = 1 indicates
+no intra-query attention is used and 38.4 AP is achieved. When we increase the volume T of the
+feature bank, the performance rises and saturates at 40.2 AP with T = 18. It is worthwhile to note
+that this lightweight yet effective intra-query attention module brings almost no speed drop.
+Temporal Propagation vs. Tracking-by-Detection
+Despite the fact that our InsPro does not
+perform explicit instance association, it still outperforms all other online methods implementing
+explicit tracking or matching. To verify that our superior performance comes from the temporal
+propagation mechanism rather than the instance segmentation model design, we compare our temporal
+propagation VIS approach to the typical ‘tracking-by-detection’ paradigm with the same instance
+segmentation baseline. We implement a ‘tracking-by-detection’ VIS system by replacing the Mask
+R-CNN part in MaskTrack R-CNN [1] with our instance segmentation model. In this case, the only
+independent variable is the object tracking method.
+As shown in Table 2 (d), our InsPro surpasses the ‘tracking-by-detection’ model by a large margin
+even if our design is simpler and faster, which soundly proves the effectiveness of our method. We
+argue again that this is because the evolving instance query-proposal pair in propagation encodes
+object temporal and spatial cues intrinsic in videos, whereas ‘tracking-by-detection’ methods are
+generally incapable of exploiting this advantage.
+Our United Query vs. Track-and-Detect Query
+We further compare our method to those MOT
+methods that adopt a similar query-propagation method for object tracking. These methods rely on
+two different query sets, i.e., a track query set and an object query set, to track seen objects and detect
+new objects respectively, while we only maintain one united query set. They need heuristic rules to
+combine these two type queries. Meanwhile, they manually select track queries with high scores
+from the previous frame to build the track query set. This makes them complex and less effective
+in tracking since occluded objects with low scores probably have broken trajectories because of the
+filtering.
+To show the superiority of our method, we compare the ‘track-and-detect query’ paradigm adopted
+in the most recent MOTR [16] to ours using the same instance segmentation baseline. We follow
+MOTR [16] exactly to set up the model and experiment settings. To exclude the influence of
+other factors, we do not use temporal feature aggregation in both methods. Table 2 (e) shows the
+comparisons on YouTube-VIS 2019. It can be seen that our InsPro achieves a higher performance
+even using a simpler query design. We attribute this advantage to our conscientiously designed
+modules described in Sec 3.
+5
+Conclusion
+In this paper, we propose a simple, fast and yet effective query-based framework for online VIS. In
+this framework, we rely on a novel instance query and proposal propagation mechanism to undertake
+VIS, where we generate object instances based on a set of evolving instance query-proposal pairs
+propagated from previous frames. This mechanism enables our model not only to associate object
+instances implicitly, but to utilize video temporal cues elegantly. To make this propagation mechanism
+work well, we develop several modules to ensure that the learned instance query-proposal pair
+keeps being bound to one object, These modules include an intra-query attention unit, a temporally
+consistent matching mechanism and a box deduplication loss. Extensive experiments on YouTube-
+VIS 2019 and 2021 verify the effectiveness of our designs, and show that our InsPro achieves superior
+VIS performance, outperforming all other online VIS methods.
+Acknowledgments
+This work is supported in part by the National Natural Science Foundation of China (Grant No.
+61721004), the Projects of Chinese Academy of Science (Grant No. QYZDB-SSW-JSC006), the
+Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA27000000),
+and the Youth Innovation Promotion Association CAS.
+10
+
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+pages 376–383. 2014.
+13
+
+A
+Appendix
+A.1
+More Details about InsPro-lite
+𝐼�
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+DIIM 3
+DIIM 2
+DIIM 1
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+⋯
+Intra-query Attn.
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+𝑟��
+DIIM 4
+DIIM 3
+DIIM 2
+DIIM 1
+DIIM 0
+DIIM 5
+Intra-query Attn.
+⋯
+𝑘𝑒𝑦 frame
+𝑛𝑜𝑛𝑘𝑒𝑦 frame
+𝑛𝑜𝑛𝑘𝑒𝑦 frame
+𝑘𝑒𝑦 frame
+⋯
+𝑡
+Figure 4: Overview of our InsPro-lite.
+Overview
+Figure 4 displays the overview of our InsPro-lite. In this framework, the video frames are
+divided into key frames and non-key frames, and we process them differently. On the key frame Ik,
+the segmentation head consists of 6 dynamic instance interaction modules (DIIM) and one intra-query
+attention module for generating VIS results rk, and both the instance query and proposal are updated.
+On the non-key frame Ik+1, the segmentation head contains only one dynamic instance interaction
+module to generate VIS results rk+1, and only the instance proposal is updated while the propagated
+instance query keeps unchanged. For more details, please refer to [32].
+Training Details
+Two-phase training is performed for InsPro-lite. In the first phase, we train the
+segmentation head processing key frames as described in Sec. 4.2. In the second phase, We train
+the segmentation head processing non-key frames while other parts in the network are fixed. Each
+training batch contains five frame images randomly sampled from the same video, with three as key
+frames and two as non-key frames.
+Effects of key frame interval
+Table 3 lists the performances of InsPro-lite using different key
+frame intervals. When K = 1, it represents the original InsPro model.
+Table 3: Performances of InsPro-lite (ResNet-50 backbone) using different key frame intervals. The
+inference speed is tested on a Nvidia RTX2080Ti GPU.
+K
+1
+5
+10
+15
+AP
+40.2
+39.4
+38.7
+37.5
+FPS
+26.3
+41.8
+45.7
+49.1
+A.2
+Additional Comparison on OVIS
+We additionally evaluate our InsPro on the occluded video instance segmentation dataset OVIS [54].
+OVIS is a very challenging dataset that contains many occlusion scenes. It consists of 296k high-
+quality instance masks (about 2× of YouTube-VIS 2019 [1]), 5.8 instances per video (about 3.4× of
+YouTube-VIS 2019), 25 semantic categories, 607 training videos and 140 validation videos.
+Table 4 shows the comparison between our InsPro and other state-of-the-art methods on the OVIS
+validation set. Due to the severe occlusion and crowded scenes, the performances of all methods
+14
+
+Table 4: Comparison of our InsPro to state-of-the-art methods on the OVIS validation. All methods
+use ResNet-50 [13] as backbone. The inference speed is tested on a Nvidia RTX2080Ti GPU.
+Method
+Online
+AP
+AP50
+AP75
+AR1
+AR10
+FPS
+STEm-Seg [7]
+
+13.8
+32.1
+11.9
+9.1
+20.0
+4.4
+MaskTrack R-CNN [1]
+
+10.9
+26.0
+8.1
+8.3
+15.2
+26.1
+MaskTrack R-CNN [1] + Calibration [54]
+
+14.9
+32.4
+12.5
+9.1
+19.5
+< 26.1
+SipMask [4]
+
+10.3
+25.4
+7.8
+7.9
+15.8
+30
+SipMask [4] + Calibration [54]
+
+13.9
+30.7
+11.9
+9.4
+19.4
+< 30
+CrossVIS [18]
+
+14.9
+32.7
+12.1
+-
+-
+25.6
+CrossVIS [18] + Calibration [54]
+
+18.1
+35.5
+16.9
+-
+-
+< 25.6
+InsPro
+
+21.1
+42.6
+19.0
+11.1
+25.6
+26.3
+Table 5: Comparison between InsPro and other object instance segmentation baselines on COCO
+validation set. All methods adopt ResNet-50 backbone, a training time of 36 epochs, and the same
+data augmentation. QueryInst [3] and InsPro both use 100 queries.
+Methods
+AP
+AP50
+AP75
+APS
+APM
+APL
+Mask R-CNN [5]
+37.5
+59.3
+40.2
+21.1
+39.6
+48.3
+QueryInst [3]
+39.8
+61.8
+43.1
+21.3
+42.7
+58.3
+InsPro
+39.4
+61.8
+41.9
+19.7
+42.9
+59.3
+on OVIS drop significantly compared to those on YouTube VIS [1]. To tackle occlusion, previous
+methods [1, 4, 18] apply temporal feature calibration [54] to align adjacent frames and complement
+the mission objective cues in severe occlusion, which increases the overhead of these systems.
+Our InsPro employs flexible instance query-proposal pair propagation and intra-query attention,
+outperforming all other popular methods without feature calibration.
+A.3
+Object Instance Segmentation Baselines Comparison
+The performance of the object instance segmentation baseline is also one of the important factors
+affecting the VIS results. Table 5 lists the performance comparison between our InsPro, Mask
+R-CNN [5] and QueryInst [3] on the COCO instance segmentation validation set. Except that they
+use different segmentation head structures, both of them adopt the same ResNet-50 backbone, same
+training time of 36 epochs, and the same data augmentation with ours. QueryInst [3] and InsPro both
+use 100 queries. It can be seen that our base model performs a bit poorer than QueryInst.
+A.4
+Qualitative Results
+We show qualitative results under some challenging scenarios on YouTube-VIS 2019 [1] and
+OVIS [54] in Figure 5 and Figure 6, respectively.
+Our InsPro is robust to many challenging situations, including fast motion, occlusion, crowd, similar
+objects, and motion blur, etc. For example, our InsPro can easy handle new objects with similar
+appearance, as shown in line 6 and 8 of Figure 5. This is because our InsPro propagates not only object
+queries but their corresponding proposals. Since those proposals have tracked objects positional
+prior encoded, when using such a query-proposal pair to predict objects, it is easy for the network to
+distinguish objects of similar appearance. However, InsPro may fail to segment some small objects
+since it has no specific design for processing small objects.
+A.5
+Additional Experiments on ImageNet VID
+To further verify the effectiveness and generality of our InsPro, we conduct additional experiments
+on ImageNet VID [55]. ImageNet VID [55] is a large-scale video dataset where the object instances
+on every frame are fully annotated with object bounding box, object category and instance ID. It
+consists of 3,862 training videos and 555 validation videos from 30 object categories. The average
+length of videos in ImageNet VID is 317 (about × 11.5 of YouTube-VIS 2019 [1]). We evaluate
+the joint detection and tracking performance of our InsPro on ImageNet VID. Following [1], we
+use video-level average precision (AP) as the evaluation metric and the video-level IoU is computed
+using box sequences instead of mask sequences.
+15
+
+Fast
+Motion
+Occlusion
+&
+Crowded 
+Failure
+cases
+time
+Figure 5: Qualitative results on the YouTube-VIS 2019 [1] validation set. Our InsPro with ResNet-
+50 [13] backbone performs well under many challenging scenes, including fast motion, occlusion and
+crowded. InsPro may fail to segment some small objects since it has no specific design for processing
+small objects.
+A.5.1
+Implementation Details
+We make minor modifications to InsPro to adapt it for joint detection and tracking on ImageNet
+VID. We remove the segmentation part in the instance segmentation module (Sec. 3.2), and the rest
+remain unchanged. During training, the segmentation losses Ldice and Lfocal are removed from the
+overall loss L (Equation 6) either. The modified InsPro, called InsPro-VID, is trained on both the
+ImageNet DET training set and the ImageNet VID training set. Each training batch contains three
+frame images. On VID, the three images are randomly sampled from the same video, while on DET,
+the three images are the same because DET contains only images. InsPro-VID is trained for 90k
+iterations and the initial learning rate is set to 2.5 × 10−5, which is divided by 10 at iteration 65k and
+80k, respectively.
+16
+
+person95%id
+surfboard78%id2person92%id2
+acket88%idersong5ld
+TPI%69MO3person93%id2
+TPI%08MOberson86%id2
+N
+cow60%id1person88%id2
+cow74%id1person92%id
+cow77%id1person.96%id
+skateboard 59%id1person96%id2
+skateboard91%id1
+erson37%id4元oerson91%
+skateboard 76%idperson94%id2
+kateboard76%idmperson95%id1
+surfboard83% id2person92%id2zebra 89%id1epre
+201%76
+zebra 87%id1zebra35%id 3
+zebra67%id2
+zebra91% id 1zebra32%id3
+zebra88%id2zebra72%id3
+zebra87%id2cow90%id4
+cow71%id1
+cow86%id3
+cow88%id2cow86%id4
+cow55%id5
+cow83%id1
+cow85%id3
+cow85%id2cow65%id5
+cow86%id4
+cow80%id1
+cow84%id3
+cow85%id2cow76%id4
+cow61%id5
+cow77%id1
+cow80%id3
+cow77%id2ersong4%
+surfboard85%id2cow46%id7
+cow76%id5
+cow84%id4
+cow63%id1
+cow79%id2
+COWperson60%id10
+person67%id2
+person69%id11
+person40%id3
+person78%id1
+persor
+oersol
+person54%id7
+person55%id9
+person27%id4person55%id10
+person75%id6
+person67%id5
+person80%id2
+person70%id11
+person36%id3
+person76%id1
+personperson28%id
+person49%id7
+person52%id9
+person28%id4person49%id2
+person40%id10
+person50%id6d5
+person48%id11
+person30%id3
+person29%i
+person58%id1
+person39%id8
+person35%id7
+person50%id9
+person28%id4person45%id2
+person33%Id10
+person43%id6rson50%id5
+person43%id11
+person32%id3
+person44%id8
+person65%id1
+person39%id7
+person50%id9
+person32%id4%id11
+person35%id10
+person47%id6
+person53%id5
+oer
+person27%id3
+person77%id1
+person49%id8
+person32%id7
+person25%person31%
+person48%id9Wurttemberg
+K27
+person79%id1
+person30%idWurttemberg
+on.25%id4Wurttemberg
+person88%id1
+erson26%id6Wurttemberg
+erson
+erson43ld
+erson
+person32%id4
+person29%idperson94%id1
+surfboard78%id2aWurttembperson87%id5
+person66%id3
+person87%id2
+skateboard 19% id 6
+skateboard42% id
+skateboard 61% id4person90%id5
+person76%id3
+person90%id2
+skateboard 32% id 6
+kateboperson94%id5
+person75%id3
+person91%id2
+ard40%id6
+skateboard62%id4person91%id5
+person82%id3
+person88%id2person88%id5
+person28%id2
+person57%id3
+9pi%0EpJe
+skateboard64%idairplane29%id4
+airplane60%id3
+airplane76%id1
+airplane47%id5
+airplane75%id2airplane64%id4
+airplane75%id3
+airplane74%id1
+airplane71%id5
+airplane78%id2airplane31%id6
+airplane72%id3
+airplane82%id1
+airplane 54%id 2airplane74%id3
+airplane76%id
+7surfboard 71%id2
+person92%id1airplane24%id4
+airplane82%id.3
+airplane85%id1
++
+airplane60%id5skateboard1g%6id6airplane60%id3
+airplane76%1d1airplane75%id3
+airplane74%id1person93%id2person93%id2
+ennisracket60%id1ennis racket 63% id 1
+person92%id2tennisracket70%id
+person95%id2time
+Figure 6: Qualitative results on the challenging OVIS [54] validation set.
+Table 6: Main results and ablation studies on ImageNet VID [55].
+(a) Comparison to other methods. All methods use
+ResNet-101 [13] as backbone.
+Methods
+AP50
+Faster R-CNN [31] + Viterbi [56]
+60.3
+D&T [57]
+60.5
+STMN [58] + Viterbi [56]
+60.4
+InsPro-VID
+64.1
+(b) Effectiveness of the temporal propagation, and
+temporally consistent matching (TCM).
+propagation
+TCM
+AP
+AP50
+AP75
+(A)
+9.7
+15.7
+9.1
+(B)
+
+29.2
+43.8
+30.6
+(C)
+
+
+33.2
+48.8
+35.5
+(c) Effectiveness of the proposed box deduplication
+loss. β is a hyperparameter in Equation 4.
+AP
+AP50
+AP75
+β=0
+33.2
+48.8
+35.5
+β=0.08
+37.8
+57.5
+40.0
+β=0.1
+38.9
+57.8
+42.6
+β=0.15
+39.5
+57.2
+42.7
+β=0.2
+39.2
+57.3
+42.7
+β=0.25
+39.1
+57.2
+43.0
+(d) Effectiveness of the intra-query attention. T is the
+length of the feature bank.
+AP
+AP50
+AP75
+T=20
+39.8
+59.1
+43.5
+T=40
+41.3
+59.9
+45.7
+T=60
+41.8
+60.7
+45.9
+T=80
+41.3
+60.0
+45.3
+T=100
+42.2
+60.6
+46.8
+T=120
+41.8
+60.1
+46.2
+A.5.2
+Main Results
+We compare our InsPro-VID with other methods in Table 6 (a). Other methods mainly use tracking
+or post-processing matching (e.g., Viterbi [56] algorithm) techniques to associate detection results
+across frames, which are more complex and unable to fully utilize temporal cues. Our InsPro-VID
+achieves instance association in parallel with detection and elegantly takes advantage of temporal
+clues in videos through the proposed instance query-proposal propagation mechanism. As shown in
+Table 6 (a), our InsPro-VID achieves superior performance compared over other methods.
+A.5.3
+Ablation Study
+We conduct extensive experiments on ImageNet VID to study the effectiveness and individual
+performance contribution of our proposed modules. All experiments use ResNet-50 [13] as backbone.
+Temporal Propagation and Matching
+Table 6 (b) shows how the proposed instance query and
+proposal temporal propagation mechanism and the temporally consistent matching strategy contribute
+to our high performance.
+17
+
+Parrot94%id2
+Parrot94%id1Sheep 95% id 4Sheep 96% id5Cow84%id3
+Cow91%id2
+Cow94%id1Cow92%id2
+Cow89%id3
+Cow96%id1Cow86%id2
+Cow87%id3
+Cow97%id1Cow69%id2
+Cow94%id3
+Cow96%id1Cow69%id2
+Cow88%id3
+Cow94%id1Cat 93% id 1
+Cat76%id2Cat89%id1
+Cat45%id2Cat 45% id 1
+Cat74%id2Parrot52%id2
+Parrot 78%id1Cat 46% id 1
+Cat67%id2Cat62%id1
+Cat47%id2Monkey40%id2
+Monkey42%id1Monkey56%id2
+Monkey56%id 1Monkey72%id2
+Monkey67%id1Monkey34% id 4
+Monkey67%id2
+Monkey73%id 1Monkey42%id4
+Monkey83%id2
+Monkey 85%id 1Palfot69%id2
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+Parrot 87%id 1n98%id
+Sheep92%id5Sheep93%id4Person96%id
+Sheep85%id5
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+Sheep95%id4Sheep 96%id
+双羊地走122
+128
+86
+67
+49
+(a) w/o BDL
+(b) w/ BDL
+𝑡𝑖𝑚𝑒
+Figure 7: Visualization of box deduplication loss effects on ImageNet VID. We display all predicted
+boxes on each frame without a score threshold, with 100 detections by default. (a) All predicted
+boxes in the first three frames are clustered around a few instances if the box deduplication loss is not
+applied, and newly emerging instances (shown in red box) on the last two frames cannot be detected.
+(b) After applying the box deduplication loss (BDL), each instance is predicted by only one accurate
+predicted box and the missing new objects in the first row are detected.
+Box Deduplication Loss
+Table 6 (c) shows the effectiveness of the proposed box deduplication
+loss. β is a hyperparameter in Equation 4, which controls the center distance between the duplicate
+predicted boxes and the ground truth box during training. β = 0 indicates no box deduplication loss
+is used during training and 33.2 AP is achieved. When we increase the value of β, the performance
+rises and saturates at 39.5 AP when β = 0.15.
+We also provide some qualitative results in Figure 7 to show the effect of the box deduplication
+loss. We display all predicted boxes on each frame without a score threshold, with 100 detections by
+default. The first row is the visualization results when β = 0. We can see that all predicted boxes
+in the first three frames are clustered around a few instances. Since in our temporal propagation
+mechanism the prediction of the next frame is generated based on the instance proposals propagated
+from the last frame, if all predicted boxes in the last frame are clustered around the existing instances,
+it will be difficult for the model to detect new objects in the next frame. As shown in Figure 7 (a), we
+can see that the newly emerging instances in red box on the last two frames are indeed not detected.
+Figure 7 (b) is the visualization results when β = 0.15. It shows that each instance has only one
+accurate predicted box and the new emerging objects are detected.
+Therefore, the proposed box deduplication loss enables our InsPro not only to alleviate the duplicate
+problem, but to detect new objects easily with the sparsely distributed unmatched proposal.
+Intra-query Attention
+Table 6 (d) shows the effectiveness of the intra-query attention. When
+increasing the size T of the feature bank, the instance query can capture more temporal cues and the
+performance improves considerably.
+A.6
+Broader Impact and Future Work
+Our InsPro introduces a novel instance query and proposal propagation mechanism to the VIS system,
+and constructs a simple, fast and yet effective online framework to achieve one-shot video instance
+segmentation. Our InsPro-lite can achieve promising accuracy while running in real-time. Due to
+its fine-grained object representation result and efficiency, we believe our VIS system InsPro can
+positively impact many applications such as autonomous driving and video editing, etc.
+For future work, we plan to verify the generality of our method in other query-based frameworks [11,
+25, 59], since our proposed temporal propagation mechanism is only applied in Sparse R-CNN [12]
+at present. Furthermore, we also intend to implement the simple yet effective temporal propagation
+mechanism in tasks that require instance association, such as multi-object tracking [60], panoptic
+segmentation [61, 62] in video, and semi-supervised video object segmentation [63].
+18
+
+iairplane3%
+tairplane51%
+airplane
+airplane4%6
+airplane2%6
+airplane2%%aiairplane1%
+airplane29
+airplane
+airplane1%
+airplane1%lane 1%
+airplane86%
+3%
+airplane1%
+airplane85%
+airplane1%
+airplane90%
+airplane3%
+tairplane2%
+airplane87%
+aiairplane1%
+airplane86%
+airplane1%
+airplane1%
+airplane1%
+airplane1%airplane2%
+airplane27%
+airplane67%
+airrairplane3%
+airplane3%
+airplane4%
+airplane19%
+airplane42%
+caairplane3%
+airnlane2
+20/
+airplane2%
+airplane2%
+airplane36%airplane2%
+airplane239
+airplane56%
+ainairplane 4%
+airplane59%
+airplane47%
+airplane10%
+airplane2%
+airplane2%
+airplane21%
+airplairplane2%
+airplane2%
+airplane1%airplane2o
+airplan
+airplane2%
+airplane 1%6
+airplane2%iairplane25%
+airplane1%
+Irplane
+airpairplane1%
+airplane
+airplane4%
+1aairplane8%%
+airplane2%%airplane1%
+airplane1%
+airpianei
+airplane10%
+airplane2%
+airplane
+airpla
+airplane
+airplane2%
+airplane84%airplane 19
+airplane1%
+airplane3%
+airplane2%
+rplane84%
+airplane3%
+airplane1%
+airplane80%
+airplane83%
+iairplane2%
+airplane81%irplane1%
+airplane1%
+airplane 1%rplar
+1.04
+airplane4%
+airplane86%
+hpiaiairplantairplane
+airplane1%
+airplane84%
+airplane87
+airplane85
+airplane89%
+airplane87%
+airplane86
\ No newline at end of file
diff --git a/udAzT4oBgHgl3EQf6_4i/content/tmp_files/load_file.txt b/udAzT4oBgHgl3EQf6_4i/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..94de0a9f386501b8d5dba7dde13d5e4c3028bb89
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf,len=1459
+page_content='InsPro: Propagating Instance Query and Proposal  for Online Video Instance Segmentation  Fei He1,2, Haoyang Zhang4, Naiyu Gao4, Jian Jia1,2, Yanhu Shan4,  Xin Zhao1,2∗, Kaiqi Huang1,2,3  1 CRISE, Institute of Automation, Chinese Academy of Sciences  2 School of Artificial Intelligence, University of Chinese Academy of Sciences  3 CAS Center for Excellence in Brain Science and Intelligence Technology 4 Horizon Robotics  {hefei2018,jiajian2018}@ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='cn, {haoyang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='zhang,naiyu01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='gao,yanhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='shan}@horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='ai  {xzhao,kaiqi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='huang}@nlpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='cn  Abstract  Video instance segmentation (VIS) aims at segmenting and tracking objects in  videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Prior methods typically generate frame-level or clip-level object instances  first and then associate them by either additional tracking heads or complex instance  matching algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This explicit instance association approach increases system  complexity and fails to fully exploit temporal cues in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In this paper, we  design a simple, fast and yet effective query-based framework for online VIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Relying on an instance query and proposal propagation mechanism with several  specially developed components, this framework can perform accurate instance  association implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Specifically, we generate frame-level object instances based  on a set of instance query-proposal pairs propagated from previous frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This  instance query-proposal pair is learned to bind with one specific object across  frames through conscientiously developed strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' When using such a pair to  predict an object instance on the current frame, not only the generated instance  is automatically associated with its precursors on previous frames, but the model  gets a good prior for predicting the same object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In this way, we naturally achieve  implicit instance association in parallel with segmentation and elegantly take  advantage of temporal clues in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' To show the effectiveness of our method  InsPro, we evaluate it on two popular VIS benchmarks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=', YouTube-VIS 2019  and YouTube-VIS 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Without bells-and-whistles, our InsPro with ResNet-50  backbone achieves 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2 AP and 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6 AP on these two benchmarks respectively,  outperforming all other online VIS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  1  Introduction  Video instance segmentation (VIS) [1] is a challenging but important computer vision task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It requires  not only segmenting object instances on each video frame but also associating them across all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Due to its fine-grained object representation form, it has got a wide range of applications in various  areas such as autonomous driving and video editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Existing VIS methods can be categorized into two groups: frame-level methods and clip-level  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Frame-level methods [1, 2, 3, 4] generally follow a ‘tracking-by-detection’ paradigm,  which first generate per-frame object instances by existing instance segmentation models [5, 6],  and then associate them across frames via additional tracking heads (as shown in Figure 1 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In  comparison, clip-level methods [7, 8, 9, 10] take a ‘clip-matching’ paradigm, which divide a video  ∗Corresponding author  36th Conference on Neural Information Processing Systems (NeurIPS 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='01882v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='CV]  5 Jan 2023  (a) Previous VIS paradigm  (b) Our InsPro  Instance  association  Ins 1 Ins 2  Ins 1 Ins 2  Ins 𝑦  Ins 𝑥  assign ID  Previous  Current  SegNet  SegNet  temporal propagation  Previous  Current  Ins 1  Ins 2  Ins 1  Ins 2  Our  SegNet  Our  SegNet  query-proposal pair  query-proposal pair  Figure 1: (a) Previous methods take a two-step approach to VIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' They first generate object instances  and then perform explicit instance association to link them across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' (b) Our InsPro implements  implicit instance association through a temporal propagation mechanism, achieving object instance  segmentation and tracking in one shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It generates frame-level object instances based on a set of  instance query and proposal pairs propagated from previous frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Since the instance query-proposal  pair is learned to represent one specific object across frames, the instance association is naturally  achieved in parallel with segmentation and video temporal clues are elegantly exploited meanwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  into multiple overlapped clips, generate instance predictions for each clip, and then associate these  clip-level predictions by some hand-crafted instance matching algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Whether frame-level or  clip-level methods, both of them inevitably need an explicit instance association step to fulfill object  tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This generally requires to design a complicated association strategy to achieve good tracking  performance, which is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' More importantly, the explicit association step increases model  complexity and slows inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Furthermore, this extra step also indicates that the temporal  clues intrinsic in videos are not well utilized, as the instance prediction is performed separately on  each frame or each clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  In this work, inspired by the recent success of query-based object detectors [11, 12], we propose a  simple, fast and yet effective query-based framework for online VIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Our system, dubbed as InsPro,  segments and tracks objects in one shot through an instance query and proposal propagation strategy  with carefully designed modules (Figure 1 (b)) , which eliminates the explicit instance association  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Specifically, our approach generates frame-level object instances based on a set of instance  query-proposal pairs propagated from previous frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In the learning process, we develop several  techniques to make sure that the generated instance query-proposal pair corresponds to one specific  object across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Thus, when an object instance is generated using such a query-proposal pair  on the current frame, it is automatically associated with its precursors on all previous frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In  this way, we achieve implicit object association without a linking step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Meanwhile, this instance  query-proposal propagation mechanism also enables our VIS system to achieve a better prediction  accuracy (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This benefits from the instance query-proposal pair’s encoding one object’s  temporal and spatial information across all previous frames, which provides a very good prior for  the model to infer the same object on the current frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In this sense, our query-based VIS method  actually implements an efficient way to exploit the intrinsic temporal clues in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  To fulfill the advantages of our VIS system, learning exclusive and expressive instance query-proposal  pairs is the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In this work, we develop several strategies to ensure the learning effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' First,  we design a temporally consistent matching mechanism to enforce the one-to-one correspondence  between the instance query-proposal pair and a specific ground truth object across frames during  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Second, we propose a box deduplication loss to enlarge the distance between instance  proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This helps suppress duplicate proposals on the same object and increase the exclusivity of  the generated instance query-proposal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' At the same time, the sparsely distributed unoccupied  query-proposal pairs can serve as candidates in the next frame to detect new objects, allowing our  system to achieve new object detection and tracking effortlessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Third, we propose an intra-query  attention module that enhances instance query with its predecessors encoding the same object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This  explicitly aggregates long-range object information into the query, augmenting its representation  capacity, which helps handle occlusion and motion blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  To validate the effectiveness and efficiency of our InsPro, we conduct extensive experiments on two  popular VIS benchmarks [1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=', YouTube-VIS 2019 and YouTube-VIS 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Without bells-and-  whistles, our InsPro with ResNet-50 [13] backbone achieves 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2 AP on YouTube-VIS 2019 and 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  AP on YouTube-VIS 2021 respectively, outperforming all other online VIS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Moreover, our  2  lite variant, InsPro-lite, reaches 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7 AP on YouTube-VIS 2019 at impressive 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7 FPS on a Nvidia  RTX2080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  In summary, we make the following contributions in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' 1) We propose a simple, fast and  yet effective query-based framework for online VIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' 2) We develop several techniques to make the  query-proposal pair propagation mechanism work smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' These techniques distinguish our work  from other query propagation-based object association methods [14, 15, 16], and make our work  simpler, more elegant and more effective than them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' 3) Our VIS system achieves the state-of-the-art  performances on two popular VIS benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  2  Related Work  Frame-level VIS Methods  mainly adopt a ‘tracking-by-detection’ paradigm and can run in an  online fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' They first generate instance predictions frame by frame and then perform explicit  instance association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' MaskTrack R-CNN [1] first proposes the VIS task and simply adds an additional  tracking head to Mask R-CNN [5] for instance association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Follow-up works [2, 3, 4, 17] improve  either the segmentation or the tracking algorithm to achieve better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' On the other  hand, some works [8, 18, 19, 20, 21, 22] attempt to perform temporal feature fusion to improve  instance segmentation and association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' For example, PCAN [22] proposes frame- and instance-level  prototypical cross-attention modules to leverage rich spatio-temporal information to facilitate better  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' All these methods require additional modules to achieve explicit instance association,  which expands model complexity and reduces inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' By contrast, our method performs  instance association implicitly through an instance query and proposal propagation mechanism, which  is simpler and naturally exploits the temporal and spatial consistency in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Clip-level VIS Methods  take a ‘clip-matching’ paradigm, which process multiple frames within  a clip simultaneously and then perform instance matching between clips to complete VIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' While  some methods [7, 23, 9] propagate instance information within a clip with well-designed propagation  modules to model temporal context, recent works [8, 10] utilize transformer [24] to model temporal  context in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' These methods normally need hand-crafted matching algorithms to  complete instance association between clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Although they usually achieve high performance, they  can only run in an offline mode, which restricts their application to limited areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In contrast, our  method achieves comparable performance but can run online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Query-based Methods  have attracted increasing attention in recent years due to their flexibility  and simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' DETR [11] first uses a set of learned queries interacting with image features to  encode objects, and then directly outputs detections by decoding the transformed queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Following  works [25, 26, 27, 28, 29] improve DETR in terms of either training efficiency or detection perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Sparse R-CNN [12] builds a query-based detector on top of R-CNN architecture [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Its  follow-up works [3, 32] extend it to instance segmentation and video object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Besides, Max-  DeepLab [33] proposes a box-free panoptic segmentation method with external query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' DAFL [34]  uses a set of queries to encode pedestrian information for pedestrian attribute recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  The success of DETR has also inspired query-based VIS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' VisTR [8] adapts DETR to the  VIS task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It takes a video clip as input and directly outputs the sequence of masks for each instance  orderly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' IFC [10] proposes inter-frame communication transformers to reduce the heavy computation  and memory usage of VisTR-like VIS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Similar to VisTR, Mask2Former [35] applies masked  attention to a video clip and directly predicts a 3D instance volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' To learn a powerful video-level  instance query, SeqFormer [36] aggregates temporal information from each frame to the instance  query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' These methods work on clips rather than frames, and achieve object association through  sharing of the queries within a clip rather than query propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Thus, they still need instance  matching between clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Instead, our method applies to frames, and can propagate query-proposal  pairs through the entire video and thereby can associate object instances over any frame length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Query Propagation-based Object Association Methods  have been recently explored in several  works, such as TransTrack [14], TrackFormer [15], MOTR [16] and EfficientVIS [37], which are also  inspired by query-based methods [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This shows the effectiveness and potential of such a new  object linking approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The differences between our InsPro and them are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  3  SegHead  SegHead  SegHead  𝑞����  𝑝����  𝑞�  𝑝�  ⋯  𝐼�  𝐼�  𝐼�  Feature  Extractor  Feature  Extractor  Feature  Extractor  Reg  𝐼�  𝑓�  ∅  ∅  𝑑𝑜𝑔  𝑝𝑒𝑟𝑠𝑜𝑛  ∅  ∅  Feature Bank  update  × 𝑀  𝑞���  𝑝���  𝑞�  𝑝�  Cls  Mask Branch  𝑓�  ����  Mask  FCN Head  Params  Intra-query  Attention  𝑜�  Temporal propagation  RoIAlign  (a) Overview of InsPro  (b) Details of SegHead  Weight  Generator  ⋯  𝑟�  𝑞�  𝑝�  𝑟�  𝑞�  𝑝�  𝑟�  Dynamic  Conv  Self-att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  DIIM  Figure 2: (a) Overview of our InsPro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It performs VIS by propagating instance query-proposal pairs  across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' qinit ∈ RN×C and pinit ∈ RN×4 are initial instance queries and proposals on the first  video frame, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' They are used in SegHead to predict instance results r0 on frame I0, and  to produce updated q0 and p0 which are propagated to the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' By repeating this process, we  complete the VIS task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' (b) Details of SegHead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It is a multi-stage network, consisting of a dynamic  instance interaction module (DIIM) and an instance segmentation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The former transforms  instance queries with RoI features of corresponding proposals and produces object features, while the  latter predicts object instances based on the object features and conditional convolution [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  First, our InsPro is different from them in the way of either tracking seen objects or detecting new  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' TransTrack is basically a ‘tracking-by-detection’ method, because it still needs to explicitly  match detection boxes to tracked boxes in each frame, while our InsPro performs implicit association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  More importantly, TransTrack, TrackFormer and MOTR adopt a track query subset to track seen  objects and an extra object query subset to detect new objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This requires additional heuristic rules  to combine two type queries, and may miss occluded or blurred objects with low scores, which can  result in object trajectory break [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Our InsPro simply propagates all object queries produced in  the previous frame to the current frame, and keeps using this set to track seen objects and detect  new objects, which is much simpler and more elegant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' As for EfficientVIS, our concurrent work, it  does not consider this new object detection problem, and its performance will probably be impacted  greatly if there are new objects in the next clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Furthermore, we design a more intelligent strategy to suppress duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' TransTrack and Track-  Former employ score filtering or NMS to reduce duplicate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' MOTR builds a temporal  aggregation network to learn more discriminative features to address this problem, while EfficientVIS  does not discuss this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' By contrast, we design a Box Deduplication Loss to suppress dupli-  cates and an Inter-query attention module to enhance queries with their predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Our solution  avoids heuristic rules and post-processing steps, and is more effective according to the experimental  results (see Table 2 (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  3  InsPro  We aim to design a simple and fast VIS system that performs instance association implicitly and  exploits video temporal clues elegantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' To this end, we take a query-based VIS approach that  predicts object instances on each frame based on a set of instance query-proposal pairs propagated  from previous frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In this section, we introduce our VIS system, InsPro, including an instance  query and proposal propagation mechanism and an instance segmentation head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Meanwhile, we also  describe those proposed techniques that make our propagation mechanism work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  Instance Query and Proposal Propagation  The instance query and proposal propagation mechanism enables our VIS system to perform object  instance association implicitly in parallel with instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Since it is inspired by the  recent query-based object detector Sparse R-CNN [12], we first briefly review Sparse R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  4  Sparse R-CNN [12] formulates object detection as a set prediction problem and achieves state-of-the-  art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It simplifies the detection pipeline and removes heuristic components like NMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Specifically, it first initializes a fixed set of learnable instance queries (N × C, N denotes the number  of queries and C the query dimension) paired with learnable instance proposals (N × 4) to describe  objects in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' As illustrated in Figure 2 (b), each instance query is convolved with the RoI  feature of the corresponding proposal to output a more discriminate feature ot [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' After multi-stage  iterative updating, the instance query encodes more accurate object appearance information while the  proposal captures more precise location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Finally, decoding the object feature ot produced using the  instance query-proposal pairs, we get the detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Inspired by this instance query-proposal representation of an object, we design a query-proposal  temporal propagation mechanism (as shown in Figure 2 (a)) to achieve implicit object instance associ-  ation and temporal cue utilization in VIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Our key insight is that there is a one-to-one correspondence  between the learned instance query-proposal pair and a specific object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' If we manage to preserve this  correspondence from the first frame to the one where the object finally disappears, then we realize  object tracking and object information propagation spontaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  To this end, we first initialize a set of instance queries qinit ∈ RN×C and proposals pinit ∈ RN×4 on  the first video frame I0, where qinit and pinit are learnable parameters and arranged in pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' After  learning, they are able to encode objects on the first frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Decoding them with the first frame image  feature inside the SegHead, we obtain instance results r0 as well as a new set of updated pairs (q0,  p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Then we propagate this pair set (q0, p0) to the next frame as input to the SegHead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Similarly,  we get the instance results r1 and another new set of (q1, p1) on this frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Among them, the object  instance produced on this frame shares the same ID with the one on the previous frame if they are  both decoded by the same slice of the instance queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In this way, we automatically link object  instances belonging to an identical object across frames and elegantly make use of object priors from  the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Repeating the above process until the last video frame, we then accomplish the VIS task on  this video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Please note that our InsPro simply propagates all object queries produced in the previous frame to  the current frame, and keeps using this set to track seen objects and detect new objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Instead,  recent works [14, 15, 16] that take a similar query-propagation mechanism use a track query set  to track seen objects and a new object query set to detect new objects respectively, which requires  additional heuristic rules to combine these two type queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Moreover, they rely on hand-crafted rules  like a score threshold to select a subset of track queries, and occluded objects with low prediction  scores are probably filtered out, which results in non-negligible true object missing and fragmented  trajectories [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In comparison, our method is obviously simpler, more elegant and more effective  (see Table 2 (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Intra-query Attention  Since frame-by-frame temporal propagation encodes only short-range  temporal cues, the instance query from just the last frame shows limitations in dealing with tough  scenarios, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=', occlusion and motion blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' To boost the representation capacity of instance query,  we augment it in practice with instance features from previous T frames [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Specifically, we  build a feature bank that caches instance features from previous T frames and perform intra-query  attention inside this bank to aggregate long-range temporal cues into the current instance query, as  shown in the upper part of Figure 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Formally, at frame It, instance features o from previous T  frames are put together to form a feature bank fb = {ot−T +1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' , ot}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Then, the enhanced instance  query is computed as:  qi  t =  �T −1  n=0 oi  t−n exp(ε(oi  t−n))  �T −1  m=0 exp(ε(oi  t−m))  + oi  t,  (1)  where i denotes the i-th query and ε(·) is a linear transformation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The enhanced qt is  basically a weighted sum of instance features inside the feature bank, and the weights are learned  upon the quality of the queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Experiments (Table 6 (c)) show that this augmentation improves the  query representation capacity greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  Segmentation Head  The segmentation head transforms the instance query and performs instance segmentation on each  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' As illustrated in Figure 2 (b), it is a multi-stage network and has two main parts: a dynamic  instance interaction module and an instance segmentation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  5  Dynamic Instance Interaction Module  transforms the instance query with the proposal RoI  feature and yields object features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It has M stages and forms an iterative structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' At the first stage,  given a pair of instance queries qt−1 ∈ RN×C and proposals pt−1 ∈ RN×4 that propagated from  the previous frame It−1, it first augments the instance queries by a self-attention module [24], which  models the inter-query relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' At the same time, it extracts the RoI feature of each proposal on the  feature map by RoIAlign [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Then, each enhanced instance query convolves with its corresponding  RoI feature through dynamic convolution [42] to get the object feature ot ∈ RN×C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Since the object  feature absorbs temporal cues encoded in instance queries, it has better representation ability than the  single-frame RoI feature (see Table 2 (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The object feature is used to predict object instances in the  following instance segmentation module, and the newly generated object boxes together with the  object features proceed as input to the next stage in the iterative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' At the final stage, the object  feature is processed together with its precursors from previous frames through the aforementioned  intra-query attention module, to produce instance queries for the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Instance Segmentation Module  decodes the object feature ot and produces VIS predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It has  three main heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' While the classification head predicts object classes, the regression one generates  object boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Another mask head is responsible for producing instance masks through a conditional  convolution [39] approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Specifically, it first uses ot to generate conditional convolution weights in  weight generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' As shown in Figure 2 (b), it inputs oi  t that represents the i-th object instance feature  to the weight generator and outputs a set of convolution parameters ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Meanwhile, it produces  mask feature maps by transforming FPN [43] feature maps through a mask branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Note that the  output mask feature maps have 8 channels and a 1  8 resolution of the input image, and are boosted by  concatenating a 2-channel relative coordinates map to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This relative coordinates map is computed  using the center of predicted object boxes and provides strong location cues for predicting instance  masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Finally, we feed the combined feature maps f mask  t  ∈ R10× H  8 × W  8 and convolution parameters  ωi to a mask FCN head, predicting the i-th object instance mask via a conditional convolution:  mi  t = CondConv(f mask  t  , ωi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  (2)  For more details about the instance segmentation module, please refer to [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  Temporally Consistent Matching  The key to the success of our InsPro is to make sure that the evolving instance query-proposal pair  corresponds to the same object across frames in a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' To ensure this, one technique we propose is  temporally consistent matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This technique matches predictions and ground truth during training,  assigns each ground truth object a proper prediction, and propagates the matching made on previous  frames to subsequent frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Specifically, given a training batch consisting of multiple consecutive frames, we first compute the  matching cost Lmatch between predictions and ground truth objects on the first frame:  Lmatch = λcls · Lcls + λL1 · LL1 + λgiou · Lgiou,  (3)  where Lcls is the focal loss [44] between predicted classifications and ground-truth labels, LL1 and  Lgiou are L1 loss and the generalized IoU loss [45] between predicted boxes and ground-truth boxes,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' λcls, λL1 and λgiou are loss weights and set as 2, 5, and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  We search for the best bipartite matching that minimizes the matching cost Lmatch with the Hungarian  algorithm [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' After finding the best matching on the first frame, we propagate this matching to other  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Concretely, if one ground truth object still exists on subsequent frames, it will be matched to  the prediction that is generated by the same instance query on the first frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' If there are new objects  emerging, new matching will be made between the new objects and yet unmatched predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' If a  ground truth object disappears, its corresponding predictions will not participate in a new matching  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Through this temporally consistent matching mechanism, we bind one ground truth object  to a single instance query during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  Loss Function  Box Deduplication Loss  Although the self-attention mechanism between queries has driven the  model to generate fewer duplication predictions [11], we still observe multiple overlapped proposal  6  time  (a) w/o BDL  (b) w/ BDL  Figure 3: (a) Multiple duplicate boxes exist on the same object across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' (b) After applying  the proposed box deduplication loss (BDL) in training, the duplicate predictions are significantly  suppressed along with temporal propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  boxes on the same object across many frames, as displayed in Figure 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We conjecture this is  caused by those unmatched queries which cannot be pushed away from those matched queries due  to lack of supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' To address this problem, we propose a box deduplication loss to push away  prediction boxes in terms of the center-to-center distance between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' As a result, not only the  duplication problem is alleviated, but the sparsely distributed unmatched query-proposal pairs can  serve as candidates in the next frame to detect and track new objects (see Figure 7 in Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The  loss is defined as:  Ldedup = 1  k  k  �  i=1  max(β − C2(b, ˆb  i  neg)  D2(b)  , 0),  (4)  where b is a ground truth box, ˆb  i  neg is a negative box that has the i-th largest IoU with b among those  unmatched predicted boxes, C(·) is the center distance calculation function, and D(·) is the diagonal  length calculation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' β is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1 and k as 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This loss penalizes the short distance between  b and ˆb  i  neg, and drags all other duplicate boxes away from b [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' With this new loss, our final box  loss function is formed as:  Lbox = λL1 · LL1 + λgiou · Lgiou + λdedup · Ldedup,  (5)  where λL1 and λgiou have the same values as in Equation 3, and λdedup is set as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Overall Loss  Given the one-to-one matching results, the final loss on each training frame is a sum  of classification, box and mask losses:  L = λcls · Lcls + λbox · Lbox + λdice · Ldice + λfocal · Lfocal,  (6)  where Ldice and Lfocal are dice loss [48] and focal loss [44] for foreground mask prediction,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We set λbox = 1, λdice = 5 and λfocal = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Finally, the losses of all training frames  inside a batch are summed together and normalized by the number of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  4  Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  Datasets and Evaluation Metrics  We evaluate our method on YouTube-VIS 2019 and 2021 benchmarks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' YouTube-VIS 2019  consists of 2,238 training videos and 302 validation videos, and labels 40 object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' YouTube-  VIS 2021 is an extended version, which comprises 2,985 training videos and 421 validation videos,  and labels improved 40 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' All videos in these two datasets are annotated every 5 frames with  object bounding box, object category, instance mask and instance ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Following [1], we report the  video-level average precision (AP) and average recall (AR) on the validation sets as the evaluation  metrics, where both accurate instance segmentation and instance association are necessary to achieve  high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  7  Table 1: Comparison of our InsPro to state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' All methods use ResNet-50 as  backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' C: additionally using COCO train2017 images that contain YouTube-VIS categories for  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The inference speed is tested on a Nvidia RTX2080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' ∗ indicates using deformable  convolution [52] in backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' ‡ indicates that the FPS is measured by parallel processing of images  in one clip rather than sequential processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  YouTube-VIS 2019 Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  YouTube-VIS 2021 Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Method  Online  AP  AP50 AP75 AR1 AR10  AP  AP50 AP75 AR1 AR10  FPS  STEm-Seg [7] (C)  \x17  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  VisTR [8]  \x17  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0‡  Propose-Reduce [9] (C)  \x17  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7 < 20  MaskProp∗ [23]  \x17  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9 < 10  IFC [10]  \x17  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
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+page_content='3  < 20  InsPro-lite  \x13  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
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+page_content='3  InsPro (C)  \x13  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
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+page_content='2  Implementation Details  We implement our InsPro with Detectron2 [49], and most hyperparameters are set following Sparse  R-CNN [12] and CondInst [39] unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' More implementation details can be found  in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Training Details  We employ AdamW [50] with an initial learning rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5 × 10−5 and weight  decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0001 as our model optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We initialize our model with parameters pre-trained on  COCO [51], and train it for 32k iterations where the learning rate is divided by 10 at iterations 24k  and 28k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The training is performed end-to-end on 8 Nvidia RTX2080Ti GPUs and each  GPU holds one mini-batch which contains three frame images randomly sampled from the same  video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Data augmentation includes only random horizontal flip and multi-scale training where the  training image is resized so that the length of its shortest side is at least 288 and at most 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Unless  otherwise noted, our InsPro adopts ResNet-50 [13] as backbone and uses 100 instance queries in our  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Inference Details  In inference, we resize the frame image size to 640 × 360, which follows  MaskTrack R-CNN [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The size of the feature bank is set to 18 by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' If the generated proposal  box exceeds the frame’s boundaries, it will be clipped to corresponding boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' No multi-scale  testing is adopted in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  InsPro-lite  We also build a lite version of our method, named InsPro-lite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In this variant, inspired  by [32], we divide video frames into key frames and non-key frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=', we select one key frame  per K frames in a video and treat other frames as non-key ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' K is 10 by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' On key frames,  we conduct the dynamic instance interaction 6 times while only once on non-key frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This takes  advantage of the redundancy of videos and helps reduce inference computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Our InsPro-lite  reaches a high inference speed of 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7 FPS at a small accuracy loss (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  Main Results  We perform a thorough comparison of our InsPro to state-of-the-art VIS methods on YouTube-VIS  2019 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Existing VIS methods can be divided into two categories according to whether they  run online or offline [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Since some methods [7, 9] use 80k transformed COCO images [51] as  extra training data to prevent overfitting to YouTube-VIS, for a fair comparison, we also report our  8  Table 2: Ablation studies on YouTube-VIS 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  (a) Effectiveness of instance query and pro-  posal propagation, and temporally consis-  tent matching (TCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  query  proposal  TCM  AP  AP50  AP75  (A)  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  (B)  \x13  \x13  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  (C)  \x13  \x13  \x13  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  (D)  \x13  \x13  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  (E)  \x13  \x13  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  (b) Effectiveness of the proposed  box deduplication loss (BDL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  AP  AP50  AP75  FPS  w/o BDL  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  w/ BDL  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  (c) Intra-query attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' T is the  length of the feature bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  AP  AP50  AP75  FPS  T=1  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  T=9  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  T=18  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  T=27  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  T=36  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  (d) Comparison between our temporal propagation paradigm  and ‘tracking-by-detection’ paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  AP  AP50  AP75  Param (M)  FLOPs (G)  FPS  Tracking-by-detection  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  Ours  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  (e) Comparison between our united query  and ‘track-and-detect query’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  AP  AP50  AP75  Track-and-detect query  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  Ours  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  results with and without extra COCO training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Table 1 presents all the results obtained with a  ResNet-50 backbone on a Nvidia RTX2080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  YouTube-VIS 2019  Table 1 (left) shows the comparison between our InsPro and other state-of-the-  art methods on YouTube-VIS 2019 validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We can see that, in the online group, our InsPro  outperforms all other popular methods under the same data setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Specifically, our InsPro achieves  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2 AP without COCO data and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2 AP with COCO data respectively, surpassing other online VIS  methods by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Even our lite version, InsPro-lite, performs better than all other online  methods trained without COCO data, reaching 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7 AP at an impressive speed of 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7 FPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  YouTube-VIS 2021  Table 1 (right) displays results on YouTube-VIS 2021 validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It shows  a similar comparison pattern to YouTube-VIS 2019 and our InsPro achieves the state-of-the-art  performance once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  Ablation Study  We conduct extensive experiments on YouTube-VIS 2019 to study the effectiveness and individual  performance contribution of our proposed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Temporal Propagation and Matching  Our InsPro is built on the proposed instance query and  proposal temporal propagation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Table 2 (a) shows how this mechanism contributes to  our high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In this table, method A represents the video instance segmentation baseline,  where each frame is processed individually without any temporal propagation, and object instances  generated on each frame are linked if they are produced from the same instance query slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Since  this method lacks the mechanism to ensure the instance query-proposal pair corresponds to the  same object across frames, it only achieves 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0 AP due to inaccurate instance association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' By  contrast, when we add the temporal propagation (method B), we can easily improve the performance  significantly to 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3 AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This evidences the importance and effectiveness of the proposed temporal  propagation technique in a query-based VIS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' If we further adopt the temporally consistent  matching strategy during training (method C), we achieve an even better performance of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4 AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  We also analyze the separate performance of propagating only instance query (method D) or instance  proposal (method E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The results show that these two settings achieve a similar performance boost  (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7 AP vs 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6 AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Applying them together yields 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4 AP, bringing further performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Box Deduplication Loss  We propose a box deduplication loss to suppress the duplicate proposal  boxes on the same object across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' With the qualitative results shown in Figure 3, we show the  quantitative comparison in Table 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We can see that supervising the learning with this loss during  training can lead 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0 AP improvement (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4 AP vs 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4 AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This performance gain is brought by  fewer duplicate boxes and fewer missed detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  9  Intra-query Attention  We perform intra-query attention inside a feature bank to augment the  instance query so that it can capture long-range temporal cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' As we can see in Table 2 (c), this  simple method works well and improves the performance considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In particular, T = 1 indicates  no intra-query attention is used and 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4 AP is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' When we increase the volume T of the  feature bank, the performance rises and saturates at 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2 AP with T = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It is worthwhile to note  that this lightweight yet effective intra-query attention module brings almost no speed drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Temporal Propagation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Tracking-by-Detection  Despite the fact that our InsPro does not  perform explicit instance association, it still outperforms all other online methods implementing  explicit tracking or matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' To verify that our superior performance comes from the temporal  propagation mechanism rather than the instance segmentation model design, we compare our temporal  propagation VIS approach to the typical ‘tracking-by-detection’ paradigm with the same instance  segmentation baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We implement a ‘tracking-by-detection’ VIS system by replacing the Mask  R-CNN part in MaskTrack R-CNN [1] with our instance segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In this case, the only  independent variable is the object tracking method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  As shown in Table 2 (d), our InsPro surpasses the ‘tracking-by-detection’ model by a large margin  even if our design is simpler and faster, which soundly proves the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We  argue again that this is because the evolving instance query-proposal pair in propagation encodes  object temporal and spatial cues intrinsic in videos, whereas ‘tracking-by-detection’ methods are  generally incapable of exploiting this advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Our United Query vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Track-and-Detect Query  We further compare our method to those MOT  methods that adopt a similar query-propagation method for object tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' These methods rely on  two different query sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=', a track query set and an object query set, to track seen objects and detect  new objects respectively, while we only maintain one united query set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' They need heuristic rules to  combine these two type queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Meanwhile, they manually select track queries with high scores  from the previous frame to build the track query set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This makes them complex and less effective  in tracking since occluded objects with low scores probably have broken trajectories because of the  filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  To show the superiority of our method, we compare the ‘track-and-detect query’ paradigm adopted  in the most recent MOTR [16] to ours using the same instance segmentation baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We follow  MOTR [16] exactly to set up the model and experiment settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' To exclude the influence of  other factors, we do not use temporal feature aggregation in both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Table 2 (e) shows the  comparisons on YouTube-VIS 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It can be seen that our InsPro achieves a higher performance  even using a simpler query design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We attribute this advantage to our conscientiously designed  modules described in Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  5  Conclusion  In this paper, we propose a simple, fast and yet effective query-based framework for online VIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In  this framework, we rely on a novel instance query and proposal propagation mechanism to undertake  VIS, where we generate object instances based on a set of evolving instance query-proposal pairs  propagated from previous frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This mechanism enables our model not only to associate object  instances implicitly, but to utilize video temporal cues elegantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' To make this propagation mechanism  work well, we develop several modules to ensure that the learned instance query-proposal pair  keeps being bound to one object, These modules include an intra-query attention unit, a temporally  consistent matching mechanism and a box deduplication loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Extensive experiments on YouTube-  VIS 2019 and 2021 verify the effectiveness of our designs, and show that our InsPro achieves superior  VIS performance, outperforming all other online VIS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Acknowledgments  This work is supported in part by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  61721004), the Projects of Chinese Academy of Science (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' QYZDB-SSW-JSC006), the  Strategic Priority Research Program of Chinese Academy of Sciences (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' XDA27000000),  and the Youth Innovation Promotion Association CAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
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+page_content=' Jia, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Panopticdepth: A unified framework for depth-aware panoptic  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern  Recognition, pages 1632–1642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  [63] Avinash Ramakanth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Venkatesh Babu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Seamseg: Video object segmentation using patch  seams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,  pages 376–383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  13  A  Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  More Details about InsPro-lite  𝐼�  𝑞�  𝑝�  𝑟�  DIIM 4  DIIM 3  DIIM 2  DIIM 1  DIIM 0  DIIM 5  CNN  𝑓�  CNN  𝑓���  𝐼���  DIIM  𝑝���  𝑟���  𝑞��  𝑝���  𝑞�  CNN  𝑓���  DIIM  𝑝���  𝑟���  𝑞�  𝐼���  CNN  𝑓��  𝑞�  𝑝����  𝐼��  ⋯  Intra-query Attn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  𝑞��  𝑝��  𝑟��  DIIM 4  DIIM 3  DIIM 2  DIIM 1  DIIM 0  DIIM 5  Intra-query Attn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  ⋯  𝑘𝑒𝑦 frame  𝑛𝑜𝑛𝑘𝑒𝑦 frame  𝑛𝑜𝑛𝑘𝑒𝑦 frame  𝑘𝑒𝑦 frame  ⋯  𝑡  Figure 4: Overview of our InsPro-lite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Overview  Figure 4 displays the overview of our InsPro-lite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In this framework, the video frames are  divided into key frames and non-key frames, and we process them differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' On the key frame Ik,  the segmentation head consists of 6 dynamic instance interaction modules (DIIM) and one intra-query  attention module for generating VIS results rk, and both the instance query and proposal are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  On the non-key frame Ik+1, the segmentation head contains only one dynamic instance interaction  module to generate VIS results rk+1, and only the instance proposal is updated while the propagated  instance query keeps unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' For more details, please refer to [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Training Details  Two-phase training is performed for InsPro-lite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In the first phase, we train the  segmentation head processing key frames as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' In the second phase, We train  the segmentation head processing non-key frames while other parts in the network are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Each  training batch contains five frame images randomly sampled from the same video, with three as key  frames and two as non-key frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Effects of key frame interval  Table 3 lists the performances of InsPro-lite using different key  frame intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' When K = 1, it represents the original InsPro model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Table 3: Performances of InsPro-lite (ResNet-50 backbone) using different key frame intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The  inference speed is tested on a Nvidia RTX2080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  K  1  5  10  15  AP  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  FPS  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  Additional Comparison on OVIS  We additionally evaluate our InsPro on the occluded video instance segmentation dataset OVIS [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  OVIS is a very challenging dataset that contains many occlusion scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It consists of 296k high-  quality instance masks (about 2× of YouTube-VIS 2019 [1]), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8 instances per video (about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4× of  YouTube-VIS 2019), 25 semantic categories, 607 training videos and 140 validation videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Table 4 shows the comparison between our InsPro and other state-of-the-art methods on the OVIS  validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Due to the severe occlusion and crowded scenes, the performances of all methods  14  Table 4: Comparison of our InsPro to state-of-the-art methods on the OVIS validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' All methods  use ResNet-50 [13] as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The inference speed is tested on a Nvidia RTX2080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Method  Online  AP  AP50  AP75  AR1  AR10  FPS  STEm-Seg [7]  \x17  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  MaskTrack R-CNN [1]  \x13  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  MaskTrack R-CNN [1] + Calibration [54]  \x13  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  SipMask [4]  \x13  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  30  SipMask [4] + Calibration [54]  \x13  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  < 30  CrossVIS [18]  \x13  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  CrossVIS [18] + Calibration [54]  \x13  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9 < 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  InsPro  \x13  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  Table 5: Comparison between InsPro and other object instance segmentation baselines on COCO  validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' All methods adopt ResNet-50 backbone, a training time of 36 epochs, and the same  data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' QueryInst [3] and InsPro both use 100 queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Methods  AP  AP50  AP75  APS  APM  APL  Mask R-CNN [5]  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  QueryInst [3]  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  InsPro  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='7  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='9  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  on OVIS drop significantly compared to those on YouTube VIS [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' To tackle occlusion, previous  methods [1, 4, 18] apply temporal feature calibration [54] to align adjacent frames and complement  the mission objective cues in severe occlusion, which increases the overhead of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Our InsPro employs flexible instance query-proposal pair propagation and intra-query attention,  outperforming all other popular methods without feature calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  Object Instance Segmentation Baselines Comparison  The performance of the object instance segmentation baseline is also one of the important factors  affecting the VIS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Table 5 lists the performance comparison between our InsPro, Mask  R-CNN [5] and QueryInst [3] on the COCO instance segmentation validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Except that they  use different segmentation head structures, both of them adopt the same ResNet-50 backbone, same  training time of 36 epochs, and the same data augmentation with ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' QueryInst [3] and InsPro both  use 100 queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It can be seen that our base model performs a bit poorer than QueryInst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='4  Qualitative Results  We show qualitative results under some challenging scenarios on YouTube-VIS 2019 [1] and  OVIS [54] in Figure 5 and Figure 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Our InsPro is robust to many challenging situations, including fast motion, occlusion, crowd, similar  objects, and motion blur, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' For example, our InsPro can easy handle new objects with similar  appearance, as shown in line 6 and 8 of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' This is because our InsPro propagates not only object  queries but their corresponding proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Since those proposals have tracked objects positional  prior encoded, when using such a query-proposal pair to predict objects, it is easy for the network to  distinguish objects of similar appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' However, InsPro may fail to segment some small objects  since it has no specific design for processing small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5  Additional Experiments on ImageNet VID  To further verify the effectiveness and generality of our InsPro, we conduct additional experiments  on ImageNet VID [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' ImageNet VID [55] is a large-scale video dataset where the object instances  on every frame are fully annotated with object bounding box, object category and instance ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It  consists of 3,862 training videos and 555 validation videos from 30 object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The average  length of videos in ImageNet VID is 317 (about × 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5 of YouTube-VIS 2019 [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We evaluate  the joint detection and tracking performance of our InsPro on ImageNet VID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Following [1], we  use video-level average precision (AP) as the evaluation metric and the video-level IoU is computed  using box sequences instead of mask sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  15  Fast  Motion  Occlusion  &  Crowded  Failure  cases  time  Figure 5: Qualitative results on the YouTube-VIS 2019 [1] validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Our InsPro with ResNet-  50 [13] backbone performs well under many challenging scenes, including fast motion, occlusion and  crowded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' InsPro may fail to segment some small objects since it has no specific design for processing  small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  Implementation Details  We make minor modifications to InsPro to adapt it for joint detection and tracking on ImageNet  VID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We remove the segmentation part in the instance segmentation module (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2), and the rest  remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' During training, the segmentation losses Ldice and Lfocal are removed from the  overall loss L (Equation 6) either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The modified InsPro, called InsPro-VID, is trained on both the  ImageNet DET training set and the ImageNet VID training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
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+page_content='9  T=80  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='0  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  T=100  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  T=120  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='8  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='1  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2  Main Results  We compare our InsPro-VID with other methods in Table 6 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Other methods mainly use tracking  or post-processing matching (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=', Viterbi [56] algorithm) techniques to associate detection results  across frames, which are more complex and unable to fully utilize temporal cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Our InsPro-VID  achieves instance association in parallel with detection and elegantly takes advantage of temporal  clues in videos through the proposed instance query-proposal propagation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' As shown in  Table 6 (a), our InsPro-VID achieves superior performance compared over other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='3  Ablation Study  We conduct extensive experiments on ImageNet VID to study the effectiveness and individual  performance contribution of our proposed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' All experiments use ResNet-50 [13] as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Temporal Propagation and Matching  Table 6 (b) shows how the proposed instance query and  proposal temporal propagation mechanism and the temporally consistent matching strategy contribute  to our high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
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+page_content='双羊地走122  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
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+page_content='(a) w/o BDL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='(b) w/ BDL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='𝑡𝑖𝑚𝑒  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='Figure 7: Visualization of box deduplication loss effects on ImageNet VID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We display all predicted  boxes on each frame without a score threshold, with 100 detections by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' (a) All predicted  boxes in the first three frames are clustered around a few instances if the box deduplication loss is not  applied, and newly emerging instances (shown in red box) on the last two frames cannot be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  (b) After applying the box deduplication loss (BDL), each instance is predicted by only one accurate  predicted box and the missing new objects in the first row are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Box Deduplication Loss  Table 6 (c) shows the effectiveness of the proposed box deduplication  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' β is a hyperparameter in Equation 4, which controls the center distance between the duplicate  predicted boxes and the ground truth box during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' β = 0 indicates no box deduplication loss  is used during training and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='2 AP is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' When we increase the value of β, the performance  rises and saturates at 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='5 AP when β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  We also provide some qualitative results in Figure 7 to show the effect of the box deduplication  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We display all predicted boxes on each frame without a score threshold, with 100 detections by  default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' The first row is the visualization results when β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' We can see that all predicted boxes  in the first three frames are clustered around a few instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Since in our temporal propagation  mechanism the prediction of the next frame is generated based on the instance proposals propagated  from the last frame, if all predicted boxes in the last frame are clustered around the existing instances,  it will be difficult for the model to detect new objects in the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' As shown in Figure 7 (a), we  can see that the newly emerging instances in red box on the last two frames are indeed not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Figure 7 (b) is the visualization results when β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' It shows that each instance has only one  accurate predicted box and the new emerging objects are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Therefore, the proposed box deduplication loss enables our InsPro not only to alleviate the duplicate  problem, but to detect new objects easily with the sparsely distributed unmatched proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  Intra-query Attention  Table 6 (d) shows the effectiveness of the intra-query attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' When  increasing the size T of the feature bank, the instance query can capture more temporal cues and the  performance improves considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='6  Broader Impact and Future Work  Our InsPro introduces a novel instance query and proposal propagation mechanism to the VIS system,  and constructs a simple, fast and yet effective online framework to achieve one-shot video instance  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Our InsPro-lite can achieve promising accuracy while running in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Due to  its fine-grained object representation result and efficiency, we believe our VIS system InsPro can  positively impact many applications such as autonomous driving and video editing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content='  For future work, we plan to verify the generality of our method in other query-based frameworks [11,  25, 59], since our proposed temporal propagation mechanism is only applied in Sparse R-CNN [12]  at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
+page_content=' Furthermore, we also intend to implement the simple yet effective temporal propagation  mechanism in tasks that require instance association, such as multi-object tracking [60], panoptic  segmentation [61, 62] in video, and semi-supervised video object segmentation [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udAzT4oBgHgl3EQf6_4i/content/2301.01882v1.pdf'}
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+Quantifying Rare Events in Stochastic Reaction-Diffusion Dynamics Using Tensor
+Networks
+Schuyler B. Nicholson1 and Todd R. Gingrich1
+1Department of Chemistry, Northwestern University,
+2145 Sheridan Road, Evanston,
+Illinois 60208, USA
+(Dated: January 11, 2023)
+The interplay between stochastic chemical reactions and diffusion can generate rich spatio-
+temporal patterns. While the timescale for individual reaction or diffusion events may be very fast,
+the timescales for organization can be much longer. That separation of timescales makes it particu-
+larly challenging to anticipate how the rapid microscopic dynamics gives rise to macroscopic rates in
+the non-equilibrium dynamics of many reacting and diffusing chemical species. Within the regime
+of stochastic fluctuations, the standard approach is to employ Monte Carlo sampling to simulate
+realizations of random trajectories. Here, we present an alternative numerically tractable approach
+to extract macroscopic rates from the full ensemble evolution of many-body reaction diffusion prob-
+lems. The approach leverages the Doi-Peliti second-quantized representation of reaction-diffusion
+master equations along with compression and evolution algorithms from tensor networks. By fo-
+cusing on a Schl¨ogl model with one-dimensional diffusion between L otherwise well-mixed sites, we
+illustrate the potential of the tensor network approach to compute rates from many-body systems,
+here with approximately 3 × 1015 microstates. Specifically, we compute the rate for switching be-
+tween metastable macrostates, with the time to compute those rates growing subexponentially in
+L. Because we directly work with ensemble evolutions, we crucially bypass many of the difficulties
+encountered by rare event sampling techniques—detailed balance and reaction coordinates are not
+needed.
+I.
+INTRODUCTION
+Connecting microscopic kinetics to emergent rates has
+a long history for equilibrium and nonequilibrium sys-
+tems.
+Complex systems often settle into metastable
+basins with rare transitions regulating the rates of switch-
+ing between the emergent states. In equilibrium, transi-
+tion state theory [1, 2], variational transition state the-
+ory [3], Grote-Hynes theory [4, 5], and transition path
+sampling (TPS) [6, 7] have given strategies to estimate
+the transition rate.
+Away from equilibrium, theories
+built upon a free energy landscape lose their applicabil-
+ity, yet trajectory sampling approaches can still extract
+rates from microscopic dynamics, either through a noise-
+guided TPS approach [8, 9] or using forward flux sam-
+pling (FFS) [10–12]. For all their merits, sampling ap-
+proaches bring intrinsic challenges. How many samples
+are required to estimate the rate? Is a good approximate
+reaction coordinate needed, and if so, how is it found
+[13]? How does the expense grow with larger or more
+complicated microscopic systems?
+Motivated by those
+ubiquitous concerns, we set out to consider a different
+approach—one that effectively samples all possible tra-
+jectories as an ensemble evolution instead of propagating
+individual realizations. Such a proposition may appear
+absurd given the exploding state space presented by the
+many-body problem, but we show that for the class of
+discrete state, continuous-time, reaction-diffusion master
+equations (RDME), we can evolve the full distribution of
+a classical many-body systems in a numerically control-
+lable manner by combining two well-developed methods,
+the Doi-Peliti (DP) formalism with tensor network (TN)
+algorithms.
+While TPS and FFS have been developed for a va-
+riety of dynamical equations of motion (Hamiltonian,
+Langevin, Markov jump, etc.), we focus our attention
+solely on the chemical master equation (CME) dynam-
+ics, the setting where FFS was first developed [10]. More
+specifically, we study discrete reaction-diffusion dynamics
+of L sites or voxels, [14, 15]. The rationale for this fo-
+cus is twofold. From a practical perspective, the RDME
+is important as a popular method to model chemical
+and biological processes [16, 17]. From a theoretical per-
+spective, the RDME is interesting because it provides a
+setting where one can analyze the influence of stochas-
+ticity on the emergence of patterns [18–20].
+In many
+pattern-formation problems, it is possible to understand
+the underlying physics in terms of a mean-field partial
+differential equation that tracks spatiotemporal evolution
+of deterministically evolving fields [21, 22]. It is, how-
+ever, well-appreciated that stochastic effects can quan-
+titatively and even qualitatively impact kinetics [23–28].
+It is therefore important that the methodology we dis-
+cuss captures more than just the typical dynamics. Ac-
+curately computing transition rates between metastable
+states requires quantitatively resolving even rare tails of
+the state space distribution since these rare fluctuations
+can be instrumental in triggering transitions [11, 29].
+Historically, similar schemes for the CME ensemble
+evolution have been hampered by the curse of dimension-
+ality. As a result, exact solutions are rare [30] and numer-
+ical solutions often resort to sampling methods built upon
+the kinetic Monte-Carlo stochastic simulation algorithm
+(SSA) [31, 32], also known as the Gillespie algorithm.
+Another approach is to evolve a subset of the state space
+through the finite state projection (FSP) method [33].
+arXiv:2301.03717v1  [cond-mat.stat-mech]  9 Jan 2023
+
+2
+FSP has previously been combined with TNs to approx-
+imately describe the CME [34, 35], resembling our use
+of TNs to tame the many-body problem. Unlike those
+projective methods, however, the approach we describe
+uses the time-dependent variational principle (TDVP) to
+evolve the dynamics with probability conservation at all
+times, akin to prior work with diffusion and no reac-
+tions [36, 37]. Since we estimate rare events by measuring
+the distribution’s small probability fluxes, conservation
+of probability and high-fidelity dynamical evolution are
+materially important.
+While the formalism we describe applies quite gener-
+ally to different well-mixed and reaction-diffusion models,
+we center our paper around a single model problem: rate
+calculations for transitions between basins of the Schl¨ogl
+model [38] in the bistable regime, both with and without
+diffusion. In Sec. II, we first illustrate the rare event prob-
+lem and extract rate constants from a propagator in the
+well-mixed regime.
+For more complicated systems (ei-
+ther systems with more chemical species or more voxels),
+the curse of dimensionality precludes the exact calcula-
+tion of the propagator, but the remainder of the paper
+shows how to employ TNs to follow the same concep-
+tual path. In Sec. III, we review the Doi-Peliti frame-
+work for writing reaction-diffusion dynamics in a second-
+quantized field theoretic form. Sec. IV then reviews the
+TN matrix product state (MPS) ansatz that enables en-
+semble evolution via TDVP. Those methods are applied
+to the many-body rate calculation in Sec. V. Finally, in
+Sec. VI, we show results for the RDME with diffusion be-
+tween L well-mixed Schl¨ogl voxels, demonstrating that
+the methodology can extend numerical transition rate
+calculations to regimes where the SSA becomes imprac-
+tical.
+The approach effectively evolves the probability
+of occupying each microstate in an unfathomably large
+many-body state space, yet the calculations scale sub-
+exponentially with system size and do not assume detail-
+balanced equilibrium.
+II.
+ENSEMBLE RATE CALCULATIONS
+Consider an archetypal two-state rate process between
+reactants and products R −−⇀
+↽−− P, initiated entirely in the
+reactant state. After some reaction timescale, the con-
+centration of products grows before eventually leveling
+off at its equilibrium value. The reaction rate can sim-
+ply be defined and measured in terms of how quickly the
+product concentration rises. An alternative formulation
+is to study the fate of an individual reactant on its path
+to transitioning into a product. The TPS and FFS path
+sampling approaches extract the reaction rate by focusing
+on the statistical properties of these individual trajecto-
+ries. Algorithms exist to propagate those single trajecto-
+ries, algorithms that can be practically implemented even
+when the system grows large and complex. By contrast,
+propagating the entire ensemble of trajectories has typ-
+ically been limited to simple low-dimensional problems.
+In this section, we illustrate an ensemble-evolution rate
+calculation for a simple case in order to develop notation
+and lay the groundwork for the many-body problem. The
+remainder of the paper will show that the methodology
+can be extended beyond such simple problems by em-
+ploying TNs.
+A.
+Well-mixed Schl¨ogl Model
+Consider the well-mixed Schl¨ogl model, a quintessen-
+tial example of bistability in chemical reaction networks
+(CRNs) [39].
+The model tends to be studied under
+chemostatted conditions that fix the concentrations of
+species A and B while allowing species X’s concentration
+to evolve according to the four reactions:
+2X + A
+˜c1
+−−⇀
+↽−−
+˜c2
+3X
+(1)
+B
+˜c3
+−−⇀
+↽−−
+˜c4
+X
+(2)
+where ˜c1, ˜c2, ˜c3 and ˜c4 are the stochastic rate coefficients.
+Our starting point is microscopic such that the rate ˜c1
+should be interpreted as the reciprocal of the waiting time
+for a particular set of 2 X molecules to react and form 3
+X molecules when mediated by a fixed concentration of A
+molecules. In the large system, well-mixed limit of chem-
+ical kinetics, it is common to also define macroscopic, or
+kinetic rate constants kr for each reaction r. The rela-
+tion between kr and the microscopic rate coefficients ˜cr
+are only equivalent for linear reactions and in the ther-
+modynamic limit [40]. By varying the number of A and B
+molecules along with the rate coefficients, the Schl¨ogl sys-
+tem can occupy either a unimodal or a bimodal regime.
+In the bimodal regime, the number n of species X fluctu-
+ates, switching between two metastable states. A repre-
+sentative trajectory, shown in Fig. 1, oscillates between
+a state with n ≈ 5 and another with n ≈ 50 molecules.
+We first set out to compute the rate of bistable switching
+between those two metastable states, making use of the
+fact that the dynamics is one-dimensional (1D)—n exe-
+cutes a random walk between 0 and some large maximum
+number M that we impose for the sake of calculation.
+Rather than track n(t) for a single trajectory as in
+Fig. 1, the ensemble approach tracks evolution of the
+distribution pt(n), measuring the probability that an in-
+dividual trajectory would have n molecules at time t.
+Foreshadowing the analogies with quantum mechanical
+methodologies, we choose to write that distribution as a
+ket: |pt⟩ = [pt(0), pt(1) . . . , pt(M)]T , where T indicates
+the transpose to give a column vector. The distribution
+evolves according to the master equation
+d |pt⟩
+dt
+= H |pt⟩ .
+(3)
+Here, H is a rate matrix whose off-diagonal elements
+Hn′n are the rates of transitioning from n X molecules to
+
+3
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0
+0.005
+0.01
+0.015
+0.02
+0
+1
+2
+3
+4
+5
+0
+0.1
+0.2
+0
+10
+20
+30
+40
+50
+0
+20
+40
+60
+80
+∝ kBA t
+n
+t
+FIG. 1.
+(Top) A typical trajectory in the bistable regime
+(c1 = 2.676, c2 = 0.040, c3 = 108.102, and c4 = 37.881) of the
+well-mixed Schl¨ogl model of Sec. II A. The system stochasti-
+cally switches between the two metastable basins, where basin
+A is the top colored region and B is the bottom colored region.
+(Bottom) Starting with a distribution localized in basin A, the
+transition rate can be extracted from the probability growth
+in the linear regime of the evolution, P(Bt|A0) ∝ kBAt. The
+inset shows that growth on a much longer timescale, revealing
+the eventual approach toward equilibrium.
+n′ and whose diagonal elements Hnn are minus the rate
+of escaping from the state with n X molecules. The rate
+matrix conserves probability, a fact compactly expressed
+as ⟨1| H = 0 ⟨1|, with ⟨1| denoting a vector of ones. The
+distribution |pt⟩ can always be formally evolved in terms
+of a propagator eHt as
+|pt⟩ = eHt |p0⟩ .
+(4)
+Due to the analogy with quantum-mechanical time evo-
+lution, we equivalently call H an effective Hamilto-
+nian.
+Assuming H is irreducible, the long-time limit
+for this distribution tends to the unique steady-state
+|π⟩ = limt→∞ eHt |p0⟩.
+H can be decomposed into contributions from the two
+different reversible mechanisms of Eqs. (1) and (2), la-
+beled HA and HB, respectively, based on whether the
+transitions are mediated by the chemostatted species A
+or B. Then, the Schl¨ogl mechanism at the level of the
+master equations corresponds to tridiagonal rate matri-
+ces with elements
+HA
+n′n =˜c1n(n − 1)
+2
+δn′−1,n
+˜c2n(n − 1)(n − 2)
+6
+δn′+1,n−
+n(n − 1)
+2
+�
+˜c1 + ˜c2(n − 2)
+3
+�
+δn′,n
+(5)
+and
+HB
+n′,n = ˜c3 δn′−1,n + ˜c4nδn′+1,n − (˜c3 + ˜c4n) δn′,n,
+(6)
+where H = HA + HB. Given the specific tridiagonal H
+and the restriction 0 ≤ n ≤ M, both H and the propa-
+gator eHt are (M + 1) × (M + 1) matrices, amenable to
+evolving the entire distribution of realizations, effectively
+averaging over all the realizations.
+To extract a rate from the evolution of distribution,
+we must initialize the distribution in one of the two
+metastable states and measure the rate of relaxation.
+In analogy with free energy basins, we will call the two
+metastable regions basins A and B (red and blue regions
+of Fig. 1 respectively) with A ∩ B = ∅.
+We write ˆPω
+for the non-negative idempotent ( ˆP2
+ω = ˆPω) operator
+that projects onto the density lying in a region ω, where
+ω ∈ {A, B}. If |pt⟩ has support over a state space Ω,
+then the projection ˆPω |pt⟩ is an unnormalized distribu-
+tion with support restricted to ω. The total probability
+in ω at time t can then be expressed as ⟨1| ˆPω |pt⟩.
+B.
+Rate Constants
+In general, basins can be arbitrary subsets of Ω. For
+the current example, A and B are basins of attraction
+for the Schl¨ogl model (see Fig. 1a), and we would like
+to measure the rate of going from A → B. Rate con-
+stants can be defined either for the unidirectional flux
+from A → B, the quantity we focus on in this work, or
+for the timescale of the relaxation to equilibrium, which
+also involves the B → A transitions [41]. Additionally,
+the precise definition of the rate depends on whether one
+seeks trajectories that move from the boundary of one
+basin to another or from the interior of one basin to an-
+other. For instance, studying the time to transition from
+the boundary of A to the boundary of B yields the Tran-
+sition Path Theory (TPT) rate related to the committor
+function [42]. Here, we will have access to the distribu-
+tion over all states in the interior of A and B, allowing
+for the calculation of the unidirectional rate kBA. That
+kBA is the rate to start in A at time zero with probability
+P(A0), and transition to any state in B at time t with
+conditional probability P(Bt|A0).
+We initialize with a
+distribution over microstates ˆPA |π⟩ / ⟨1| ˆPA |π⟩, formed
+by projecting the equilibrium distribution onto A and
+normalizing, making P(A0) = 1 by construction. With
+this initialization, the rate for transitioning from A to B
+can then be written in terms of the time derivative of the
+
+4
+conditional probability of occupying B a time t later [10]:
+kBA ≡ d
+dtP(Bt|A0)P(A0)
+����
+t>τmol
+,
+= d
+dt
+⟨1| ˆPB eHt ˆPA |π⟩
+⟨1| ˆPA |π⟩
+����
+t>τmol
+.
+(7)
+That derivative is evaluated at some timescale exceeding
+τmol, a molecular time scale of the system that is long
+enough for the first trajectories to reach B from A but
+much shorter than the typical transit time k−1
+BA. To the
+extent that there is a separation of timescales between
+the time to explore the basin and the time to transit be-
+tween basins, distinctions between this rate and a TPT
+rate are insignificant. For rare events, no transitions can
+occur faster than the molecular time scale τmol, beyond
+that point, the number of transitions accumulates over
+time as P(Bt|A0) ∝ kBAt. At times long compared to
+k−1
+BA, P(Bt|A0) must eventually stop growing linearly to
+level off at the stationary value ⟨1| ˆPB |π⟩, but our inter-
+est is in extracting kBA from the growth at much shorter
+times. A typical realization of the process will not require
+observation much beyond τmol units of time, to ascertain
+the rate, meaning our ensemble evolutions must be prop-
+agated for roughly τmol.
+Our ability to compute kBA from the ensemble evo-
+lution traces back to the fact that it is practical to nu-
+merically calculate eHt for the well-mixed model with a
+single species and a modest M.
+While Eq. 7 general-
+izes naturally for harder problems (those with more than
+one species or with diffusion between multiple well-mixed
+voxels), the state space for |pt⟩ grows exponentially with
+additional stochastic degrees of freedom. We next show
+that the Doi-Peliti framework re-expresses the rate ma-
+trix in a second-quantized form that allows us to gener-
+alize beyond the case of a single well-mixed species.
+III.
+DOI-PELITI
+The Doi-Peliti (DP) formalism is a classical version
+of the second quantization methods from quantum field
+theory [43]. Originally introduced by Doi [44] as a cri-
+tique to the assumption of the Smoluchowski equation,
+it was later developed by Peliti [45] and others [46] as
+a path integral formalism for solving chemical reaction
+networks.
+The DP framework has also found success
+in other areas such as studying kinetically constrained
+lattice models in the context of glassy dynamics [47, 48].
+The DP representation builds operators which encode
+the dynamics onto the high-dimensional space that
+accompanies many-body problems.
+Crucially, writing
+the chemical mechanism in an operator form guarantees
+it can be easily converted into a matrix product operator
+(MPO) [49, 50], as we will do in Sec. IV.
+To start, we go beyond Sec. II by considering well-
+mixed reactions with multiple dynamic species as well as
+heterogeneous systems where molecules can diffuse be-
+tween neighboring voxels. Let X1, X2, . . . XL represent
+each chemical species in each voxel. We could be talk-
+ing about a single species X with different values in L
+different voxels, a single well-mixed voxel with L distinct
+dynamic species, or a mixture with multiple species and
+multiple voxels. The number of molecules in a given voxel
+can change due to a chemical reaction or a diffusion event,
+with both events being cast as,
+�
+j
+ηr
+j Xj
+˜cr
+−−→
+�
+j
+µr
+j Xj
+(8)
+where ˜cr are again the rate coefficients for reaction r. ηr
+j
+and µr
+j reflect the stochiometry of the reactions and give
+the stochiometric vector νr = µr − ηr which evolves the
+system from state n ∈ Ω in state space Ω, to n′ = n+νr.
+In the previous section, these reactions induced changes
+between the M +1 different microstates, specified by the
+scalar n. Now, the microstates n are specified by n =
+[n1, n2, . . . , nL], counting the occupation of each species
+within each voxel.
+For a dilute well-mixed solution, where well-mixed
+means molecules are evenly distributed around a given
+sub-volume, the change in probability of microstate n
+follows the gain-loss equation [14],
+dpt(n)
+dt
+=
+N
+�
+r
+�
+αr
+n−νrpt(n − νr) − αr
+npt(n)
+�
+,
+(9)
+where the sum is taken over N different reactions of the
+form Eq. (8), each with their own stochiometric vector
+νr. The transition rate between any two states is the sum
+of contributions from each reaction Hn′n = �
+r αr
+n. Each
+propensity αr is a product of a stochastic rate coefficient
+˜cr and the combinatorial number of ways to select either
+ηj or µr molecules from reaction r, depending on the
+direction of the reaction. For a “forward” reaction the
+propensity is, αr(n) = ˜cr
+�
+nj∈r
+�nj
+ηj
+�
+.
+Encoding the reactions into a matrix form allows us to
+evolve the elements of the joint distribution pt(n) analo-
+gously to the previous section. While it is appealing to
+try and directly solve these more complicated dynamics
+as we did before, it is impractical. The curse of dimen-
+sionality demands a less onerous way to work with both
+H and pt(n).
+Since particles of the same species in the same voxel are
+indistinguishable, it is beneficial to work in an occupation
+number representation with
+|pt⟩ =
+�
+n
+pt(n) |n⟩ .
+(10)
+The basis vectors
+|n⟩ = |n1, n2, . . . nL⟩
+= |n1⟩ ⊗ |n2⟩ ⊗ . . . ⊗ |nL⟩
+(11)
+corresponds to a possible microstate, and the set of basis
+vectors forms a tensor product space called a Fock space
+
+5
+B
+˜c1
+−−→ X
+X
+˜c2
+−−→ B
+2X + A
+˜c3
+−−→ 3X
+3X
+˜c4
+−−→ 2X + A
+HAfor
+l
+= c1x†2
+l (x†
+l − I)x2
+l ,
+HArev
+l
+= c2x†2
+l (I − x†
+l)x3
+l ,
+HBfor
+l
+= c3(x†
+l − I)
+HBrev
+l
+= c4(xl − x†
+lxl)
+Xi, Xi+1
+d
+−−→ Xi ±1, Xi+1 ∓1
+Hdiff
+l→l+1 = d
+�
+x†
+l+1xl − x†
+lxl
+�
+Hdiff
+l+1→l = d
+�
+x†
+lxl+1 − x†
+l+1xl+1
+�
+· · ·
+· · ·
+H = Hdiff
+1→2 + Hdiff
+2→1 + · · · + HAfor
+1
++ HBrev
+1
++ · · ·
+Transition
+(a)
+(b)
+(c)
+FIG. 2.
+The reaction-diffusion master equation (RDME) consists of voxels in which well-mixed chemical reactions occur (a)
+and between which species can diffuse (b). This work centers around a one-dimensional chain of L such voxels, each with a
+fluctuating number of molecules of species X in a Schl¨ogl chemical reaction. The Doi-Peliti (DP) framework turns the reaction-
+diffusion evolution into a second-quantized effective Hamiltonian that governs the evolution of the distribution over classical
+reaction-diffusion microstates. (c) That effective Hamiltonian is represented by a high-dimensional tensor, with each vertical
+leg taking one of (M +1) possible values corresponding to the number of molecules of species X which could occupy each voxel.
+The DP form of the effective Hamiltonian lends itself to a compact representation of the effective Hamiltonian tensor of rates
+for transitioning from one microstate to another.
+FL with |n⟩ ∈ FL and ⟨m|n⟩ = δmn [51]. Without loss
+of generality, we will discuss the case of a single chem-
+ical species and L physical sites, in which case nl has
+the interpretation of the number of molecules at site l.
+The second-quantization procedure rewrites H in terms
+of creation and annihilation operators, x†
+l and xl, that
+respectively raise and lower the number of X molecules
+at site l. That is to say,
+x†
+l |nl⟩ = |nl + 1⟩ ,
+xl |nl⟩ = nl |nl − 1⟩ ,
+xl |0⟩ = 0.
+(12)
+where |0⟩ is the vacuum state, i.e., no molecules. By xl
+or x†
+l we are implying that the operator only acts on the
+lth site:
+xl = I1 ⊗ I2 ⊗ . . . ⊗ Il−1 ⊗ x ⊗ Il+1 ⊗ . . . ⊗ IL
+(13)
+Because they act on the single-site space, the general one-
+site operators x and x† can be written as the M+1×M+1
+matrices
+x† =
+�
+����
+0
+0
+0
+. . .
+1
+0
+0
+. . .
+0
+1
+0
+...
+...
+...
+�
+���� and
+x =
+�
+����
+0
+1
+0
+. . .
+0
+0
+2
+. . .
+0
+0
+0
+...
+...
+...
+�
+���� .
+(14)
+That these matrices yield Eq. (12) can be readily con-
+firmed.
+More abstractly, the existence of such raising
+and lowering operators in this classical setting originates
+from the classical commutation relation [xix†
+j] = δij and
+the existence of a positive semi-definite number operator
+ˆnl = x†
+l xl, so named because ˆnl |nl⟩ = nl |nl⟩. Together,
+the set {Ω, FL, x, x†} are also sometimes called an inter-
+acting Fock space [43]. Our use of the raising and low-
+ering operators is less formal and more pragmatic. As in
+the construction of the number operator, we will be able
+to decompose contributions to ˆH in terms of the x† and
+x’s.
+One benefit of defining operators in terms of creation
+and annihilation operators is that expectation values for
+observables can be expressed easily in terms of inner
+products with operators. Consider, for example, the ob-
+servable that counts all particles in all voxels, ˆn = �
+l ˆnl.
+The operator ˆn acts on a many-body state |n⟩, yet each
+ˆnl is a single-body operator acting on site l and leaving
+all other sites unchanged. It is therefore understood that
+when local operators such as ˆnl act on |n⟩, it is written
+as a shorthand for
+ˆnl = I ⊗ . . . ⊗ x†
+l xl ⊗ . . . ⊗ I.
+(15)
+In terms of that shorthand, ˆnl |n⟩ = nl |n⟩ counts the
+number of particles in site l for the many-body state |n⟩.
+The time-dependent expected number of particles in the
+whole system is thus ⟨n(t)⟩ = �
+l ⟨1| ˆnl |pt⟩, where the
+vector of ones ⟨1| serves to sum over all microstates. The
+time-dependence of a distribution can also be extracted
+from the action of a second-quantized ˆH on |pt⟩, and
+this operation can be practically computed when ˆH is
+expressed in terms of the single-site operators.
+Mirroring the discussion in Sec. II, the generator ˆHl
+for the Schl¨ogl model reactions within the lth voxel is
+a sum of contributions from the elementary reactions
+ˆHl = �
+r ˆHr. Fig. 2a shows each reaction, expressed as
+a second-quantized contribution to the effective Hamil-
+tonian, as derived in App. A. Combining together the
+forward and reverse reactions mediated by species A, we
+see that the reversible reaction Eq. (1) contributes
+ˆHA
+l = x†2
+l
+��
+c1x†
+l + c2
+�
+I − x†
+l
+�
+xl
+�
+x2
+l − c1x2
+l ˆPc
+Ml
+�
+.
+(16)
+to the effective Hamiltonian. For convenience we have
+absorbed the combinatorial terms into that rate coeffi-
+
+6
+pt(nX1, nX2, nX3, · · · , nXNs) ≈
+· · ·
+nXNs
+nX1
+nX2
+nX3
+pt(nX
+1 , nX
+2 , nX
+3 , · · · , nX
+L) ≈
+· · ·
+nX
+L
+nX
+1
+nX
+2
+nX
+3
+pt(nX1
+1 , nX2
+1 , nX1
+2 , n2
+2, · · · , nX1
+L/2, nX2
+L/2) ≈
+· · ·
+nX1
+1
+nX2
+1
+nX1
+2
+nX2
+2
+nX1
+L/2 nX2
+L/2
+H ≈
+nX
+1
+mX
+1
+nX
+2
+mX
+2
+...
+nX
+L−1
+mX
+L−1
+nX
+L
+mX
+L
+(a)
+(b)
+FIG. 3.
+The ensemble evolution of the many-body state requires that one solve for the probability of every microstate at
+all times. That high-dimensional joint probability distribution can be well approximated by a matrix product state (MPS).
+An MPS approximation for the joint distribution can be advantageous whether there are Ns species in a well-mixed reaction
+[(a) top], one dynamical (unchemostatted) species with diffusion [(a) middle], or multiple species and diffusion [(a) bottom].
+In all three cases, each circle represents a tensor, either a rank-2 at the first and last site or rank-3 in the body. Horizontal
+lines represent the bond-indices connecting tensors and vertical lines are physical indices. For an RDME, the physical indices
+represent the possible number of molecules at a given tensor. If two lines are connected, then the index is summed over. (b)
+An effective Hamiltonian H can be likewise decomposed into lower-rank tensors to form a matrix product operator (MPO).
+Now there are two physical indices per tensor, representing transitions from state nX
+l
+to mX
+l . Vertical connected lines are the
+bond indexes which are contracted over.
+cient, defining cr ≡ ˜cr/γ!, where γ is either ηr or µr for
+forward and reversed reactions, respectively. Likewise,
+the contribution
+ˆHB
+l = x†
+l (c3 − c4)xl + c4xl − c3 ˆPc
+Ml.
+(17)
+comes from Eq. (2). We show in App. A that the trunca-
+tion of the state space at finite M requires the addition
+of a projector, ˆPc
+Ml to prevent ˆHl from promoting any
+probability from the M th state into an (M + 1)th that
+has been truncated away. In the M → ∞ limit, the pro-
+jector can be discarded, as reflected by the expressions
+in Fig. 2a.
+The Hamiltonian operator for L independent, well-
+mixed Schl¨ogl voxels is thus ˆHrxn = �L
+l ˆHA
+l + ˆHB
+l . To
+add diffusion, consider the well-mixed voxels arranged in
+a 1D lattice as in Fig. 2b. Each chemical species is able
+to move between neighboring voxels as regulated by a
+diffusion operator
+ˆHdiff =
+L−1
+�
+l=1
+d
+��
+x†
+l+1 − x†
+l
+�
+(xl − xl+1)
+�
+,
+(18)
+with hopping rate d = D/h2, consistent with macroscopic
+diffusion constant D for voxel width h [17]. Combining
+reaction with diffusion:
+ˆH = ˆHrxn + ˆHdiff,
+(19)
+is straightforward at the level of the effective Hamilto-
+nian, but the diffusive coupling between neighbors has
+the consequence of introducing nontrivial interactions be-
+tween voxels.
+IV.
+TENSOR NETWORKS
+We have now seen how to transform the matrix-form
+rate operator H of Sec. II into a Doi-Peliti form ˆH, ex-
+pressed in terms of local creation and annihilation oper-
+ators. That change of representation is not merely cos-
+metic.
+As the number of chemical species or number
+of lattice sites grows, the size of H grows exponentially.
+For N dynamical (unchemostatted) species and L lattice
+sites, the state space has (M + 1)NL states. To put this
+scaling in perspective, the largest numerical result that
+follows uses N = 1, M = 85 and L = 8, meaning the
+system contains 2.99 × 1015 microstates. Calculations on
+a matrix H, even a very sparse matrix H, are untenable.
+By contrast, it is practical to construct a TN decompo-
+sition of the second quantized form of ˆH in terms of an
+MPO, as shown in Fig. 3b, and demonstrated explicitly
+for the Schl¨ogl mechanism in Appendix B. Given the sec-
+ond quantized form, the MPO can be readily constructed
+in a practical form through SVD decomposition using au-
+tomated packages such as autoMPO in ITensor [52].
+Much like the operator ˆH can be compressed as an
+MPO, the states that operators act upon can also be
+expressed more compactly in terms of TNs.
+Consider
+|pt⟩ = pt(n) = pt(n1, n2, . . . , nL), the probability of mi-
+crostate n at time t. This |pt⟩ can be thought of as a rank
+L tensor. By specifying the occupancy of each of the L
+sites, |pt⟩ outputs a number—a probability—associated
+to that microstate.
+By leveraging a matrix product
+ansatz, |pt⟩ can be approximated as a partial contraction
+over L different low-rank tensors (see Fig. 3a). The low
+rank tensors of an MPS [53] are organized in a 1D chain of
+
+7
+tensors Qn1
+s1 , Qn2
+s1s2, . . . , QnL−1
+sL−2sL−1, and QnL
+sL−1 whose up-
+per index nl specifies the occupancy of site l and whose
+lower indices are so-called bond indices, nonphysical in-
+dices which are contracted over to generate correlations
+between nearby sites. For suitably chosen Q tensors, we
+therefore approximate
+|pt⟩ =
+�
+n1,n2,...nL
+pt(n1, n2, . . . , nL) |n1, n2, . . . nL⟩
+≈
+�
+s1,...,sL−1
+Qn1
+s1 Qn2
+s1s2 . . . QnL
+sL−1 |n1, n2, . . . nL⟩
+≡ |qt⟩ ,
+(20)
+where |qt⟩ is the approximate distribution over mi-
+crostates generated from the MPS. Though not explicitly
+written, the Q tensors are time-dependent.
+There is a diagrammatic representation of the tensors,
+shown in Fig. 3, which aids in visualizing tensor opera-
+tions. The tensors Q are given by circles, one per site,
+and vertical lines feeding into these circles are the phys-
+ical indices that pick out a particular slice of the tensor
+based on the occupation number of that site. Circles also
+have horizontal lines emanating from them, representing
+the non-physical bond indices.
+Connection of two cir-
+cles by a line represents a shared index being contracted
+over. Fig. 3 emphasizes that the TN decomposition is
+flexible enough to address varied CME’s. The MPS of
+Fig. 3a middle, has been the focus. Each site from 1 to
+L reflects the number of X molecules in that voxel, but
+Fig. 3a (top) and (bottom) show that the MPS decom-
+position can equally well be constructed when multiple
+species Xi, i = 1, 2, . . . , Ns can occupy a single voxel or
+when Ns different species diffuse between L/Ns voxels to
+yield an MPS with L sites. What is practically important
+is that Q’s be arranged such that the physical dynamics
+correlates the neighbors.
+For example, for the RDME
+Schl¨ogl model, neighboring Q’s correspond to neighbor-
+ing sites in the 1D Schl¨ogl lattice.
+An attractive feature of the MPS ansatz is that the
+size and accuracy of the MPS are controllable through
+the bond dimension of the s auxiliary indices. If the di-
+mension of those indices is allowed to grow exponentially
+from s1 through sL/2, the ansatz in Eq. (20) can be exact,
+but even when the bond dimension of the indices s are
+capped at a maximum dimension of κ, the ansatz can be a
+very good approximation. Crucially, capping that maxi-
+mum bond dimension makes it practical to work with |qt⟩
+because the rank-L tensor |pt⟩ has been replaced by the
+L different Q tensors, each with no more than (M +1)κ2
+elements.
+Having approximated both |qt⟩ and ˆH by an MPS and
+MPO, respectively, we can now revisit the otherwise in-
+tractable dynamics problem of Eq. (3), only now we wish
+to evolve |qt⟩ in lieu of |pt⟩. As |qt⟩ is an MPS state, we
+wish for the dynamics to be constrained for all times to
+the manifold M of possible MPS states. Merely replacing
+|pt⟩ by |qt⟩ in Eq. (3):
+d |qt⟩
+dt
+= ˆH |qt⟩
+(21)
+would not impose such a constraint because ˆH in general
+evolves |qt⟩ off M. An appealing resolution is given by
+the Time-Dependent Variational Principle (TDVP) [54,
+55], which proposes to first project the right-hand side
+onto the tangent space of the MPS manifold:
+d |qt⟩
+dt
+= PTM ˆH |qt⟩ .
+(22)
+In this way, dynamics is restricted to the desired sub-
+space of MPS states [56].
+It turns out that the time-discretized numerical inte-
+gration of this |qt⟩ is amenable to efficient TN opera-
+tions. The technical details of the time-evolution algo-
+rithm have been reported elsewhere [55, 57–59]. We give
+only a brief, high-level overview of that procedure (see
+also App. C for more details). The central step in de-
+riving the TDVP time-evolution algorithm is to demon-
+strate that the projector PTM can be decomposed in
+terms of a sum over L terms (associated to each site
+of the MPS) and L − 1 terms (associated to each bond
+connecting sites). In the limit of a small time step δt, the
+propagator ePTM ˆ
+Hδt can thus be evaluated as a compo-
+sition of time-evolutions involving the 2L − 1 individual
+terms. That composition of time-evolutions is executed
+by a one-site TDVP algorithm that “sweeps” from lattice
+site 1 through L, advancing each term for the small time
+step.
+The efficiency of the TDVP time-evolution algorithm
+relies on the flexibility to represent the same MPS state
+in redundant ways due to a gauge freedom. Consider,
+for example, neighboring tensors Qnl
+sl−1sl and Qnl+1
+slsl+1 in
+an MPS. For an invertible R, replacing those tensors by
+Qnl
+sl−1slR and R−1Qnl+1
+slsl+1, respectively, does not change
+the MPS state; upon contracting over the neighboring
+sites, the identity RR−1 would fall out. By leveraging
+the gauge freedoms, it is always possible to transform
+Eq. (20) into a so-called mixed canonical form [60]:
+|qt⟩ = An1 . . . Anl−1QnlBnl+1 . . . Bnl |n1, . . . , nL⟩ , (23)
+where each A is a left-orthogonal tensor satisfying AT A =
+I and each B is a right-orthogonal tensor satisfying
+BBT
+= I, where here, T represents the transpose.
+For notational compactness, we have suppressed implied
+summations over the bond indices between tensors in
+Eq. (23). The mixed canonical form is said to be cen-
+tered about the tensor Qnl. From that tensor’s perspec-
+tive, the chain of A’s and chain of B’s constitute the en-
+vironment, and it is beneficial to contract over the links
+between A’s and links between B’s to obtain so-called
+environment tensors. These environment tensors inherit
+an orthogonality from the orthogonality of the A’s and
+B’s, significantly reducing the computational complex-
+ity to advance Qnl in time by δt provided the MPS was
+already centered about the Qnl we aim to evolve.
+
+8
+A
+B
+t
+t = 0.01
+t = 0.14
+t = 0.44
+t = 0.75
+FIG. 4.
+Snapshots of the joint density in two neighboring voxels from an L = 3 site lattice of Schl¨ogl voxels.
+Using
+c1 = 2.676, c2 = 0.040, c3 = 108.102, c4 = 37.881 and d = 8.2207, the steady-state distribution was computed with κ = 30 by
+evolving an initial uniform distribution according to ˆH via TDVP. That steady-state distribution was projected into A at time
+0, renormalized, then propagated according to the modified no-recrossing dynamics ˜H using TDVP. For clarity of visualization,
+the density within A is not plotted as it would overwhelm the small probability that leaks into B. By t = 0.01 (left image),
+probability has begun to leak from A but none has reached B. Probability has just begun to reach B at at t = 0.14, roughly
+the τmol timescale for this problem. By t = 0.44, probability is collecting in B, and by t = 0.75, the flux passing from A to B
+has attained an approximately constant steady-state value corresponding to the linear growth phase where P(Bt|A) ∝ kBAt.
+As made more explicit in App. C, the one-site TDVP
+algorithm initially centers the MPS around the first ten-
+sor, propagates that tensor forward in time by δt, per-
+forms a QR decomposition to change the gauge, propa-
+gates backward in time by δt the R term of that QR de-
+composition then passes that term to the neighboring site
+to re-center the MPS around the neighboring site. The
+neighboring site is then propagated forward in time as
+the algorithm iterates. After the complete sweep through
+the MPS, the probability distribution |qt⟩ has been prop-
+agated by a time step δt. Whether that propagation well
+approximates the evolution of the full distribution |pt⟩
+depends on the time-step error but also on the severity
+of the MPS approximation, which can be systematically
+controlled through the maximum bond dimension κ.
+Notably, the time-evolved MPS state |qt⟩ allows us
+to estimate the probability of any microstate of inter-
+est, even though it would be impossible to enumerate
+probabilities of all microstates. Alternatively, sets of mi-
+crostates can be summed over to yield tractable marginal
+distributions. As an example, Fig. 4 shows the results
+of calculations on a 1D chain of L = 3 Schl¨ogl voxels.
+Though there are three voxels, site one is traced over to
+leave the two-dimensional joint probability of finding nX
+2
+X molecules in site 2 and nX
+3 X molecules in site 3. At
+time zero, the distribution is initialized within region A
+according to ˆPA |π⟩ / ⟨1| ˆPA |π⟩. As time progresses, a
+small flux of probability leaks from A to B. Because the
+overwhelming preference is to stay in A, Fig. 4 highlights
+the rare transitions by only plotting the joint distribution
+outside of A. In the next section, we illustrate the fact
+that the MPS state’s ability to capture the rare time-
+dependent flux is key to calculating the rate of traversing
+between basins.
+V.
+MANY-BODY RATE CALCULATION
+Having established the TDVP dynamics of an MPS
+state, we can now repeat the strategy of Eq. (7) to com-
+pute a transition rate between states A and B:
+kBA = d
+dt
+⟨1| ˆPB ePTM ˆ
+Ht ˆPA |π⟩
+⟨1| ˆPA |π⟩
+����
+t>τmol
+.
+(24)
+In doing so, there are a few important differences from
+Eq. (24). Most obviously, the TDVP dynamics includes
+the projection onto the MPS tangent space and |π⟩ must
+now be an MPS approximation to the steady-state dis-
+tribution. Additionally, the meaning of ˆPA and ˆPB must
+be adapted to now define many-body analogs of the pro-
+jectors onto the two metastable states. The single-voxel
+Schl¨ogl model had an A state defined as all microstates
+with occupancy n above a threshold q∗
+A.
+Similarly, B
+included all microstates with occupancy less than some
+other threshold q∗
+B. The L-voxel Schl¨ogl model general-
+izes to involve hypercubic metastable states; A requires
+that each voxel satisfies n ≥ q∗
+A and B requires that each
+voxel satisfies n ≤ q∗
+B.
+With these generalizations, the transition rate of
+Eq. (24) would measure the flux crossing into B through
+its boundary, but if we are to compare with a transition
+path theory rate, we must exclude re-crossing events. In
+other words, we want to include trajectories which exit A
+then some time later enter B without having re-entered
+A, but we want to exclude the flux that leaves B and
+re-enters B some time later without having returned to
+A. In Sec. II B, kBA was sufficiently large that those re-
+crossings could be ignored, but as we generalize to handle
+smaller kBA, those recrossings can become significant.
+We note that the TDVP dynamics can just as well be
+executed with a modified absorbing-boundary-condition
+
+70X
+1(
+3-40
+020406000nX
+3
+3X
+240250600
+0102(30405060710L30X
+n
+214050500
+0102(30405060710L30X
+n
+214050500
+0102(30405060710730X
+n
+214050509
+dynamics with effective Hamiltonian
+˜H = ˜Hrxn + ˜Hdiff,
+(25)
+where the reaction and diffusion Hamiltonians exclude
+events that exit B. An explicit construction of that mod-
+ified dynamics is provided in App. D, allowing us to di-
+rectly remove the re-crossing events.
+VI.
+NUMERICAL EVALUATION OF THE
+L-VOXEL SCHL¨OGL MODEL
+The formalism of the preceding sections has laid out
+a controllable approximation to extract rates between
+metastable states of RDMEs.
+Numerical experiments
+are necessary to evaluate whether that formalism can
+be practically useful.
+We set out to demonstrate that
+utility by computing the transition rates between high-
+occupancy and low-occupancy states of a L-voxel Schl¨ogl
+model as a function of L.
+The model is parameter-
+ized by the four stochastic rate coefficients discussed
+in Sec. II: c1 = 2.676, c2 = 0.040, c3 = 108.102, and
+c4 = 37.881, chosen so the L = 1 Schl¨ogl model will
+exhibit the bistablity reflected in Fig. 1. With those cho-
+sen rates, A and B are identified by choosing na = 15
+and nb = 25, and the dynamics lends itself to a finite
+maximum-occupancy truncation of M = 85. By linking
+multiple voxels together in a 1D chain with a diffusive
+hopping rate d = 8.2207, it becomes rarer to switch be-
+tween the two metastable states than it would be with
+a single isolated voxel. With multiple voxels, a switch-
+ing event must flip the state of each voxel, and diffusion
+from a neighbor can stabilize the original state of each
+voxel. Increasing L thus gives a way to systematically
+probe rarer transition rates.
+The most straightforward way to numerically detect
+the L-dependence of the transition rate is via sampling of
+trajectories. Realizations of the continuous-time RDME,
+subject to the M = 85 maximum voxel occupancy, were
+generated using the SSA. First, 5×105 independent SSA
+realizations were initialized from a uniform distribution
+and propagated up to t = 5 to ensure the steady state dis-
+tribution was reached. Sampled configurations that fell
+within A were then uniformly drawn as an initial con-
+figuration for 400 independent trajectories which were
+evolved via the SSA until the trajectory first reached B.
+From each of those 400 trajectories, the time between
+the final exit of A and the first entry of B, τBA was
+recorded and the transition rate was computed from the
+mean of those waiting times: kBA = ⟨τBA⟩−1. Fig. 5a
+(red line) shows that the rate exponentially decreases as
+L is increased, a result consistent with the exponential
+dependence of rate on volume in the well-mixed Schl¨ogl
+model [61]. While the rate can be computed by trajec-
+tory sampling, the difficulty to do so (via brute force
+sampling) grows in proportion to the typical τBA, mean-
+ing the computational cost grows exponentially with L
+(see Fig. 5c). The poor scaling of brute-force SSA sam-
+pled rate calculations has motivated so-called advanced
+sampling methods like FFS, which focus computational
+effort on the progress along a reaction coordinate. Fig. 5
+shows that the DP/TDVP approach provides a radically
+different strategy for circumventing the scaling problem,
+reproducing the SSA rates at lower computational ex-
+pense.
+The SSA rate calculations can be made more accurate
+by collecting more trajectories, and the DP/TDVP cal-
+culations likewise offer a systematic way to improve accu-
+racy by increasing the computational expense. The two
+most prominent ways to adjust that accuracy are through
+the time step δt for the TDVP evolution and through
+the maximum bond dimension κ for the MPS state. In
+this work, a time step of δt = 10−4 was used through-
+out, and the trade-off between expense and accuracy was
+entirely tuned through the choice of κ. The single-site
+TDVP algorithm we used to propagate an MPS state
+does not alter the bond dimension of the MPS, so κ was
+entirely set by the bond dimension of the initial MPS.
+We therefore needed to approximate the steady state |π⟩
+with different values of κ. Approximations were gener-
+ated by starting with an initially uniform state using a
+bond dimension κ set to several different values between
+5 and 30. Those uniform distributions were evolved with
+single-site TDVP for 10, 000 time-steps allowing for the
+natural dynamics of ˆH to approach the limit |π⟩. We
+found that evolving that 10, 000 time steps was sufficient
+to approximate the steady-state distribution, as reflected
+by the average rate of change of the distribution. That
+is to say, ⟨pt| ˙pt⟩ = ⟨pt| ˆH |pt⟩ ≈ 0 by time t = 1, indicat-
+ing proximity to the actual steady state, which satisfies
+| ˙π⟩ = |0⟩.
+To get rates, the κ-dependent MPS approximation to
+the steady state was projected onto A and renormalized.
+That renormalized distribution was then propagated un-
+der single-site TDVP with the modified dynamics for an-
+other 10, 000 time steps, and projected onto B to add up
+the total probability to have reached B. Note that these
+10,000 steps are not nearly enough to reach the steady
+state |π⟩ because the slow rate of transitioning from A to
+B is a bottleneck, one that did not need to be surmounted
+to relax from the uniform distribution to |π⟩ in 10, 000
+steps. As additional time steps of dynamics were taken,
+the growth of the transition flux from A to B was mea-
+sured and plotted in Fig. 5b. That figure shows that at
+very short times (t < τmol), the flux from A to B has not
+reached a steady state value because there hasn’t been
+enough time for any transitions. After that initial time
+tmol, the plot plateaus at a value corresponding to the
+rate kBA, a plateau that is reached orders of magnitude
+faster than the time for a typical SSA trajectory to make
+a transition. We find that if the bond dimension is insuf-
+ficient (i.e., κ = 5 for L = 3), then the dynamics becomes
+unstable, reflected in erratic estimates for the conditional
+probability P(Bt|A0). As the bond dimension is grown
+to κ = 9, the estimates converge, allowing a stable rate to
+
+10
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.005
+0.01
+0.015
+0.02
+2
+3
+4
+5
+6
+7
+8
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+2
+3
+4
+5
+6
+7
+8
+100
+101
+102
+103
+104
+(a)
+(b)
+(c)
+FIG. 5.
+(a) The rate kBA for switching between metastable states of the L-voxel Schl¨ogl model is calculated using the SSA
+(Red line) and using TDVP (dashed lines) with varying bond-dimensions κ. For sufficiently large bond-dimensions, TDVP
+agrees with a SSA rate estimate built using 400 transitions from A to B at each L. (b) As L grows a larger bond-dimension is
+required to capture the dynamics. For L = 3 we see that for κ = 5 and κ = 6 the flow of probability from A to B is unstable.
+As κ grows, the solutions converge. For this system, κ = 20 is stable up to L = 6. While κ = 30 gives stable solutions up
+to L = 8. (c) The total computation time in hours to estimate kBA from both the SSA and TDVP. Despite using different
+bond-dimensions to estimate rates on larger lattices, we see that the total time to evolve the TDVP ensemble evolution scales
+sub-exponentially while the SSA scales exponentially.
+be extracted. Larger values of L demand an even greater
+value of bond dimension. Fig. 5a shows that SSA rates
+agree with κ = 20 calculations up to L = 6, but κ = 30
+was required to push up to L = 8. Provided that larger
+bond dimension is used, the DP/TDVP rates reproduce
+the SSA rates across five orders of magnitude, up to and
+beyond the point that the brute-force SSA rates are prac-
+tical.
+Though the required bond dimension grows with L,
+Fig. 5c shows that the total time to estimate the rate
+grows sub-exponentially. The blue and black dashed lines
+in that plot are the total computational time to estimate
+the rate using TDVP with κ = 30 and κ = 20 respec-
+tively. Both SSA and TDVP were run using codes writ-
+ten in the Julia programming language [62] on equivalent
+CPUs. While the exact wall time, of course, depends on
+details of the hardware and the implementation, the plot
+clearly demonstrates the difference in scaling between
+brute-force SSA and DP/TDVP. An increase in bond di-
+mension increases the computation time, but it does not
+fundamentally alter the favorable scaling in L.
+In addition to the favorable scaling at large L, we
+highlight that the DP/TDVP formalism allows us to ex-
+tract many observables with similar ease. For example,
+Fig. 6 shows ⟨nl(t)⟩ = ⟨1| ˆnl |pt⟩, the average number
+of molecules in site l with l ranging from 1 through 4
+for an L = 8 site lattice. The state |pt⟩ was initialized
+in the normalized steady-state confined to A, the high-
+occupation-number state. For t < τmol, a rapid loss of
+particles near the boundary of A leads to a sharp drop.
+After that τmol timescale, the rate of decrease in ⟨nl(t)⟩
+becomes linear, reflecting the steady-state flux from A to
+B.
+0
+0.2
+0.4
+0.6
+0.8
+1
+43.5
+44
+44.5
+45
+45.5
+46
+FIG. 6.
+Average number of molecules for different lattice
+sites when L = 8.
+Colors correspond to a different lattice
+sites. The expected number of molecules ⟨nl(t)⟩ = ⟨1| ˆnl |pt⟩
+for sites 1 to ⌊L/2⌋ (shown) are mirrored by sites ⌊L/2+1⌋ to
+L. The initial non-linear flow of probability out of A is shown
+by the steep drop in average number of molecules. The system
+passing τmol is shown by the transition to a linear change in
+particle number ⟨nl(t)⟩.
+VII.
+DISCUSSION
+In this work we have shown how macroscopic rates can
+be efficiently calculated from the microscopic dynamics of
+a RDME. We have illustrated that capability in a min-
+imal toy model, the L-voxel Schl¨ogl model, with each
+voxel chemostatted such that it contains a single dynam-
+ical species (X). In actuality, the formalism and numer-
+ical methodology extend far beyond this initial demon-
+stration. Any reaction-diffusion mechanism which can be
+written as a set of elementary reactions will have a DP
+representation, one which can involve raising and low-
+ering operators for different particle types, not only for
+X. For example, Table I shows the effective Hamiltoni-
+
+11
+ans for several common (unchemostatted) reactions: uni-
+molecular, bimolecular, and auto-catalytic.
+Given the
+set of reactions, it is comparatively straightforward to
+derive the DP representation, and computational tools
+already exist to generate an MPO [49] and propagate via
+TDVP from that DP form of the effective Hamiltonian.
+As those computational tools become increasingly well
+Reaction
+Hamiltonian
+X
+˜c
+−−→ Y
+c[(y† − x†)x]
+X + Y
+˜c
+−−→ Z
+c[(z† − x†y†)xy]
+X + Y
+˜c
+−−→ 2 X c[x†(x† − y†)xy]
+TABLE I.
+Examples of the mapping between a common
+chemical reaction and the Doi-Peliti form of the contribution
+to an effective Hamiltonian.
+The procedure to derive such
+correspondences is systematic [63] and typically discussed for
+a Fock space with occupation number ranging from 0 to infin-
+ity. To truncate to a finite Fock space, one must additionally
+add a projector to each effective Hamiltonian, as in this work,
+to ensure conservation of probability.
+developed, one can imagine that the TN tools will offer
+generic “turn the crank” analysis of microscopic models
+of reaction-diffusion at the ensemble level. That vision
+motivates more future work at the interface of reaction-
+diffusion kinetics and tensor network methods. Here, we
+have approximated the microscopic distribution with an
+MPS, allowing us to capitalize on the efficient TDVP al-
+gorithm for MPS states. The downside is that an RDME
+with many dynamic species will not always be easily ap-
+proximated by a 1D chain of tensors. If two chemical
+species are strongly correlated but far apart in the MPS
+structure, the MPS calculation will demand a very high
+bond dimension. It will be important for future work ei-
+ther to avoid the need for high bond dimension, for exam-
+ple by leveraging TN topologies which mirror that of an
+underlying CRN or to adapt the bond dimension through
+subspace expansion [64]. Those technical advances may
+eventually lay the groundwork to move the methodol-
+ogy from this Schl¨ogl study to the complex, biologically-
+motivated models where SSA realizations presently reign
+supreme.
+VIII.
+ACKNOWLEDGMENTS
+We gratefully acknowledge Hadrien Vroylandt, Nils
+Strand, and Luis Pedro Garcia-Pintos for many insightful
+discussions. We are also grateful to Miles Stoudenmire,
+Matthew Fishman, Steven White, and other developers
+of ITensor, a library for implementing tensor network cal-
+culations, upon which this work was built. The material
+presented in this manuscript is based upon work sup-
+ported by the National Science Foundation under Grant
+No. 2141385. TRG is thankful that Phill Geissler could
+see this work in a preliminary form, but is deeply sad-
+dened that Phill’s much-too-soon passing prevented him
+from sharing more of his wisdom.
+Appendix A: Doi-Peliti encoding
+Assume a set of elementary reactions are given. These
+can be encoded into a Doi-Peliti representation which
+converts the set of elementary reactions to a set of
+effective Hamiltonians.
+The operator representations
+can then be exactly written as an MPO, as shown below.
+Consider a 1D lattice in which each voxel can undergo
+reactions from the Schl¨ogl model. A state of the system
+will be written with respect to the Fock space FL which
+consists of basis vectors |n⟩ ∈ FL, ⟨m|n⟩ = δm,n, being
+clear that |n⟩ = |n1, n2, . . . , nL⟩. The probability of a
+microstate is given by pn(t). The distribution over the
+system is p(n, t) = p(n1, n2, . . . , nL, t). When focusing
+on an individual voxel we will use the shorthand,
+p(n, t) := p(nc, nl, t), where nc = {nk |k ̸= l}. Finally,
+powers of ladder operators are written as, x†2
+l
+= x†
+l x†
+l ,
+for example.
+2X + A
+˜c1
+−−−→ 3X: For the reaction 2X + A
+˜c1
+−−→ 3X
+within site l the change in probability is,
+dpt(n)
+dt
+= ˜c1
+2 [(nl − 1)(nl − 2)pt(nc, nl − 1)
+− nl(nl − 1)pt(nc, nl)] .
+(A1)
+We can expand with respect to the Fock states in order
+to write out the distribution in the Fock-basis represen-
+tation, |pt⟩ = �
+n pt(n) |n⟩ which gives,
+d |pt⟩
+dt
+= ˜c1
+2
+�
+n
+[(nl − 1)(nl − 2)pt(nc, nl − 1) |n⟩
+− nl(nl − 1)pt(nc, nl) |n⟩]
+(A2)
+For the second term we can substitute in ladder operators
+for the number of molecules x†2
+l x2
+l = nl(nl −1). The first
+term can be written with respect to the vacuum states
+|0⟩, x†3
+l x2
+l x†(nl−1)
+l
+|0⟩ = (nl − 1)(nl − 2) |n⟩,
+d |pt⟩
+dt
+= ˜c1
+2
+�
+n
+�
+x†3
+l x2
+l pt(nc, nl − 1) |nc, nl − 1⟩
+− x†2
+l x2
+l pt(nc, nl) |n⟩
+�
+(A3)
+Distributing the summation we see that there are two
+separate sets of states being summed over. To make it
+clearer, substitute ml = nl − 1 to give,
+d |pt⟩
+dt
+= ˜c1
+2
+�
+nc
+� M−1
+�
+ml=−1
+x†3
+l x2
+l pt(nc, ml) |nc, ml⟩
+− x†2
+l x2
+l
+M
+�
+nl=0
+pt(nc, nl) |n⟩
+�
+.
+(A4)
+
+12
+We would like both summations to be over the same set
+of states. For the first summation, the ml = −1 term
+is zero due to |nc, −1⟩ = xl |nc, 0⟩ = 0 and thus can be
+excluded from the first summation. Likewise, the upper
+limit can be extended since x†3
+l x2
+l |nc, Ml⟩ = 0. In order
+to conserve probability we require the effective Hamil-
+tonian satisfy ⟨1n| ˆHn,n′ = 0, since this guarantees ˆH
+has a zero-eigenvalue. Conservation of probability man-
+ifests in Eq.
+(A4) through a balance between the es-
+cape rates given by the second term and the transitions
+into a state through the first terms. Looking at states
+|nc, Ml⟩, the second term is non-zero, but the first term
+is x†3
+l x2
+l |nc, Ml⟩ = 0 due to the finite state space. Result-
+ing in negative column sums and violation of probability
+conservation. A solution is to project out the offending
+states.
+d |pt⟩
+dt
+⇒ ˜c1
+2
+�
+nc
+�
+M
+�
+ml=0
+x†3
+l x†2
+l pt(nc, ml) |nc, ml⟩
+− x†2
+l x2
+l
+M
+�
+nl=0
+pt(nc, nl) |n⟩
+�
++ ˜c1
+2
+�
+nc
+�
+nl
+x†2
+l x2
+l pt(nc, nl) |nc, Ml⟩ ⟨nc, Ml|n⟩ ,
+= ˜c1
+2
+�
+nc
+�
+M
+�
+ml=0
+x†3
+l x2
+l pt(nc, ml) |nc, ml⟩
+−
+M
+�
+n=0
+x†2
+l x2
+l
+�
+Il − ˆPMl
+�
+pt(nc, nl) |nc, nl⟩
+�
+,
+= ˜c1
+2
+�
+x†3
+l x2
+l − x†2
+l x2
+l ˆPc
+Ml
+�
+|pt⟩ ,
+(A5)
+where we used the projectors, ˆPMl = |nc, Ml⟩ ⟨nc, Ml|,
+ˆPc
+Ml = I − ˆPMl. Since probability is now conserved we
+can set m → n and ˜c1/2 = c1, and identify the Effective
+Hamiltonian contribution from the reaction,
+HAfor
+l
+= c1
+�
+x†3
+l x2
+l − x†2
+l x2
+l ˆPc
+Ml
+�
+.
+(A6)
+3X
+˜c2
+−−−→ 2X + A: The change in probability for the
+back reaction is
+dpt(n)
+dt
+= ˜c2
+6 [n(n − 1)(n + 1)pt(nc, nl + 1)
+− n(n − 1)(n − 2)pt(nc, nl)]
+(A7)
+Writing in terms of the Fock space and substituting in the
+number operators, x†2
+l x3
+l x†
+l |nc, nl⟩ = n(n − 1)(n + 1) |n⟩
+and x†3
+l x3
+l |n⟩ = n(n − 1)(n − 2) |n⟩ gives,
+d |pt⟩
+dt
+= ˜c2
+6
+�
+n
+[n(n − 1)(n + 1)pt(nc, nl + 1) |n⟩
+− n(n − 1)(n − 2)pt(n) |n⟩] ,
+= ˜c2
+6
+�
+nc
+� M
+�
+nl=0
+x†2
+l x3
+l pt(nc, nl + 1) |nc, nl + 1⟩
+−
+M
+�
+nl=0
+x†3
+l x3
+l pt(nc, nl) |nc, nl⟩
+�
+.
+(A8)
+Writing the indices of the first term as, m = n + 1
+M+1
+�
+m=1
+x†2
+l x3
+l pt(nc, ml) |nc, ml⟩
+(A9)
+We have to check the upper and lower bounds. The lower
+term ml = 1 can be see to be zero both by looking
+at the reaction, or by directly checking x†2
+l x3
+l |nc, 1⟩ =
+0.
+When m = M + 1 the imposed chemostat makes
+x†2
+l x3
+l |nc, M + 1⟩ = x†2
+l x3
+l x†
+l |nc, M⟩ = 0. So we see that
+unlike the previous reaction, the boundary terms which
+differ between the first and second summations are zero.
+Thus we do not need to project out any states and can
+immediately write,
+d |pt⟩
+dt
+= c2
+�
+x†2
+l
+�
+Il − x†
+l
+�
+x3
+l
+�
+|pt⟩ ,
+= HArev
+l
+|pt⟩ ,
+(A10)
+again using ˜c2/6 = c2. By summing ˆHAfor
+l
+and ˆHArev
+l
+we
+have ˆHA
+l , Eq. (16) from the manuscript
+ˆHA = x†2
+l
+��
+c1x†
+l + c2(xl − x†
+l xl)
+�
+x2
+l − c1x2
+l ˆPc
+Ml
+�
+.
+B
+˜c3
+−−−→ X: Following the previous steps, Start with
+the change in probability for state n,
+dpt(n)
+dt
+= ˜c3pt(nc, nl − 1) − c3pt(n),
+d |pt⟩
+dt
+= ˜c3
+�
+nc
+��
+nl
+x†
+l pt(nc, nl − 1) |nc, nl − 1⟩
+−
+�
+nl
+pt(nc, nl) |nc, nl⟩
+�
+.
+(A11)
+Again, using ml = nl − 1 with |nc, −1⟩ = xl |nc, 0⟩ =
+0, renaming ˜c3 = c3 and projecting out non-conserving
+
+13
+states we have,
+d |pt⟩
+dt
+⇒ c3
+�
+nc
+�
+x†
+l
+M−1
+�
+ml=0
+pt(nc, ml) |nc, ml⟩
+−
+M
+�
+nl=0
+pt(nc, nl) |nc, nl⟩
+�
++
+M
+�
+nl=0
+pt(nc, nl) |nc, Ml⟩ ⟨nc, Ml|nc, nl⟩ ,
+= c3
+�
+x†
+l − ˆPc
+Ml
+�
+|pt⟩ ,
+= HBfor
+l
+|pt⟩ .
+(A12)
+X
+˜c4
+−−−→ B: Following the above process we start with,
+dpt(n)
+dt
+= ˜c4(nl +1)pt(nc, nl +1)−c4nlpt(nc, nl). (A13)
+Expanding and substituting operators gives,
+d |pt⟩
+dt
+= ˜c4
+�
+nc
+M
+�
+nl=0
+�
+xlx†
+l pt(nc, nl + 1) |nc, nl⟩ ,
+− x†
+l xlpt(n) |n⟩
+�
+,
+= ˜c4
+�
+nc
+M
+�
+nl=0
+[xl pt(nc, nl + 1) |nc, nl + 1⟩
+− x†
+l xlpt(n) |n⟩
+�
+,
+= ˜c4
+�
+nc
+�M+1
+�
+ml=1
+xl pt(nc, ml) |nc, ml⟩
+−
+M
+�
+nl=0
+x†
+l xlpt(n) |n⟩
+�
+,
+(A14)
+With ˜c4 = c4 and using the same logic used for reaction
+2, the upper and lower bounds of the first summation are
+zero, giving,
+d |pt⟩
+dt
+= c4
+�
+Il − x†
+l
+�
+xl |pt⟩ ,
+= HBrev
+l
+|pt⟩ .
+(A15)
+By combining HBfor
+l
+and HBrev
+l
+we arrive at HB
+l
+in the
+manuscript
+HB
+l = x†
+l (c3 − c4xl) + c4xl − c3 ˆPc
+Ml.
+Appendix B: Matrix Product Operators
+Any set of reactions which can be put in a “product”
+form i.e. Eq. (16), (17) and Eq. (18) can be converted
+to a MPO form and compressed if need be through an
+automated package such as AUTOMPO in ITensor [52].
+For systems with local and nearest neighbor interactions
+there exists relatively simple representations of the
+effective Hamiltonian.
+Writing out an exact representation for the Schl¨ogl
+model, each reaction only acts on one local subspace at
+a time ˆHrxn = �L
+l=1 ˆHl. Diffusion is composed of next
+nearest neighbor terms Eq. (18). These can be combined
+into three matrix terms. A bulk operator
+W [l] =
+�
+��������
+I
+x†
+l
+−x†
+l xl
+0
+xl
+D
+Hl,
+Dxl, D, Dx†
+l , −x†
+l xl, I
+�
+��������
+,
+(B1)
+and two boundary terms
+W [1] =
+�
+H1, Dx1, D, Dx†
+1, −x†
+1x1, I
+�
+,
+W [L] =
+�
+��������
+I
+x†
+L
+−x†
+LxL
+xL
+D
+HL
+�
+��������
+.
+(B2)
+The product then reproduces the Hamiltonian
+ˆH
+=
+�L
+l=1 W [l].
+Appendix C: TDVP algorithm
+We wish to solve the Schr¨odinger equation
+��� ˙ψ
+�
+= H |ψ⟩
+(C1)
+Since the dimension of |ψ⟩ grows exponentially with
+molecule number we approximate the state by an MPS
+|ψ⟩ ≈ |ψ⟩Q where we have used the Q subscript to em-
+phasize the dependence on the set of tensors {Qnl
+sl−1sl},
+l = 1, 2, . . . , L. In the main text, the equivalent state
+would be |ψ⟩Q = |qt⟩. Two important definitions we will
+need are,
+Manifold of matrix product states (Mκ): is the
+set of states |ψ⟩Q which can be represented by a matrix
+product state with bond-dimension κ [65].
+Tangent states and manifold: Write the set of ten-
+sors in |ψ⟩Q as Q = {Qn1, Qn2, . . . , QnL}. Use the in-
+clusive index l = (nl, sl−1, sl) in the shorthand Ql. Let
+Γ be a set of tensors where the tensor ΓL has the same
+dimensions as Ql ∀ l.
+A tangent state |Θ⟩ of |ψ⟩Q is
+
+14
+= eδt ˆ
+P +
+2 |ψ⟩Q = �∞
+k=0
+δtk
+k! (P +
+2 ˆH)k |ψ⟩Q =
+=
+· · ·
++ δt
+· · ·
+· · ·
+· · ·
+· · ·
++ δt2
+2
+· · ·
+· · ·
+· · ·
+· · ·
+· · ·
+· · ·
+· · ·
++ · · ·
+=
+· · ·
++ δt
+· · ·
+H(c)
++ δt2
+2
+· · ·
+H(c)
+H(c)
++ · · ·
+= �
+n
+�∞
+k=0
+δtk
+k! [H(2)k]n2,n′
+2An′
+1(t + δt)Qn2(t)Bn′
+3(t) + . . . + Bn′
+L(t) |n⟩
+= �
+n An′
+1(t + δt)Qn2(t + δt)Bn′
+3(t) + . . . + Bn′
+L(t) |n⟩
+FIG. 7.
+Applying each operator P ±
+l
+to |ψ⟩Q results in a local equation of motion as shown here for the P +
+2
+operator.
+Writing out the infinite series in the second line, we can partially contract around the center site (black diamond). These
+partial contractions form the local effective Hamiltonian H(l). The rest of the MPS remains unchanged. Combing these local
+Hamiltonians forms the matrix exponential only for the centered site.
+defined as,
+|Θ(Γ, Q)⟩ =
+�
+l
+�
+α,β,nl
+Φ[1:l−1]
+A,α
+Γnl
+αβΦ[l+1:L]
+B,β
+|nl⟩
+(C2)
+where Φ[1:l−1]
+A,α
+and Φ[l+1:L]
+B,β
+are the left and right environ-
+ment tensors. The corresponding tangent space for the
+manifold of MPS, Mκ is defined as,
+TMκ ≡ {|Θ(Γ, Q)⟩ | ∀ l : dim(Γl) = dim(Ql)}
+(C3)
+Approximating |ψ⟩ with an MPS results in a second
+equation of motion,
+|ψ⟩Q = ˆH |ψ⟩Q
+(C4)
+since in general |ψ⟩ ̸= |ψ⟩Q.
+We would like to solve
+Eq. C1 but cannot for all but toy problems.
+Instead
+we seek a solution to Eq. C4 which will be a good ap-
+proximation to |ψ⟩.
+There are two issues, one is that a direct solution to
+ˆH |ψ⟩Q requires dealing with the entire MPS/MPO equa-
+tion simultaneously which is prohibitively expensive. The
+second and more pressing issue is that ˆH |ψ⟩Q is no longer
+an MPS, but instead a tensor with L legs, meaning we
+have left the manifold Mκ.
+The solution that TDVP
+proposes is to project ˆH |ψ⟩Q onto the manifold of MPS
+and then evolve |ψ⟩Q. In this way the state never leaves
+Mκ. The projector we seek is that which projects a state
+|ψ⟩ onto the closest tangent vector |Θ(Γ, Q)⟩ = ˆPMκ |ψ⟩.
+The tensors which minimize the distance between the
+tangent vector and |ψ⟩ is found through the solution to,
+min
+Γ
+∥ |Θ(Γ, Q)⟩ − |ψ⟩ ∥2 .
+(C5)
+
+15
+The variationally optimal projector is given by, [59]
+ˆPT =
+L
+�
+l=1
+P [1:l]
+L
+⊗ 1l ⊗ P [l+1:L]
+R
+−
+L−1
+�
+l=1
+P [1:l]
+L
+⊗ P [l+1:L]
+R
+,
+=
+L
+�
+l=1
+P +
+l +
+L−1
+�
+l=1
+P −
+l .
+(C6)
+Where, P [1:n]
+L
+= �
+α
+���Φ[n1:nl]
+L,α
+� �
+Φ[n1:nl]
+L,α
+���, and P [l+1:L]
+R
+=
+�
+β
+���Φ[nl+1:nL]
+R,β
+� �
+Φ[nl+1:nL]
+R,β
+��� are the projectors built from
+the environment tensors. The solution to Eq. C4 can be
+written as a sum of terms,
+|ψ(t + δt)⟩Q =
+exp
+�
+δt
+�
+P +
+L ˆH +
+L−1
+�
+l=1
+P +
+l
+ˆH + P −
+l
+ˆH
+��
+|ψ(t)⟩Q .
+(C7)
+If δt ≪ 1 then we can pull out each pair of terms in the
+exponential and apply them to |ψ(t)⟩Q one at a time. For
+example, starting with l = 1 we have,
+|ψ(t + δt)⟩Q =
+exp
+�
+dt
+�
+P +
+L H +
+L−1
+�
+l=2
+P +
+l H + P −
+l
+ˆH
+��
+edtP −
+1 ˆ
+HedtP +
+1 ˆ
+H |ψ(t)⟩Q .
+(C8)
+For a given l we need to evaluate, edtP +
+l H |ψ(t)⟩ which
+when expanded is,
+edtP +
+l H |ψ(t)⟩ =
+∞
+�
+k=0
+dtk
+k!
+�
+P +
+l H
+�k |ψ(t)⟩Q
+(C9)
+Now, the matrix exponential exp
+�
+dtP +
+l H
+�
+is acting glob-
+ally on |ψ(t)⟩Q, which would prove to be very expensive
+for many-body systems. Luckily the projector combined
+with allowed gauge transformations reduces the global
+matrix exponential to a local matrix exponential. Fig.
+(7) shows the local nature for the P +
+2 term, where |ψ⟩Q
+is centered at site two. We see each tensor around the
+center site is unchanged while the center site is evolved
+forward in time. The P −
+l
+terms have a similar local na-
+ture to them but before acting on |ψ⟩Q we use a gauge
+transformation Qnl �→ AnlCl where Cl only consists of
+non-physical bond indices.
+P −
+l
+then induces the local
+exponential term exp(−δtK(l))Cl(t + δt) which evolves
+Cl(t + δt) back by δt.
+Absorbing with the next term
+Cl(t)Bnl+1 = Qnl+1 shifts the orthogonality center al-
+lowing for the P +
+l+1 term to be applied. By first center-
+ing |ψ(t)⟩Q at site one, and then applying sequences of
+gauge transformations with P +
+l
+and P −
+l
+terms in Eq. C7,
+we can “sweep” from l = 1 to l = L, which amounts to
+evolving |ψ(t)⟩Q → |ψ(t + δt)⟩Q.
+Appendix D: Modified Dynamics
+To measure a rate from A into B without multiple
+crossings of B’s boundary, we define a modified dynamics
+via an effective Hamiltonian that zeroes out the rates of
+any reaction or diffusion event that would depart the re-
+gion B. Deriving the forms of ˜Hrxn and ˜Hdiff in Eq. (25)
+requires that we first identify the set of states and reac-
+tions (here meaning reaction or diffusion events) as
+Bout ≡ {n, r : n + νr ∈ Bc | n ∈ B}
+(D1)
+where Bc is the complement of B. For the Sch¨ogl model,
+only diffusion and the forward reactions can map out of
+B. The states in Bout are those where particles in the lth
+voxel are at the boundary to B, nl = q∗
+B and nk ∈ B ∀ k ̸=
+l. The local boundary for B can be projected onto using
+Q = |q∗
+B⟩ ⟨q∗
+B|. Combining Q with the projector Nl =
+�
+nl∈B |nl⟩ ⟨nl|, lets us define the projection operator
+ˆPout
+l
+= N1 ⊗ N2 ⊗ · · · Nl−1 ⊗ Ql ⊗ Nl+1 ⊗ . . . NL, (D2)
+The joint projector built from non-independent projec-
+tors is given by the powerset expansion
+ˆPout =
+L
+�
+l=1
+(−1)l+1 �
+{sl}
+�
+sl
+ˆPout
+sl
+(D3)
+where {sl} is the set of combinations from L elements
+taken l at a time.
+For L = 3 Eq. (D3) becomes
+ˆPout = [ ˆPout
+1
++ ˆPout
+2
++ ˆPout
+3
+]
+− [ ˆPout
+1
+ˆPout
+2
++ ˆPout
+1
+ˆPout
+3
++ ˆPout
+2
+ˆPout
+3
+]
++ ˆPout
+1
+ˆPout
+2
+ˆPout
+3
+.
+Each product over ˆPout
+sl
+can be simplified noting that,
+NlQl = Ql which leads to
+ˆPout =
+L
+�
+i=1
+(−1)l+1 �
+{ck}
+�
+ck
+{N, Q}ck
+(D4)
+where the sum over ck is now all combinations of N and
+Q where Q appears k times. The advantage of Eq. (D4)
+is that effective Hamiltonians can be projected onto only
+using local operators.
+In this work, B consists of all
+molecule numbers less than q∗
+B+1. Then, projecting onto
+reactions that can map out of B the effective Hamiltoni-
+ans for the Schl¨ogl model become
+˜Hrxn =
+L
+�
+l
+ˆHArev
+l
++ ˆHBrev
+l
++ ( ˆHAfor
+l
++ ˆHBfor
+l
+)(I − ˆPout),
+(D5)
+˜Hdiff = ˆHdiff(I − ˆPout).
+(D6)
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diff --git a/xNE2T4oBgHgl3EQfLwZ4/content/tmp_files/load_file.txt b/xNE2T4oBgHgl3EQfLwZ4/content/tmp_files/load_file.txt
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@@ -0,0 +1,1021 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf,len=1020
+page_content='Quantifying Rare Events in Stochastic Reaction-Diffusion Dynamics Using Tensor  Networks  Schuyler B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Nicholson1 and Todd R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Gingrich1  1Department of Chemistry, Northwestern University,  2145 Sheridan Road, Evanston,  Illinois 60208, USA  (Dated: January 11, 2023)  The interplay between stochastic chemical reactions and diffusion can generate rich spatio-  temporal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' While the timescale for individual reaction or diffusion events may be very fast,  the timescales for organization can be much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' That separation of timescales makes it particu-  larly challenging to anticipate how the rapid microscopic dynamics gives rise to macroscopic rates in  the non-equilibrium dynamics of many reacting and diffusing chemical species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Within the regime  of stochastic fluctuations, the standard approach is to employ Monte Carlo sampling to simulate  realizations of random trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Here, we present an alternative numerically tractable approach  to extract macroscopic rates from the full ensemble evolution of many-body reaction diffusion prob-  lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The approach leverages the Doi-Peliti second-quantized representation of reaction-diffusion  master equations along with compression and evolution algorithms from tensor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By fo-  cusing on a Schl¨ogl model with one-dimensional diffusion between L otherwise well-mixed sites, we  illustrate the potential of the tensor network approach to compute rates from many-body systems,  here with approximately 3 × 1015 microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Specifically, we compute the rate for switching be-  tween metastable macrostates, with the time to compute those rates growing subexponentially in  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Because we directly work with ensemble evolutions, we crucially bypass many of the difficulties  encountered by rare event sampling techniques—detailed balance and reaction coordinates are not  needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  INTRODUCTION  Connecting microscopic kinetics to emergent rates has  a long history for equilibrium and nonequilibrium sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Complex systems often settle into metastable  basins with rare transitions regulating the rates of switch-  ing between the emergent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In equilibrium, transi-  tion state theory [1, 2], variational transition state the-  ory [3], Grote-Hynes theory [4, 5], and transition path  sampling (TPS) [6, 7] have given strategies to estimate  the transition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Away from equilibrium, theories  built upon a free energy landscape lose their applicabil-  ity, yet trajectory sampling approaches can still extract  rates from microscopic dynamics, either through a noise-  guided TPS approach [8, 9] or using forward flux sam-  pling (FFS) [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For all their merits, sampling ap-  proaches bring intrinsic challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' How many samples  are required to estimate the rate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Is a good approximate  reaction coordinate needed, and if so, how is it found  [13]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' How does the expense grow with larger or more  complicated microscopic systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Motivated by those  ubiquitous concerns, we set out to consider a different  approach—one that effectively samples all possible tra-  jectories as an ensemble evolution instead of propagating  individual realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Such a proposition may appear  absurd given the exploding state space presented by the  many-body problem, but we show that for the class of  discrete state, continuous-time, reaction-diffusion master  equations (RDME), we can evolve the full distribution of  a classical many-body systems in a numerically control-  lable manner by combining two well-developed methods,  the Doi-Peliti (DP) formalism with tensor network (TN)  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  While TPS and FFS have been developed for a va-  riety of dynamical equations of motion (Hamiltonian,  Langevin, Markov jump, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' ), we focus our attention  solely on the chemical master equation (CME) dynam-  ics, the setting where FFS was first developed [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' More  specifically, we study discrete reaction-diffusion dynamics  of L sites or voxels, [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The rationale for this fo-  cus is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' From a practical perspective, the RDME  is important as a popular method to model chemical  and biological processes [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' From a theoretical per-  spective, the RDME is interesting because it provides a  setting where one can analyze the influence of stochas-  ticity on the emergence of patterns [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  In many  pattern-formation problems, it is possible to understand  the underlying physics in terms of a mean-field partial  differential equation that tracks spatiotemporal evolution  of deterministically evolving fields [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' It is, how-  ever, well-appreciated that stochastic effects can quan-  titatively and even qualitatively impact kinetics [23–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  It is therefore important that the methodology we dis-  cuss captures more than just the typical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Ac-  curately computing transition rates between metastable  states requires quantitatively resolving even rare tails of  the state space distribution since these rare fluctuations  can be instrumental in triggering transitions [11, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Historically, similar schemes for the CME ensemble  evolution have been hampered by the curse of dimension-  ality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' As a result, exact solutions are rare [30] and numer-  ical solutions often resort to sampling methods built upon  the kinetic Monte-Carlo stochastic simulation algorithm  (SSA) [31, 32], also known as the Gillespie algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Another approach is to evolve a subset of the state space  through the finite state projection (FSP) method [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='03717v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='stat-mech]  9 Jan 2023  2  FSP has previously been combined with TNs to approx-  imately describe the CME [34, 35], resembling our use  of TNs to tame the many-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Unlike those  projective methods, however, the approach we describe  uses the time-dependent variational principle (TDVP) to  evolve the dynamics with probability conservation at all  times, akin to prior work with diffusion and no reac-  tions [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Since we estimate rare events by measuring  the distribution’s small probability fluxes, conservation  of probability and high-fidelity dynamical evolution are  materially important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  While the formalism we describe applies quite gener-  ally to different well-mixed and reaction-diffusion models,  we center our paper around a single model problem: rate  calculations for transitions between basins of the Schl¨ogl  model [38] in the bistable regime, both with and without  diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' II, we first illustrate the rare event prob-  lem and extract rate constants from a propagator in the  well-mixed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  For more complicated systems (ei-  ther systems with more chemical species or more voxels),  the curse of dimensionality precludes the exact calcula-  tion of the propagator, but the remainder of the paper  shows how to employ TNs to follow the same concep-  tual path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' III, we review the Doi-Peliti frame-  work for writing reaction-diffusion dynamics in a second-  quantized field theoretic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' IV then reviews the  TN matrix product state (MPS) ansatz that enables en-  semble evolution via TDVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Those methods are applied  to the many-body rate calculation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Finally, in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' VI, we show results for the RDME with diffusion be-  tween L well-mixed Schl¨ogl voxels, demonstrating that  the methodology can extend numerical transition rate  calculations to regimes where the SSA becomes imprac-  tical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The approach effectively evolves the probability  of occupying each microstate in an unfathomably large  many-body state space, yet the calculations scale sub-  exponentially with system size and do not assume detail-  balanced equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  ENSEMBLE RATE CALCULATIONS  Consider an archetypal two-state rate process between  reactants and products R −−⇀  ↽−− P, initiated entirely in the  reactant state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' After some reaction timescale, the con-  centration of products grows before eventually leveling  off at its equilibrium value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The reaction rate can sim-  ply be defined and measured in terms of how quickly the  product concentration rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' An alternative formulation  is to study the fate of an individual reactant on its path  to transitioning into a product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The TPS and FFS path  sampling approaches extract the reaction rate by focusing  on the statistical properties of these individual trajecto-  ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Algorithms exist to propagate those single trajecto-  ries, algorithms that can be practically implemented even  when the system grows large and complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By contrast,  propagating the entire ensemble of trajectories has typ-  ically been limited to simple low-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  In this section, we illustrate an ensemble-evolution rate  calculation for a simple case in order to develop notation  and lay the groundwork for the many-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The  remainder of the paper will show that the methodology  can be extended beyond such simple problems by em-  ploying TNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Well-mixed Schl¨ogl Model  Consider the well-mixed Schl¨ogl model, a quintessen-  tial example of bistability in chemical reaction networks  (CRNs) [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The model tends to be studied under  chemostatted conditions that fix the concentrations of  species A and B while allowing species X’s concentration  to evolve according to the four reactions:  2X + A  ˜c1  −−⇀  ↽−−  ˜c2  3X  (1)  B  ˜c3  −−⇀  ↽−−  ˜c4  X  (2)  where ˜c1, ˜c2, ˜c3 and ˜c4 are the stochastic rate coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Our starting point is microscopic such that the rate ˜c1  should be interpreted as the reciprocal of the waiting time  for a particular set of 2 X molecules to react and form 3  X molecules when mediated by a fixed concentration of A  molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In the large system, well-mixed limit of chem-  ical kinetics, it is common to also define macroscopic, or  kinetic rate constants kr for each reaction r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The rela-  tion between kr and the microscopic rate coefficients ˜cr  are only equivalent for linear reactions and in the ther-  modynamic limit [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By varying the number of A and B  molecules along with the rate coefficients, the Schl¨ogl sys-  tem can occupy either a unimodal or a bimodal regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  In the bimodal regime, the number n of species X fluctu-  ates, switching between two metastable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' A repre-  sentative trajectory, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 1, oscillates between  a state with n ≈ 5 and another with n ≈ 50 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  We first set out to compute the rate of bistable switching  between those two metastable states, making use of the  fact that the dynamics is one-dimensional (1D)—n exe-  cutes a random walk between 0 and some large maximum  number M that we impose for the sake of calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Rather than track n(t) for a single trajectory as in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 1, the ensemble approach tracks evolution of the  distribution pt(n), measuring the probability that an in-  dividual trajectory would have n molecules at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Foreshadowing the analogies with quantum mechanical  methodologies, we choose to write that distribution as a  ket: |pt⟩ = [pt(0), pt(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , pt(M)]T , where T indicates  the transpose to give a column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The distribution  evolves according to the master equation  d |pt⟩  dt  = H |pt⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (3)  Here, H is a rate matrix whose off-diagonal elements  Hn′n are the rates of transitioning from n X molecules to  3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='02  0  1  2  3  4  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='2  0  10  20  30  40  50  0  20  40  60  80  ∝ kBA t  n  t  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (Top) A typical trajectory in the bistable regime  (c1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='676, c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='040, c3 = 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='102, and c4 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='881) of the  well-mixed Schl¨ogl model of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The system stochasti-  cally switches between the two metastable basins, where basin  A is the top colored region and B is the bottom colored region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (Bottom) Starting with a distribution localized in basin A, the  transition rate can be extracted from the probability growth  in the linear regime of the evolution, P(Bt|A0) ∝ kBAt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The  inset shows that growth on a much longer timescale, revealing  the eventual approach toward equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  n′ and whose diagonal elements Hnn are minus the rate  of escaping from the state with n X molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The rate  matrix conserves probability, a fact compactly expressed  as ⟨1| H = 0 ⟨1|, with ⟨1| denoting a vector of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The  distribution |pt⟩ can always be formally evolved in terms  of a propagator eHt as  |pt⟩ = eHt |p0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (4)  Due to the analogy with quantum-mechanical time evo-  lution, we equivalently call H an effective Hamilto-  nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Assuming H is irreducible, the long-time limit  for this distribution tends to the unique steady-state  |π⟩ = limt→∞ eHt |p0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  H can be decomposed into contributions from the two  different reversible mechanisms of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (1) and (2), la-  beled HA and HB, respectively, based on whether the  transitions are mediated by the chemostatted species A  or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Then, the Schl¨ogl mechanism at the level of the  master equations corresponds to tridiagonal rate matri-  ces with elements  HA  n′n =˜c1n(n − 1)  2  δn′−1,n  ˜c2n(n − 1)(n − 2)  6  δn′+1,n−  n(n − 1)  2  �  ˜c1 + ˜c2(n − 2)  3  �  δn′,n  (5)  and  HB  n′,n = ˜c3 δn′−1,n + ˜c4nδn′+1,n − (˜c3 + ˜c4n) δn′,n,  (6)  where H = HA + HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Given the specific tridiagonal H  and the restriction 0 ≤ n ≤ M, both H and the propa-  gator eHt are (M + 1) × (M + 1) matrices, amenable to  evolving the entire distribution of realizations, effectively  averaging over all the realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  To extract a rate from the evolution of distribution,  we must initialize the distribution in one of the two  metastable states and measure the rate of relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  In analogy with free energy basins, we will call the two  metastable regions basins A and B (red and blue regions  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 1 respectively) with A ∩ B = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  We write ˆPω  for the non-negative idempotent ( ˆP2  ω = ˆPω) operator  that projects onto the density lying in a region ω, where  ω ∈ {A, B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' If |pt⟩ has support over a state space Ω,  then the projection ˆPω |pt⟩ is an unnormalized distribu-  tion with support restricted to ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The total probability  in ω at time t can then be expressed as ⟨1| ˆPω |pt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Rate Constants  In general, basins can be arbitrary subsets of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For  the current example, A and B are basins of attraction  for the Schl¨ogl model (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 1a), and we would like  to measure the rate of going from A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Rate con-  stants can be defined either for the unidirectional flux  from A → B, the quantity we focus on in this work, or  for the timescale of the relaxation to equilibrium, which  also involves the B → A transitions [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Additionally,  the precise definition of the rate depends on whether one  seeks trajectories that move from the boundary of one  basin to another or from the interior of one basin to an-  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For instance, studying the time to transition from  the boundary of A to the boundary of B yields the Tran-  sition Path Theory (TPT) rate related to the committor  function [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Here, we will have access to the distribu-  tion over all states in the interior of A and B, allowing  for the calculation of the unidirectional rate kBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' That  kBA is the rate to start in A at time zero with probability  P(A0), and transition to any state in B at time t with  conditional probability P(Bt|A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  We initialize with a  distribution over microstates ˆPA |π⟩ / ⟨1| ˆPA |π⟩, formed  by projecting the equilibrium distribution onto A and  normalizing, making P(A0) = 1 by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' With  this initialization, the rate for transitioning from A to B  can then be written in terms of the time derivative of the  4  conditional probability of occupying B a time t later [10]:  kBA ≡ d  dtP(Bt|A0)P(A0)  ����  t>τmol  ,  = d  dt  ⟨1| ˆPB eHt ˆPA |π⟩  ⟨1| ˆPA |π⟩  ����  t>τmol  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (7)  That derivative is evaluated at some timescale exceeding  τmol, a molecular time scale of the system that is long  enough for the first trajectories to reach B from A but  much shorter than the typical transit time k−1  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' To the  extent that there is a separation of timescales between  the time to explore the basin and the time to transit be-  tween basins, distinctions between this rate and a TPT  rate are insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For rare events, no transitions can  occur faster than the molecular time scale τmol, beyond  that point, the number of transitions accumulates over  time as P(Bt|A0) ∝ kBAt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' At times long compared to  k−1  BA, P(Bt|A0) must eventually stop growing linearly to  level off at the stationary value ⟨1| ˆPB |π⟩, but our inter-  est is in extracting kBA from the growth at much shorter  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' A typical realization of the process will not require  observation much beyond τmol units of time, to ascertain  the rate, meaning our ensemble evolutions must be prop-  agated for roughly τmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Our ability to compute kBA from the ensemble evo-  lution traces back to the fact that it is practical to nu-  merically calculate eHt for the well-mixed model with a  single species and a modest M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  While Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 7 general-  izes naturally for harder problems (those with more than  one species or with diffusion between multiple well-mixed  voxels), the state space for |pt⟩ grows exponentially with  additional stochastic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' We next show  that the Doi-Peliti framework re-expresses the rate ma-  trix in a second-quantized form that allows us to gener-  alize beyond the case of a single well-mixed species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  DOI-PELITI  The Doi-Peliti (DP) formalism is a classical version  of the second quantization methods from quantum field  theory [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Originally introduced by Doi [44] as a cri-  tique to the assumption of the Smoluchowski equation,  it was later developed by Peliti [45] and others [46] as  a path integral formalism for solving chemical reaction  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The DP framework has also found success  in other areas such as studying kinetically constrained  lattice models in the context of glassy dynamics [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The DP representation builds operators which encode  the dynamics onto the high-dimensional space that  accompanies many-body problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Crucially, writing  the chemical mechanism in an operator form guarantees  it can be easily converted into a matrix product operator  (MPO) [49, 50], as we will do in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  To start, we go beyond Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' II by considering well-  mixed reactions with multiple dynamic species as well as  heterogeneous systems where molecules can diffuse be-  tween neighboring voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Let X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' XL represent  each chemical species in each voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' We could be talk-  ing about a single species X with different values in L  different voxels, a single well-mixed voxel with L distinct  dynamic species, or a mixture with multiple species and  multiple voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The number of molecules in a given voxel  can change due to a chemical reaction or a diffusion event,  with both events being cast as,  �  j  ηr  j Xj  ˜cr  −−→  �  j  µr  j Xj  (8)  where ˜cr are again the rate coefficients for reaction r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' ηr  j  and µr  j reflect the stochiometry of the reactions and give  the stochiometric vector νr = µr − ηr which evolves the  system from state n ∈ Ω in state space Ω, to n′ = n+νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  In the previous section, these reactions induced changes  between the M +1 different microstates, specified by the  scalar n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Now, the microstates n are specified by n =  [n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , nL], counting the occupation of each species  within each voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  For a dilute well-mixed solution, where well-mixed  means molecules are evenly distributed around a given  sub-volume, the change in probability of microstate n  follows the gain-loss equation [14],  dpt(n)  dt  =  N  �  r  �  αr  n−νrpt(n − νr) − αr  npt(n)  �  ,  (9)  where the sum is taken over N different reactions of the  form Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (8), each with their own stochiometric vector  νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The transition rate between any two states is the sum  of contributions from each reaction Hn′n = �  r αr  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Each  propensity αr is a product of a stochastic rate coefficient  ˜cr and the combinatorial number of ways to select either  ηj or µr molecules from reaction r, depending on the  direction of the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For a “forward” reaction the  propensity is, αr(n) = ˜cr  �  nj∈r  �nj  ηj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Encoding the reactions into a matrix form allows us to  evolve the elements of the joint distribution pt(n) analo-  gously to the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' While it is appealing to  try and directly solve these more complicated dynamics  as we did before, it is impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The curse of dimen-  sionality demands a less onerous way to work with both  H and pt(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Since particles of the same species in the same voxel are  indistinguishable, it is beneficial to work in an occupation  number representation with  |pt⟩ =  �  n  pt(n) |n⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (10)  The basis vectors  |n⟩ = |n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nL⟩  = |n1⟩ ⊗ |n2⟩ ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' ⊗ |nL⟩  (11)  corresponds to a possible microstate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' and the set of basis  vectors forms a tensor product space called a Fock space  5  B  ˜c1  −−→ X  X  ˜c2  −−→ B  2X + A  ˜c3  −−→ 3X  3X  ˜c4  −−→ 2X + A  HAfor  l  = c1x†2  l (x†  l − I)x2  l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  HArev  l  = c2x†2  l (I − x†  l)x3  l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  HBfor  l  = c3(x†  l − I)  HBrev  l  = c4(xl − x†  lxl)  Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Xi+1  d  −−→ Xi ±1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Xi+1 ∓1  Hdiff  l→l+1 = d  �  x†  l+1xl − x†  lxl  �  Hdiff  l+1→l = d  �  x†  lxl+1 − x†  l+1xl+1  �  · ·  · ·  H = Hdiff  1→2 + Hdiff  2→1 + · · · + HAfor  1  + HBrev  1  + · · ·  Transition  (a)  (b)  (c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The reaction-diffusion master equation (RDME) consists of voxels in which well-mixed chemical reactions occur (a)  and between which species can diffuse (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' This work centers around a one-dimensional chain of L such voxels, each with a  fluctuating number of molecules of species X in a Schl¨ogl chemical reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The Doi-Peliti (DP) framework turns the reaction-  diffusion evolution into a second-quantized effective Hamiltonian that governs the evolution of the distribution over classical  reaction-diffusion microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (c) That effective Hamiltonian is represented by a high-dimensional tensor, with each vertical  leg taking one of (M +1) possible values corresponding to the number of molecules of species X which could occupy each voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The DP form of the effective Hamiltonian lends itself to a compact representation of the effective Hamiltonian tensor of rates  for transitioning from one microstate to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  FL with |n⟩ ∈ FL and ⟨m|n⟩ = δmn [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Without loss  of generality, we will discuss the case of a single chem-  ical species and L physical sites, in which case nl has  the interpretation of the number of molecules at site l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The second-quantization procedure rewrites H in terms  of creation and annihilation operators, x†  l and xl, that  respectively raise and lower the number of X molecules  at site l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' That is to say,  x†  l |nl⟩ = |nl + 1⟩ ,  xl |nl⟩ = nl |nl − 1⟩ ,  xl |0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (12)  where |0⟩ is the vacuum state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=', no molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By xl  or x†  l we are implying that the operator only acts on the  lth site:  xl = I1 ⊗ I2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' ⊗ Il−1 ⊗ x ⊗ Il+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' ⊗ IL  (13)  Because they act on the single-site space, the general one-  site operators x and x† can be written as the M+1×M+1  matrices  x† =  �  ����  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  1  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  0  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  �  ���� and  x =  �  ����  0  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  0  0  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  �  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (14)  That these matrices yield Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (12) can be readily con-  firmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  More abstractly, the existence of such raising  and lowering operators in this classical setting originates  from the classical commutation relation [xix†  j] = δij and  the existence of a positive semi-definite number operator  ˆnl = x†  l xl, so named because ˆnl |nl⟩ = nl |nl⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Together,  the set {Ω, FL, x, x†} are also sometimes called an inter-  acting Fock space [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Our use of the raising and low-  ering operators is less formal and more pragmatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' As in  the construction of the number operator, we will be able  to decompose contributions to ˆH in terms of the x† and  x’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  One benefit of defining operators in terms of creation  and annihilation operators is that expectation values for  observables can be expressed easily in terms of inner  products with operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Consider, for example, the ob-  servable that counts all particles in all voxels, ˆn = �  l ˆnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The operator ˆn acts on a many-body state |n⟩, yet each  ˆnl is a single-body operator acting on site l and leaving  all other sites unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' It is therefore understood that  when local operators such as ˆnl act on |n⟩, it is written  as a shorthand for  ˆnl = I ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' ⊗ x†  l xl ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' ⊗ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (15)  In terms of that shorthand, ˆnl |n⟩ = nl |n⟩ counts the  number of particles in site l for the many-body state |n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The time-dependent expected number of particles in the  whole system is thus ⟨n(t)⟩ = �  l ⟨1| ˆnl |pt⟩, where the  vector of ones ⟨1| serves to sum over all microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The  time-dependence of a distribution can also be extracted  from the action of a second-quantized ˆH on |pt⟩, and  this operation can be practically computed when ˆH is  expressed in terms of the single-site operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Mirroring the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' II, the generator ˆHl  for the Schl¨ogl model reactions within the lth voxel is  a sum of contributions from the elementary reactions  ˆHl = �  r ˆHr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 2a shows each reaction, expressed as  a second-quantized contribution to the effective Hamil-  tonian, as derived in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Combining together the  forward and reverse reactions mediated by species A, we  see that the reversible reaction Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (1) contributes  ˆHA  l = x†2  l  ��  c1x†  l + c2  �  I − x†  l  �  xl  �  x2  l − c1x2  l ˆPc  Ml  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (16)  to the effective Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For convenience we have  absorbed the combinatorial terms into that rate coeffi-  6  pt(nX1, nX2, nX3, · · · , nXNs) ≈  · ·  nXNs  nX1  nX2  nX3  pt(nX  1 , nX  2 , nX  3 , · · · , nX  L) ≈  · ·  nX  L  nX  1  nX  2  nX  3  pt(nX1  1 , nX2  1 , nX1  2 , n2  2, · · · , nX1  L/2, nX2  L/2) ≈  · ·  nX1  1  nX2  1  nX1  2  nX2  2  nX1  L/2 nX2  L/2  H ≈  nX  1  mX  1  nX  2  mX  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  nX  L−1  mX  L−1  nX  L  mX  L  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The ensemble evolution of the many-body state requires that one solve for the probability of every microstate at  all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' That high-dimensional joint probability distribution can be well approximated by a matrix product state (MPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  An MPS approximation for the joint distribution can be advantageous whether there are Ns species in a well-mixed reaction  [(a) top], one dynamical (unchemostatted) species with diffusion [(a) middle], or multiple species and diffusion [(a) bottom].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  In all three cases, each circle represents a tensor, either a rank-2 at the first and last site or rank-3 in the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Horizontal  lines represent the bond-indices connecting tensors and vertical lines are physical indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For an RDME, the physical indices  represent the possible number of molecules at a given tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' If two lines are connected, then the index is summed over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (b)  An effective Hamiltonian H can be likewise decomposed into lower-rank tensors to form a matrix product operator (MPO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Now there are two physical indices per tensor, representing transitions from state nX  l  to mX  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Vertical connected lines are the  bond indexes which are contracted over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  cient, defining cr ≡ ˜cr/γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=', where γ is either ηr or µr for  forward and reversed reactions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Likewise,  the contribution  ˆHB  l = x†  l (c3 − c4)xl + c4xl − c3 ˆPc  Ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (17)  comes from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' We show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' A that the trunca-  tion of the state space at finite M requires the addition  of a projector, ˆPc  Ml to prevent ˆHl from promoting any  probability from the M th state into an (M + 1)th that  has been truncated away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In the M → ∞ limit, the pro-  jector can be discarded, as reflected by the expressions  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The Hamiltonian operator for L independent, well-  mixed Schl¨ogl voxels is thus ˆHrxn = �L  l ˆHA  l + ˆHB  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' To  add diffusion, consider the well-mixed voxels arranged in  a 1D lattice as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Each chemical species is able  to move between neighboring voxels as regulated by a  diffusion operator  ˆHdiff =  L−1  �  l=1  d  ��  x†  l+1 − x†  l  �  (xl − xl+1)  �  ,  (18)  with hopping rate d = D/h2, consistent with macroscopic  diffusion constant D for voxel width h [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Combining  reaction with diffusion:  ˆH = ˆHrxn + ˆHdiff,  (19)  is straightforward at the level of the effective Hamilto-  nian, but the diffusive coupling between neighbors has  the consequence of introducing nontrivial interactions be-  tween voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  TENSOR NETWORKS  We have now seen how to transform the matrix-form  rate operator H of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' II into a Doi-Peliti form ˆH, ex-  pressed in terms of local creation and annihilation oper-  ators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' That change of representation is not merely cos-  metic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  As the number of chemical species or number  of lattice sites grows, the size of H grows exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  For N dynamical (unchemostatted) species and L lattice  sites, the state space has (M + 1)NL states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' To put this  scaling in perspective, the largest numerical result that  follows uses N = 1, M = 85 and L = 8, meaning the  system contains 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='99 × 1015 microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Calculations on  a matrix H, even a very sparse matrix H, are untenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  By contrast, it is practical to construct a TN decompo-  sition of the second quantized form of ˆH in terms of an  MPO, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 3b, and demonstrated explicitly  for the Schl¨ogl mechanism in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Given the sec-  ond quantized form, the MPO can be readily constructed  in a practical form through SVD decomposition using au-  tomated packages such as autoMPO in ITensor [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Much like the operator ˆH can be compressed as an  MPO, the states that operators act upon can also be  expressed more compactly in terms of TNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Consider  |pt⟩ = pt(n) = pt(n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , nL), the probability of mi-  crostate n at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' This |pt⟩ can be thought of as a rank  L tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By specifying the occupancy of each of the L  sites, |pt⟩ outputs a number—a probability—associated  to that microstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  By leveraging a matrix product  ansatz, |pt⟩ can be approximated as a partial contraction  over L different low-rank tensors (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The low  rank tensors of an MPS [53] are organized in a 1D chain of  7  tensors Qn1  s1 , Qn2  s1s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , QnL−1  sL−2sL−1, and QnL  sL−1 whose up-  per index nl specifies the occupancy of site l and whose  lower indices are so-called bond indices, nonphysical in-  dices which are contracted over to generate correlations  between nearby sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For suitably chosen Q tensors, we  therefore approximate  |pt⟩ =  �  n1,n2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='nL  pt(n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , nL) |n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nL⟩  ≈  �  s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=',sL−1  Qn1  s1 Qn2  s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' QnL  sL−1 |n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nL⟩  ≡ |qt⟩ ,  (20)  where |qt⟩ is the approximate distribution over mi-  crostates generated from the MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Though not explicitly  written, the Q tensors are time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  There is a diagrammatic representation of the tensors,  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 3, which aids in visualizing tensor opera-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The tensors Q are given by circles, one per site,  and vertical lines feeding into these circles are the phys-  ical indices that pick out a particular slice of the tensor  based on the occupation number of that site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Circles also  have horizontal lines emanating from them, representing  the non-physical bond indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Connection of two cir-  cles by a line represents a shared index being contracted  over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 3 emphasizes that the TN decomposition is  flexible enough to address varied CME’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The MPS of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 3a middle, has been the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Each site from 1 to  L reflects the number of X molecules in that voxel, but  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 3a (top) and (bottom) show that the MPS decom-  position can equally well be constructed when multiple  species Xi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , Ns can occupy a single voxel or  when Ns different species diffuse between L/Ns voxels to  yield an MPS with L sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' What is practically important  is that Q’s be arranged such that the physical dynamics  correlates the neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  For example, for the RDME  Schl¨ogl model, neighboring Q’s correspond to neighbor-  ing sites in the 1D Schl¨ogl lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  An attractive feature of the MPS ansatz is that the  size and accuracy of the MPS are controllable through  the bond dimension of the s auxiliary indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' If the di-  mension of those indices is allowed to grow exponentially  from s1 through sL/2, the ansatz in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (20) can be exact,  but even when the bond dimension of the indices s are  capped at a maximum dimension of κ, the ansatz can be a  very good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Crucially, capping that maxi-  mum bond dimension makes it practical to work with |qt⟩  because the rank-L tensor |pt⟩ has been replaced by the  L different Q tensors, each with no more than (M +1)κ2  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Having approximated both |qt⟩ and ˆH by an MPS and  MPO, respectively, we can now revisit the otherwise in-  tractable dynamics problem of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (3), only now we wish  to evolve |qt⟩ in lieu of |pt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' As |qt⟩ is an MPS state, we  wish for the dynamics to be constrained for all times to  the manifold M of possible MPS states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Merely replacing  |pt⟩ by |qt⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (3):  d |qt⟩  dt  = ˆH |qt⟩  (21)  would not impose such a constraint because ˆH in general  evolves |qt⟩ off M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' An appealing resolution is given by  the Time-Dependent Variational Principle (TDVP) [54,  55], which proposes to first project the right-hand side  onto the tangent space of the MPS manifold:  d |qt⟩  dt  = PTM ˆH |qt⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (22)  In this way, dynamics is restricted to the desired sub-  space of MPS states [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  It turns out that the time-discretized numerical inte-  gration of this |qt⟩ is amenable to efficient TN opera-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The technical details of the time-evolution algo-  rithm have been reported elsewhere [55, 57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' We give  only a brief, high-level overview of that procedure (see  also App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' C for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The central step in de-  riving the TDVP time-evolution algorithm is to demon-  strate that the projector PTM can be decomposed in  terms of a sum over L terms (associated to each site  of the MPS) and L − 1 terms (associated to each bond  connecting sites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In the limit of a small time step δt, the  propagator ePTM ˆ  Hδt can thus be evaluated as a compo-  sition of time-evolutions involving the 2L − 1 individual  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' That composition of time-evolutions is executed  by a one-site TDVP algorithm that “sweeps” from lattice  site 1 through L, advancing each term for the small time  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The efficiency of the TDVP time-evolution algorithm  relies on the flexibility to represent the same MPS state  in redundant ways due to a gauge freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Consider,  for example, neighboring tensors Qnl  sl−1sl and Qnl+1  slsl+1 in  an MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For an invertible R, replacing those tensors by  Qnl  sl−1slR and R−1Qnl+1  slsl+1, respectively, does not change  the MPS state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' upon contracting over the neighboring  sites, the identity RR−1 would fall out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By leveraging  the gauge freedoms, it is always possible to transform  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (20) into a so-called mixed canonical form [60]:  |qt⟩ = An1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Anl−1QnlBnl+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Bnl |n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , nL⟩ , (23)  where each A is a left-orthogonal tensor satisfying AT A =  I and each B is a right-orthogonal tensor satisfying  BBT  = I, where here, T represents the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  For notational compactness, we have suppressed implied  summations over the bond indices between tensors in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The mixed canonical form is said to be cen-  tered about the tensor Qnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' From that tensor’s perspec-  tive, the chain of A’s and chain of B’s constitute the en-  vironment, and it is beneficial to contract over the links  between A’s and links between B’s to obtain so-called  environment tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' These environment tensors inherit  an orthogonality from the orthogonality of the A’s and  B’s, significantly reducing the computational complex-  ity to advance Qnl in time by δt provided the MPS was  already centered about the Qnl we aim to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  8  A  B  t  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='01  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='14  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='44  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='75  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Snapshots of the joint density in two neighboring voxels from an L = 3 site lattice of Schl¨ogl voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Using  c1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='676, c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='040, c3 = 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='102, c4 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='881 and d = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='2207, the steady-state distribution was computed with κ = 30 by  evolving an initial uniform distribution according to ˆH via TDVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' That steady-state distribution was projected into A at time  0, renormalized, then propagated according to the modified no-recrossing dynamics ˜H using TDVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For clarity of visualization,  the density within A is not plotted as it would overwhelm the small probability that leaks into B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='01 (left image),  probability has begun to leak from A but none has reached B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Probability has just begun to reach B at at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='14, roughly  the τmol timescale for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='44, probability is collecting in B, and by t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='75, the flux passing from A to B  has attained an approximately constant steady-state value corresponding to the linear growth phase where P(Bt|A) ∝ kBAt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  As made more explicit in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' C, the one-site TDVP  algorithm initially centers the MPS around the first ten-  sor, propagates that tensor forward in time by δt, per-  forms a QR decomposition to change the gauge, propa-  gates backward in time by δt the R term of that QR de-  composition then passes that term to the neighboring site  to re-center the MPS around the neighboring site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The  neighboring site is then propagated forward in time as  the algorithm iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' After the complete sweep through  the MPS, the probability distribution |qt⟩ has been prop-  agated by a time step δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Whether that propagation well  approximates the evolution of the full distribution |pt⟩  depends on the time-step error but also on the severity  of the MPS approximation, which can be systematically  controlled through the maximum bond dimension κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Notably, the time-evolved MPS state |qt⟩ allows us  to estimate the probability of any microstate of inter-  est, even though it would be impossible to enumerate  probabilities of all microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Alternatively, sets of mi-  crostates can be summed over to yield tractable marginal  distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 4 shows the results  of calculations on a 1D chain of L = 3 Schl¨ogl voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Though there are three voxels, site one is traced over to  leave the two-dimensional joint probability of finding nX  2  X molecules in site 2 and nX  3 X molecules in site 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' At  time zero, the distribution is initialized within region A  according to ˆPA |π⟩ / ⟨1| ˆPA |π⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' As time progresses, a  small flux of probability leaks from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Because the  overwhelming preference is to stay in A, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 4 highlights  the rare transitions by only plotting the joint distribution  outside of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In the next section, we illustrate the fact  that the MPS state’s ability to capture the rare time-  dependent flux is key to calculating the rate of traversing  between basins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  MANY-BODY RATE CALCULATION  Having established the TDVP dynamics of an MPS  state, we can now repeat the strategy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (7) to com-  pute a transition rate between states A and B:  kBA = d  dt  ⟨1| ˆPB ePTM ˆ  Ht ˆPA |π⟩  ⟨1| ˆPA |π⟩  ����  t>τmol  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (24)  In doing so, there are a few important differences from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Most obviously, the TDVP dynamics includes  the projection onto the MPS tangent space and |π⟩ must  now be an MPS approximation to the steady-state dis-  tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Additionally, the meaning of ˆPA and ˆPB must  be adapted to now define many-body analogs of the pro-  jectors onto the two metastable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The single-voxel  Schl¨ogl model had an A state defined as all microstates  with occupancy n above a threshold q∗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Similarly, B  included all microstates with occupancy less than some  other threshold q∗  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The L-voxel Schl¨ogl model general-  izes to involve hypercubic metastable states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' A requires  that each voxel satisfies n ≥ q∗  A and B requires that each  voxel satisfies n ≤ q∗  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  With these generalizations, the transition rate of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (24) would measure the flux crossing into B through  its boundary, but if we are to compare with a transition  path theory rate, we must exclude re-crossing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In  other words, we want to include trajectories which exit A  then some time later enter B without having re-entered  A, but we want to exclude the flux that leaves B and  re-enters B some time later without having returned to  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' II B, kBA was sufficiently large that those re-  crossings could be ignored, but as we generalize to handle  smaller kBA, those recrossings can become significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  We note that the TDVP dynamics can just as well be  executed with a modified absorbing-boundary-condition  70X  1(  3-40  020406000nX  3  3X  240250600  0102(30405060710L30X  n  214050500  0102(30405060710L30X  n  214050500  0102(30405060710730X  n  214050509  dynamics with effective Hamiltonian  ˜H = ˜Hrxn + ˜Hdiff,  (25)  where the reaction and diffusion Hamiltonians exclude  events that exit B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' An explicit construction of that mod-  ified dynamics is provided in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' D, allowing us to di-  rectly remove the re-crossing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  NUMERICAL EVALUATION OF THE  L-VOXEL SCHL¨OGL MODEL  The formalism of the preceding sections has laid out  a controllable approximation to extract rates between  metastable states of RDMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Numerical experiments  are necessary to evaluate whether that formalism can  be practically useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  We set out to demonstrate that  utility by computing the transition rates between high-  occupancy and low-occupancy states of a L-voxel Schl¨ogl  model as a function of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The model is parameter-  ized by the four stochastic rate coefficients discussed  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' II: c1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='676, c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='040, c3 = 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='102, and  c4 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='881, chosen so the L = 1 Schl¨ogl model will  exhibit the bistablity reflected in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' With those cho-  sen rates, A and B are identified by choosing na = 15  and nb = 25, and the dynamics lends itself to a finite  maximum-occupancy truncation of M = 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By linking  multiple voxels together in a 1D chain with a diffusive  hopping rate d = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='2207, it becomes rarer to switch be-  tween the two metastable states than it would be with  a single isolated voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' With multiple voxels, a switch-  ing event must flip the state of each voxel, and diffusion  from a neighbor can stabilize the original state of each  voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Increasing L thus gives a way to systematically  probe rarer transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The most straightforward way to numerically detect  the L-dependence of the transition rate is via sampling of  trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Realizations of the continuous-time RDME,  subject to the M = 85 maximum voxel occupancy, were  generated using the SSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' First, 5×105 independent SSA  realizations were initialized from a uniform distribution  and propagated up to t = 5 to ensure the steady state dis-  tribution was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Sampled configurations that fell  within A were then uniformly drawn as an initial con-  figuration for 400 independent trajectories which were  evolved via the SSA until the trajectory first reached B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  From each of those 400 trajectories, the time between  the final exit of A and the first entry of B, τBA was  recorded and the transition rate was computed from the  mean of those waiting times: kBA = ⟨τBA⟩−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 5a  (red line) shows that the rate exponentially decreases as  L is increased, a result consistent with the exponential  dependence of rate on volume in the well-mixed Schl¨ogl  model [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' While the rate can be computed by trajec-  tory sampling, the difficulty to do so (via brute force  sampling) grows in proportion to the typical τBA, mean-  ing the computational cost grows exponentially with L  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The poor scaling of brute-force SSA sam-  pled rate calculations has motivated so-called advanced  sampling methods like FFS, which focus computational  effort on the progress along a reaction coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 5  shows that the DP/TDVP approach provides a radically  different strategy for circumventing the scaling problem,  reproducing the SSA rates at lower computational ex-  pense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The SSA rate calculations can be made more accurate  by collecting more trajectories, and the DP/TDVP cal-  culations likewise offer a systematic way to improve accu-  racy by increasing the computational expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The two  most prominent ways to adjust that accuracy are through  the time step δt for the TDVP evolution and through  the maximum bond dimension κ for the MPS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In  this work, a time step of δt = 10−4 was used through-  out, and the trade-off between expense and accuracy was  entirely tuned through the choice of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The single-site  TDVP algorithm we used to propagate an MPS state  does not alter the bond dimension of the MPS, so κ was  entirely set by the bond dimension of the initial MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  We therefore needed to approximate the steady state |π⟩  with different values of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Approximations were gener-  ated by starting with an initially uniform state using a  bond dimension κ set to several different values between  5 and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Those uniform distributions were evolved with  single-site TDVP for 10, 000 time-steps allowing for the  natural dynamics of ˆH to approach the limit |π⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' We  found that evolving that 10, 000 time steps was sufficient  to approximate the steady-state distribution, as reflected  by the average rate of change of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' That  is to say, ⟨pt| ˙pt⟩ = ⟨pt| ˆH |pt⟩ ≈ 0 by time t = 1, indicat-  ing proximity to the actual steady state, which satisfies  | ˙π⟩ = |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  To get rates, the κ-dependent MPS approximation to  the steady state was projected onto A and renormalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  That renormalized distribution was then propagated un-  der single-site TDVP with the modified dynamics for an-  other 10, 000 time steps, and projected onto B to add up  the total probability to have reached B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Note that these  10,000 steps are not nearly enough to reach the steady  state |π⟩ because the slow rate of transitioning from A to  B is a bottleneck, one that did not need to be surmounted  to relax from the uniform distribution to |π⟩ in 10, 000  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' As additional time steps of dynamics were taken,  the growth of the transition flux from A to B was mea-  sured and plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' That figure shows that at  very short times (t < τmol), the flux from A to B has not  reached a steady state value because there hasn’t been  enough time for any transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' After that initial time  tmol, the plot plateaus at a value corresponding to the  rate kBA, a plateau that is reached orders of magnitude  faster than the time for a typical SSA trajectory to make  a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' We find that if the bond dimension is insuf-  ficient (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=', κ = 5 for L = 3), then the dynamics becomes  unstable, reflected in erratic estimates for the conditional  probability P(Bt|A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' As the bond dimension is grown  to κ = 9, the estimates converge, allowing a stable rate to  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='8  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='02  2  3  4  5  6  7  8  10-6  10-5  10-4  10-3  10-2  10-1  2  3  4  5  6  7  8  100  101  102  103  104  (a)  (b)  (c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (a) The rate kBA for switching between metastable states of the L-voxel Schl¨ogl model is calculated using the SSA  (Red line) and using TDVP (dashed lines) with varying bond-dimensions κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For sufficiently large bond-dimensions, TDVP  agrees with a SSA rate estimate built using 400 transitions from A to B at each L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (b) As L grows a larger bond-dimension is  required to capture the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For L = 3 we see that for κ = 5 and κ = 6 the flow of probability from A to B is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  As κ grows, the solutions converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For this system, κ = 20 is stable up to L = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' While κ = 30 gives stable solutions up  to L = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (c) The total computation time in hours to estimate kBA from both the SSA and TDVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Despite using different  bond-dimensions to estimate rates on larger lattices, we see that the total time to evolve the TDVP ensemble evolution scales  sub-exponentially while the SSA scales exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Larger values of L demand an even greater  value of bond dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 5a shows that SSA rates  agree with κ = 20 calculations up to L = 6, but κ = 30  was required to push up to L = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Provided that larger  bond dimension is used, the DP/TDVP rates reproduce  the SSA rates across five orders of magnitude, up to and  beyond the point that the brute-force SSA rates are prac-  tical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Though the required bond dimension grows with L,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 5c shows that the total time to estimate the rate  grows sub-exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The blue and black dashed lines  in that plot are the total computational time to estimate  the rate using TDVP with κ = 30 and κ = 20 respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Both SSA and TDVP were run using codes writ-  ten in the Julia programming language [62] on equivalent  CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' While the exact wall time, of course, depends on  details of the hardware and the implementation, the plot  clearly demonstrates the difference in scaling between  brute-force SSA and DP/TDVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' An increase in bond di-  mension increases the computation time, but it does not  fundamentally alter the favorable scaling in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  In addition to the favorable scaling at large L, we  highlight that the DP/TDVP formalism allows us to ex-  tract many observables with similar ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For example,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 6 shows ⟨nl(t)⟩ = ⟨1| ˆnl |pt⟩, the average number  of molecules in site l with l ranging from 1 through 4  for an L = 8 site lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The state |pt⟩ was initialized  in the normalized steady-state confined to A, the high-  occupation-number state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For t < τmol, a rapid loss of  particles near the boundary of A leads to a sharp drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  After that τmol timescale, the rate of decrease in ⟨nl(t)⟩  becomes linear, reflecting the steady-state flux from A to  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='8  1  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='5  44  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='5  45  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='5  46  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Average number of molecules for different lattice  sites when L = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Colors correspond to a different lattice  sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The expected number of molecules ⟨nl(t)⟩ = ⟨1| ˆnl |pt⟩  for sites 1 to ⌊L/2⌋ (shown) are mirrored by sites ⌊L/2+1⌋ to  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The initial non-linear flow of probability out of A is shown  by the steep drop in average number of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The system  passing τmol is shown by the transition to a linear change in  particle number ⟨nl(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  DISCUSSION  In this work we have shown how macroscopic rates can  be efficiently calculated from the microscopic dynamics of  a RDME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' We have illustrated that capability in a min-  imal toy model, the L-voxel Schl¨ogl model, with each  voxel chemostatted such that it contains a single dynam-  ical species (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In actuality, the formalism and numer-  ical methodology extend far beyond this initial demon-  stration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Any reaction-diffusion mechanism which can be  written as a set of elementary reactions will have a DP  representation, one which can involve raising and low-  ering operators for different particle types, not only for  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For example, Table I shows the effective Hamiltoni-  11  ans for several common (unchemostatted) reactions: uni-  molecular, bimolecular, and auto-catalytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Given the  set of reactions, it is comparatively straightforward to  derive the DP representation, and computational tools  already exist to generate an MPO [49] and propagate via  TDVP from that DP form of the effective Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  As those computational tools become increasingly well  Reaction  Hamiltonian  X  ˜c  −−→ Y  c[(y† − x†)x]  X + Y  ˜c  −−→ Z  c[(z† − x†y†)xy]  X + Y  ˜c  −−→ 2 X c[x†(x† − y†)xy]  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Examples of the mapping between a common  chemical reaction and the Doi-Peliti form of the contribution  to an effective Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The procedure to derive such  correspondences is systematic [63] and typically discussed for  a Fock space with occupation number ranging from 0 to infin-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' To truncate to a finite Fock space, one must additionally  add a projector to each effective Hamiltonian, as in this work,  to ensure conservation of probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  developed, one can imagine that the TN tools will offer  generic “turn the crank” analysis of microscopic models  of reaction-diffusion at the ensemble level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' That vision  motivates more future work at the interface of reaction-  diffusion kinetics and tensor network methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Here, we  have approximated the microscopic distribution with an  MPS, allowing us to capitalize on the efficient TDVP al-  gorithm for MPS states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The downside is that an RDME  with many dynamic species will not always be easily ap-  proximated by a 1D chain of tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' If two chemical  species are strongly correlated but far apart in the MPS  structure, the MPS calculation will demand a very high  bond dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' It will be important for future work ei-  ther to avoid the need for high bond dimension, for exam-  ple by leveraging TN topologies which mirror that of an  underlying CRN or to adapt the bond dimension through  subspace expansion [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Those technical advances may  eventually lay the groundwork to move the methodol-  ogy from this Schl¨ogl study to the complex, biologically-  motivated models where SSA realizations presently reign  supreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We gratefully acknowledge Hadrien Vroylandt, Nils  Strand, and Luis Pedro Garcia-Pintos for many insightful  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' We are also grateful to Miles Stoudenmire,  Matthew Fishman, Steven White, and other developers  of ITensor, a library for implementing tensor network cal-  culations, upon which this work was built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The material  presented in this manuscript is based upon work sup-  ported by the National Science Foundation under Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 2141385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' TRG is thankful that Phill Geissler could  see this work in a preliminary form, but is deeply sad-  dened that Phill’s much-too-soon passing prevented him  from sharing more of his wisdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Appendix A: Doi-Peliti encoding  Assume a set of elementary reactions are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' These  can be encoded into a Doi-Peliti representation which  converts the set of elementary reactions to a set of  effective Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The operator representations  can then be exactly written as an MPO, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Consider a 1D lattice in which each voxel can undergo  reactions from the Schl¨ogl model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' A state of the system  will be written with respect to the Fock space FL which  consists of basis vectors |n⟩ ∈ FL, ⟨m|n⟩ = δm,n, being  clear that |n⟩ = |n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , nL⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The probability of a  microstate is given by pn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The distribution over the  system is p(n, t) = p(n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , nL, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' When focusing  on an individual voxel we will use the shorthand,  p(n, t) := p(nc, nl, t), where nc = {nk |k ̸= l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Finally,  powers of ladder operators are written as, x†2  l  = x†  l x†  l ,  for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  2X + A  ˜c1  −−−→ 3X: For the reaction 2X + A  ˜c1  −−→ 3X  within site l the change in probability is,  dpt(n)  dt  = ˜c1  2 [(nl − 1)(nl − 2)pt(nc, nl − 1)  − nl(nl − 1)pt(nc, nl)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (A1)  We can expand with respect to the Fock states in order  to write out the distribution in the Fock-basis represen-  tation, |pt⟩ = �  n pt(n) |n⟩ which gives,  d |pt⟩  dt  = ˜c1  2  �  n  [(nl − 1)(nl − 2)pt(nc, nl − 1) |n⟩  − nl(nl − 1)pt(nc, nl) |n⟩]  (A2)  For the second term we can substitute in ladder operators  for the number of molecules x†2  l x2  l = nl(nl −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The first  term can be written with respect to the vacuum states  |0⟩, x†3  l x2  l x†(nl−1)  l  |0⟩ = (nl − 1)(nl − 2) |n⟩,  d |pt⟩  dt  = ˜c1  2  �  n  �  x†3  l x2  l pt(nc, nl − 1) |nc, nl − 1⟩  − x†2  l x2  l pt(nc, nl) |n⟩  �  (A3)  Distributing the summation we see that there are two  separate sets of states being summed over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' To make it  clearer, substitute ml = nl − 1 to give,  d |pt⟩  dt  = ˜c1  2  �  nc  � M−1  �  ml=−1  x†3  l x2  l pt(nc, ml) |nc, ml⟩  − x†2  l x2  l  M  �  nl=0  pt(nc, nl) |n⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (A4)  12  We would like both summations to be over the same set  of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For the first summation, the ml = −1 term  is zero due to |nc, −1⟩ = xl |nc, 0⟩ = 0 and thus can be  excluded from the first summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Likewise, the upper  limit can be extended since x†3  l x2  l |nc, Ml⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In order  to conserve probability we require the effective Hamil-  tonian satisfy ⟨1n| ˆHn,n′ = 0, since this guarantees ˆH  has a zero-eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Conservation of probability man-  ifests in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (A4) through a balance between the es-  cape rates given by the second term and the transitions  into a state through the first terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Looking at states  |nc, Ml⟩, the second term is non-zero, but the first term  is x†3  l x2  l |nc, Ml⟩ = 0 due to the finite state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Result-  ing in negative column sums and violation of probability  conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' A solution is to project out the offending  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  d |pt⟩  dt  ⇒ ˜c1  2  �  nc  �  M  �  ml=0  x†3  l x†2  l pt(nc, ml) |nc, ml⟩  − x†2  l x2  l  M  �  nl=0  pt(nc, nl) |n⟩  �  + ˜c1  2  �  nc  �  nl  x†2  l x2  l pt(nc, nl) |nc, Ml⟩ ⟨nc, Ml|n⟩ ,  = ˜c1  2  �  nc  �  M  �  ml=0  x†3  l x2  l pt(nc, ml) |nc, ml⟩  −  M  �  n=0  x†2  l x2  l  �  Il − ˆPMl  �  pt(nc, nl) |nc, nl⟩  �  ,  = ˜c1  2  �  x†3  l x2  l − x†2  l x2  l ˆPc  Ml  �  |pt⟩ ,  (A5)  where we used the projectors, ˆPMl = |nc, Ml⟩ ⟨nc, Ml|,  ˆPc  Ml = I − ˆPMl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Since probability is now conserved we  can set m → n and ˜c1/2 = c1, and identify the Effective  Hamiltonian contribution from the reaction,  HAfor  l  = c1  �  x†3  l x2  l − x†2  l x2  l ˆPc  Ml  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (A6)  3X  ˜c2  −−−→ 2X + A: The change in probability for the  back reaction is  dpt(n)  dt  = ˜c2  6 [n(n − 1)(n + 1)pt(nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl + 1)  − n(n − 1)(n − 2)pt(nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl)]  (A7)  Writing in terms of the Fock space and substituting in the  number operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' x†2  l x3  l x†  l |nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl⟩ = n(n − 1)(n + 1) |n⟩  and x†3  l x3  l |n⟩ = n(n − 1)(n − 2) |n⟩ gives,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  d |pt⟩  dt  = ˜c2  6  �  n  [n(n − 1)(n + 1)pt(nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl + 1) |n⟩  − n(n − 1)(n − 2)pt(n) |n⟩] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  = ˜c2  6  �  nc  � M  �  nl=0  x†2  l x3  l pt(nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl + 1) |nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl + 1⟩  −  M  �  nl=0  x†3  l x3  l pt(nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl) |nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (A8)  Writing the indices of the first term as, m = n + 1  M+1  �  m=1  x†2  l x3  l pt(nc, ml) |nc, ml⟩  (A9)  We have to check the upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The lower  term ml = 1 can be see to be zero both by looking  at the reaction, or by directly checking x†2  l x3  l |nc, 1⟩ =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  When m = M + 1 the imposed chemostat makes  x†2  l x3  l |nc, M + 1⟩ = x†2  l x3  l x†  l |nc, M⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' So we see that  unlike the previous reaction, the boundary terms which  differ between the first and second summations are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Thus we do not need to project out any states and can  immediately write,  d |pt⟩  dt  = c2  �  x†2  l  �  Il − x†  l  �  x3  l  �  |pt⟩ ,  = HArev  l  |pt⟩ ,  (A10)  again using ˜c2/6 = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By summing ˆHAfor  l  and ˆHArev  l  we  have ˆHA  l , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (16) from the manuscript  ˆHA = x†2  l  ��  c1x†  l + c2(xl − x†  l xl)  �  x2  l − c1x2  l ˆPc  Ml  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  B  ˜c3  −−−→ X: Following the previous steps, Start with  the change in probability for state n,  dpt(n)  dt  = ˜c3pt(nc, nl − 1) − c3pt(n),  d |pt⟩  dt  = ˜c3  �  nc  ��  nl  x†  l pt(nc, nl − 1) |nc, nl − 1⟩  −  �  nl  pt(nc, nl) |nc, nl⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (A11)  Again, using ml = nl − 1 with |nc, −1⟩ = xl |nc, 0⟩ =  0, renaming ˜c3 = c3 and projecting out non-conserving  13  states we have,  d |pt⟩  dt  ⇒ c3  �  nc  �  x†  l  M−1  �  ml=0  pt(nc, ml) |nc, ml⟩  −  M  �  nl=0  pt(nc, nl) |nc, nl⟩  �  +  M  �  nl=0  pt(nc, nl) |nc, Ml⟩ ⟨nc, Ml|nc, nl⟩ ,  = c3  �  x†  l − ˆPc  Ml  �  |pt⟩ ,  = HBfor  l  |pt⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (A12)  X  ˜c4  −−−→ B: Following the above process we start with,  dpt(n)  dt  = ˜c4(nl +1)pt(nc, nl +1)−c4nlpt(nc, nl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (A13)  Expanding and substituting operators gives,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  d |pt⟩  dt  = ˜c4  �  nc  M  �  nl=0  �  xlx†  l pt(nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl + 1) |nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  − x†  l xlpt(n) |n⟩  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  = ˜c4  �  nc  M  �  nl=0  [xl pt(nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl + 1) |nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' nl + 1⟩  − x†  l xlpt(n) |n⟩  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  = ˜c4  �  nc  �M+1  �  ml=1  xl pt(nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' ml) |nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' ml⟩  −  M  �  nl=0  x†  l xlpt(n) |n⟩  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (A14)  With ˜c4 = c4 and using the same logic used for reaction  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' the upper and lower bounds of the first summation are  zero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' giving,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  d |pt⟩  dt  = c4  �  Il − x†  l  �  xl |pt⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  = HBrev  l  |pt⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (A15)  By combining HBfor  l  and HBrev  l  we arrive at HB  l  in the  manuscript  HB  l = x†  l (c3 − c4xl) + c4xl − c3 ˆPc  Ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Appendix B: Matrix Product Operators  Any set of reactions which can be put in a “product”  form i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (16), (17) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (18) can be converted  to a MPO form and compressed if need be through an  automated package such as AUTOMPO in ITensor [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  For systems with local and nearest neighbor interactions  there exists relatively simple representations of the  effective Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Writing out an exact representation for the Schl¨ogl  model, each reaction only acts on one local subspace at  a time ˆHrxn = �L  l=1 ˆHl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Diffusion is composed of next  nearest neighbor terms Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' These can be combined  into three matrix terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' A bulk operator  W [l] =  �  ��������  I  x†  l  −x†  l xl  0  xl  D  Hl,  Dxl, D, Dx†  l , −x†  l xl, I  �  ��������  ,  (B1)  and two boundary terms  W [1] =  �  H1, Dx1, D, Dx†  1, −x†  1x1, I  �  ,  W [L] =  �  ��������  I  x†  L  −x†  LxL  xL  D  HL  �  ��������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (B2)  The product then reproduces the Hamiltonian  ˆH  =  �L  l=1 W [l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Appendix C: TDVP algorithm  We wish to solve the Schr¨odinger equation  ��� ˙ψ  �  = H |ψ⟩  (C1)  Since the dimension of |ψ⟩ grows exponentially with  molecule number we approximate the state by an MPS  |ψ⟩ ≈ |ψ⟩Q where we have used the Q subscript to em-  phasize the dependence on the set of tensors {Qnl  sl−1sl},  l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In the main text, the equivalent state  would be |ψ⟩Q = |qt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Two important definitions we will  need are,  Manifold of matrix product states (Mκ): is the  set of states |ψ⟩Q which can be represented by a matrix  product state with bond-dimension κ [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Tangent states and manifold: Write the set of ten-  sors in |ψ⟩Q as Q = {Qn1, Qn2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' , QnL}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Use the in-  clusive index l = (nl, sl−1, sl) in the shorthand Ql.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Let  Γ be a set of tensors where the tensor ΓL has the same  dimensions as Ql ∀ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  A tangent state |Θ⟩ of |ψ⟩Q is  14  = eδt ˆ  P +  2 |ψ⟩Q = �∞  k=0  δtk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (P +  2 ˆH)k |ψ⟩Q =  =  · ·  + δt  · ·  · ·  · ·  · ·  + δt2  2  · ·  · ·  · ·  · ·  · ·  · ·  · ·  + · · ·  =  · ·  + δt  · ·  H(c)  + δt2  2  · ·  H(c)  H(c)  + · · ·  = �  n  �∞  k=0  δtk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' [H(2)k]n2,n′  2An′  1(t + δt)Qn2(t)Bn′  3(t) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' + Bn′  L(t) |n⟩  = �  n An′  1(t + δt)Qn2(t + δt)Bn′  3(t) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' + Bn′  L(t) |n⟩  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Applying each operator P ±  l  to |ψ⟩Q results in a local equation of motion as shown here for the P +  2  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Writing out the infinite series in the second line, we can partially contract around the center site (black diamond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' These  partial contractions form the local effective Hamiltonian H(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The rest of the MPS remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Combing these local  Hamiltonians forms the matrix exponential only for the centered site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  defined as,  |Θ(Γ, Q)⟩ =  �  l  �  α,β,nl  Φ[1:l−1]  A,α  Γnl  αβΦ[l+1:L]  B,β  |nl⟩  (C2)  where Φ[1:l−1]  A,α  and Φ[l+1:L]  B,β  are the left and right environ-  ment tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The corresponding tangent space for the  manifold of MPS, Mκ is defined as,  TMκ ≡ {|Θ(Γ, Q)⟩ | ∀ l : dim(Γl) = dim(Ql)}  (C3)  Approximating |ψ⟩ with an MPS results in a second  equation of motion,  |ψ⟩Q = ˆH |ψ⟩Q  (C4)  since in general |ψ⟩ ̸= |ψ⟩Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  We would like to solve  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' C1 but cannot for all but toy problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Instead  we seek a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' C4 which will be a good ap-  proximation to |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  There are two issues, one is that a direct solution to  ˆH |ψ⟩Q requires dealing with the entire MPS/MPO equa-  tion simultaneously which is prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The  second and more pressing issue is that ˆH |ψ⟩Q is no longer  an MPS, but instead a tensor with L legs, meaning we  have left the manifold Mκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The solution that TDVP  proposes is to project ˆH |ψ⟩Q onto the manifold of MPS  and then evolve |ψ⟩Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' In this way the state never leaves  Mκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The projector we seek is that which projects a state  |ψ⟩ onto the closest tangent vector |Θ(Γ, Q)⟩ = ˆPMκ |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  The tensors which minimize the distance between the  tangent vector and |ψ⟩ is found through the solution to,  min  Γ  ∥ |Θ(Γ, Q)⟩ − |ψ⟩ ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (C5)  15  The variationally optimal projector is given by, [59]  ˆPT =  L  �  l=1  P [1:l]  L  ⊗ 1l ⊗ P [l+1:L]  R  −  L−1  �  l=1  P [1:l]  L  ⊗ P [l+1:L]  R  ,  =  L  �  l=1  P +  l +  L−1  �  l=1  P −  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (C6)  Where, P [1:n]  L  = �  α  ���Φ[n1:nl]  L,α  � �  Φ[n1:nl]  L,α  ���, and P [l+1:L]  R  =  �  β  ���Φ[nl+1:nL]  R,β  � �  Φ[nl+1:nL]  R,β  ��� are the projectors built from  the environment tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' C4 can be  written as a sum of terms,  |ψ(t + δt)⟩Q =  exp  �  δt  �  P +  L ˆH +  L−1  �  l=1  P +  l  ˆH + P −  l  ˆH  ��  |ψ(t)⟩Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (C7)  If δt ≪ 1 then we can pull out each pair of terms in the  exponential and apply them to |ψ(t)⟩Q one at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For  example, starting with l = 1 we have,  |ψ(t + δt)⟩Q =  exp  �  dt  �  P +  L H +  L−1  �  l=2  P +  l H + P −  l  ˆH  ��  edtP −  1 ˆ  HedtP +  1 ˆ  H |ψ(t)⟩Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (C8)  For a given l we need to evaluate, edtP +  l H |ψ(t)⟩ which  when expanded is,  edtP +  l H |ψ(t)⟩ =  ∞  �  k=0  dtk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  �  P +  l H  �k |ψ(t)⟩Q  (C9)  Now, the matrix exponential exp  �  dtP +  l H  �  is acting glob-  ally on |ψ(t)⟩Q, which would prove to be very expensive  for many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Luckily the projector combined  with allowed gauge transformations reduces the global  matrix exponential to a local matrix exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  (7) shows the local nature for the P +  2 term, where |ψ⟩Q  is centered at site two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' We see each tensor around the  center site is unchanged while the center site is evolved  forward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The P −  l  terms have a similar local na-  ture to them but before acting on |ψ⟩Q we use a gauge  transformation Qnl �→ AnlCl where Cl only consists of  non-physical bond indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  P −  l  then induces the local  exponential term exp(−δtK(l))Cl(t + δt) which evolves  Cl(t + δt) back by δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Absorbing with the next term  Cl(t)Bnl+1 = Qnl+1 shifts the orthogonality center al-  lowing for the P +  l+1 term to be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' By first center-  ing |ψ(t)⟩Q at site one, and then applying sequences of  gauge transformations with P +  l  and P −  l  terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' C7,  we can “sweep” from l = 1 to l = L, which amounts to  evolving |ψ(t)⟩Q → |ψ(t + δt)⟩Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Appendix D: Modified Dynamics  To measure a rate from A into B without multiple  crossings of B’s boundary, we define a modified dynamics  via an effective Hamiltonian that zeroes out the rates of  any reaction or diffusion event that would depart the re-  gion B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Deriving the forms of ˜Hrxn and ˜Hdiff in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (25)  requires that we first identify the set of states and reac-  tions (here meaning reaction or diffusion events) as  Bout ≡ {n, r : n + νr ∈ Bc | n ∈ B}  (D1)  where Bc is the complement of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' For the Sch¨ogl model,  only diffusion and the forward reactions can map out of  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The states in Bout are those where particles in the lth  voxel are at the boundary to B, nl = q∗  B and nk ∈ B ∀ k ̸=  l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The local boundary for B can be projected onto using  Q = |q∗  B⟩ ⟨q∗  B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' Combining Q with the projector Nl =  �  nl∈B |nl⟩ ⟨nl|, lets us define the projection operator  ˆPout  l  = N1 ⊗ N2 ⊗ · · · Nl−1 ⊗ Ql ⊗ Nl+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' NL, (D2)  The joint projector built from non-independent projec-  tors is given by the powerset expansion  ˆPout =  L  �  l=1  (−1)l+1 �  {sl}  �  sl  ˆPout  sl  (D3)  where {sl} is the set of combinations from L elements  taken l at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  For L = 3 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (D3) becomes  ˆPout = [ ˆPout  1  + ˆPout  2  + ˆPout  3  ]  − [ ˆPout  1  ˆPout  2  + ˆPout  1  ˆPout  3  + ˆPout  2  ˆPout  3  ]  + ˆPout  1  ˆPout  2  ˆPout  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  Each product over ˆPout  sl  can be simplified noting that,  NlQl = Ql which leads to  ˆPout =  L  �  i=1  (−1)l+1 �  {ck}  �  ck  {N, Q}ck  (D4)  where the sum over ck is now all combinations of N and  Q where Q appears k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' The advantage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content=' (D4)  is that effective Hamiltonians can be projected onto only  using local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
+page_content='  In this work, B consists of all  molecule numbers less than q∗  B+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE2T4oBgHgl3EQfLwZ4/content/2301.03717v1.pdf'}
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+1 
+ 
+Layer-number-dependent spin Hall effects in transition metal 
+monocarbides M2C (M=V, Nb, Ta) 
+Xi Zuo1, Yulin Feng1, Na Liu1, Bing Huang2, Desheng Liu3*, Bin Cui3* 
+ 
+1College of Physics and Electronic Science, Hubei Normal University, Huangshi 
+435002, China. 
+2Beijing Computational Science Research Center, Beijing 100193, China. 
+3School of Physics, National Demonstration Center for Experimental Physics 
+Education, Shandong University, Jinan 250100, China. 
+ 
+*emails: liuds@sdu.edu.cn; cuibin@sdu.edu.cn 
+Abstract 
+   The recent discovery of strong spin Hall effects (SHE) in 2D layered topological 
+semimetals has attracted intensive attention due to its exotic electronic properties and 
+potential applications in spintronic devices. In this paper, we systematically study the 
+topological properties and intrinsic SHE of layered transition metal carbides M2C (M=V, 
+Nb, Ta). The results show that both bulk and monolayer M2C have symmetry-protected 
+nodal points (NPs) and lines (NLs) originating from the d band crossing near the Fermi 
+level (EF). The inclusion of SOC breaks the degeneracy of NLs and NPs, contributing 
+to large spin Hall conductivity (SHC) up to ~1100 and ~200 (ℏ/𝑒)(Ω cm)−1 for bulk 
+and monolayer Ta2C, respectively. Remarkably, we find that magnitude of SHC exhibits 
+a significant enhancement by increasing the layer number. For eight-layer Ta2C, the 
+maximum value of SHC can reach up to ~600 (ℏ/𝑒)(Ω cm)−1, comparable to many 
+reported 3D topological materials. Analysis of spin Berry curvature reveals that the 
+large SHC originates from layer-number-dependent nodal line structure near the EF, in 
+which the repeated crossover between valence and conduction bands creates large 
+amounts of NPs along the Г-K route. Our findings not only provide a new platform for 
+experimental research of low-dimensional SHE, but also suggest an effective way of 
+realizing giant SHE by controlling layer thickness. 
+
+2 
+ 
+I. INTRODUCTION 
+Spin Hall effects (SHE), a relativistic phenomenon that a transverse spin current 
+generated by an applied electric field in the absence of a magnetic field, has become an 
+important topic in recent years [1,2]. Two key factors, intrinsic spin Hall conductivity 
+(SHC) and spin Hall angle (SHA), influence the performance of SHE-based devices. 
+The intrinsic SHC can be accurately calculated by integrating the spin Berry curvature 
+(SBC) of the occupied bands in the Brillouin zone (BZ) [3]. And the SHA of a SHE 
+system is the ratio of the SHC to the charge conductivity (GC), which represents the 
+charge-to-spin interconversion efficiency at room temperature [4]. Therefore, exploring 
+SHE systems with large SHC and SHA is one of the main goals in this field. 
+   With the rapid development of topological matters, strong SHE is observed in 
+topological systems such as topological insulators (TIs) and topological semimetals 
+(TSMs) [5–7]. Due to their spin-momentum-locked surface states, TIs are considered 
+to be ideal systems for generating pure spin currents. However, various experiments 
+show that the SHA of the Bi2Se3 family can vary from 0.01 to 425 due to the bulk 
+doping problem induced by the hybridization between surface states and bulk 
+states [5,8,9]. Recently, increasing attention has been focused on the intrinsic SHE in 
+TSMs. For example, nodal point semimetals ZrSiTe [10], TaAs [11], and LaAlGe [12] 
+have been predicted to host large intrinsic SHC exceeding 500 (ℏ/𝑒)(Ω cm)−1 due 
+to large SBC around the spin-orbit coupling (SOC) induced small gapped Dirac or Weyl 
+nodal points. On the other hand, large intrinsic SHEs have also been reported in nodal 
+line semimetals. Different from nodal point semimetals, the gapped nodal lines can 
+induce many band anticrossing points distributed with large and continuous SBC in the 
+entire BZ, contributing to a remarkable SHC [13]. It is predicted that NLSMs such as 
+W3Ta [14], Ta3As [15], ThAl2 [16], and InBi [17] with multiple nodal lines around the 
+Fermi level (EF) exhibit a giant SHC exceeding 1000 (ℏ/𝑒)(Ω cm)−1. To realize large 
+intrinsic SHC and SHA as well as to design new devices for spin-charge current 
+conversion, it is important to explore new TSMs and study the interplay between the 
+SHC and the band topology. 
+
+3 
+ 
+Recently, the 2D transition metal dichalcogenides (TMDs) family are reported to 
+exhibit large SHE due to their tunable SOC, GC, and band topology [18–20]. For 
+example, the Weyl semimetals MoTe2 and WTe2 are reported to own large SHA of 
+0.32 [21] and 0.17 [22], respectively. Spin-orbit torque in few-layers Dirac semimetal 
+PtTe2 [23] reveals that the SHC exhibits a monotonical increment with a maximum 
+value of about 1000 (ℏ/𝑒)(Ω cm)−1 as the layer thickness increases from 3 nm to 20 
+nm. Correspondingly, first-principle calculations on bilayer PtSe2[24] and tri-layer 
+MoTe2 [21] reveal that its intrinsic SHC is about 27 and 200 (ℏ/𝑒)(Ω cm)−1 at the 
+EF, respectively. This raises the question of whether we can find large intrinsic SHC in 
+layered 2D materials comparable to familiar heavy metals or 3D topological materials. 
+On the other hand, the influence of layer thickness on the SHE in 2D materials still calls 
+for further theoretical investigation. 
+In this paper, we systematically investigate electronic structures and intrinsic SHE 
+in 2D layered transition metal carbides M2C (M=V, Nb, Ta, known as MXenes), which 
+exhibit a fascinating combination of properties such as controllable minimum layer 
+thickness, large electric conductivity, rich electronic structures, and can be easily 
+synthesized in the laboratory [25–30]. Both bulk and monolayer M2C exhibit large SHC 
+owing to strong SOC and contributions of multiple nodal lines in the band structures. 
+Taking Ta2C as an example, we observe a significant enhancement of SHC by varying 
+the thickness from one layer (1L) to eight layers (8L), which originates from a layer-
+number-dependent nodal line structure near the EF. The rest of this article is organized 
+as follows. In Sec. II A, we first report the calculated band structures, SHC, and SHA 
+for bulk M2C. In Sec. II B, we then present the calculated SHC (𝜎𝑥𝑦
+𝑧 ) by varying the 
+layer number from 1L up to 8L. To understand the origin of layer-number-dependent 
+SHC, we also give the electronic band structure, nodal line structure, and k-resolved 
+SBC at the EF. Finally, the conclusion drawn from this work is summarized in Sec. III. 
+ 
+ 
+ 
+
+4 
+ 
+II. RESULTS AND DISCUSSION 
+A. The electronic band structures and SHE in bulk M2C 
+As shown in Fig. 1(a), the bulk M2C (M=V, Nb, Ta) crystallize in a trigonal crystal 
+structure with the space group 𝑃3̅𝑚1 (164). The C atom locates at the corner sites of 
+the unit cell, while two M atoms locate at equivalent sites with the Wyckoff position of 
+2d (1/3, 2/3, 1/4). Each C atom is bonded to six equivalent M atoms to form an edge-
+sharing octahedral, and each M is bonded in a distorted T-shaped geometry to three 
+equivalent C atoms. The optimized lattice parameters and bond lengths are summarized 
+in Table S1 in the Supplemental Material (SM) [31], which agree well with previous 
+experimental and theoretical values [29,30]. The BZ is shown in Fig. 1(b). 
+ 
+FIG. 1. (a) Crystal structure of bulk M2C (M=V, Nb, Ta) compounds. (b) Brillouin zone (BZ) of 
+a primitive cell of M2C.  
+Based on the optimized structures, the calculated band structures in the absence of 
+SOC for Ta2C are shown in Fig. 2(a), respectively, and those of other compounds are 
+displayed in Fig. S1 in the SM [31]. The orbital character analysis shows that the Ta 5d 
+orbitals dominate the bands near the EF. It is interesting that several linear bands cross 
+each other and form the nodal points P1 and P2 and nodal line L1. Here, we use N to 
+replace the number of occupied bands, shown as the red line in Fig. 2(a). Along K-Г 
+and L-H route, P1 and P2 nodal points are formed by the band crossing between the 
+singlet B and A states under the C2 point group. On the other hand, it is observed that 
+the Nth and (N+1)th bands degenerate and induce the nodal line L1, which belongs to 
+the doublet E states under C3v point group symmetry. The multiple band degeneracies 
+
+(a)
+(b)5 
+ 
+around the EF imply the nodal-line structures in Ta2C. To confirm our physical intuition, 
+we perform a systemic nodal-line searching in the first BZ (see Methods section in the 
+SM [31]). Here, we first identify all the k points with zero energy gap between the Nth 
+and (N+1)th bands and plot them with green lines in Fig. 2(b) and Fig. 2(c). Then similar 
+treatment is used in the nodal line searching between the (N-1)th and Nth bands,  
+shown with blue lines. There are totally 17 nodal lines that can be classified into four 
+classes: Class I, the closed nodal ring around the Г point in the 𝜎ℎ plane; Class II, three 
+curved nodal lines and one straight nodal line crossing the Г point along the Г-A route; 
+Class III, six parabolic-like nodal lines connecting horizontal edges of the BZ; Class IV, 
+six straight nodal lines along vertical edges of the BZ. 
+ 
+FIG. 2. (a) band structure and projected density of states of Ta2C without SOC. P1, P2 and L1 
+represent the two nodal points and the nodal line in the band structure, respectively. (b) Band-
+crossing-formed nodal lines in the first BZ of Ta2C. (c) top view of (b). The EF is set to zero in (a). 
+When the SOC effect is included, the spin-rotation symmetry is destroyed, which 
+breaks the nodal line structure protected by PT symmetry [32,33]. In Fig. 3(a), the SOC 
+lifts the degeneracies of most nodal points in band structures of Ta2C. The SOC-induced 
+bandgap is 25 meV and 360 meV at P1 and P2, and a maximum value of 550 meV at Г 
+point along Г-A route, which is significantly larger than V2C (from 8 to 47 meV) and 
+Nb2C (from 11 to 142 meV) (see detailed band structure of M2C systems in Fig. S1 in 
+
+(a)
+2
+Energy (eV)
+total
+ Ta-d
+C-p
+-1 
+2
+M
+K
+r
+A
+L
+H
+A
+PDOS
+(b)
+(c)6 
+ 
+the SM [31]). This means Ta has a much stronger SOC than V and Nb. Since strong 
+SOC-induced band anticrossings around the nodal lines can generate large local SBC, 
+it is expected that M2C systems have strong SHE. To evaluate the SHE in M2C, we have 
+estimated their SHC values at EF and listed them in Table I (see SM [31] for 
+computational details). At first glance, three SHC components are near isotropic for 
+V2C, Nb2C, and Ta2C. Taking 𝜎𝑥𝑦
+𝑧   as an example, the magnitude of SHC values 
+increases rapidly from top to bottom, which is in accord with the variation of SOC 
+strength from V to Ta. The SHC at EF reaches -1084 (ℏ/𝑒)(Ω cm)−1 for Ta2C, which 
+is comparable to 5d transition metal β−W ~ 1255 (ℏ/𝑒)(Ω cm)−1 [34], and recently 
+reported nodal-line systems, such as InBi ~ 1100 (ℏ/𝑒)(Ω cm)−1 [17], HfH2 ~ 1100 
+(ℏ/𝑒)(Ω cm)−1 [35] and Ta3Bi ~ 1700 (ℏ/𝑒)(Ω cm)−1 [15]. We also show energy-
+dependent SHC (𝜎𝑥𝑦
+𝑧 ) of Ta2C in Fig. 3(c). Interestingly, for a wider range of EF shifting, 
+the magnitude of SHC values for Ta2C can still keep larger than 103 (ℏ/𝑒)(Ω cm)−1. 
+Based on the definition of SHA described in SM [31], we estimate the GC of M2C and 
+calculate the SHA. Due to their rather small GC values, the intrinsic SHA of Ta2C can 
+reach 0.22, which is larger than that of Pt (0.068) [36] and comparable to that of Ta 
+(0.15) [37]. 
+TABLE I. Intrinsic SHC for three independent tensor elements for bulk M2C (M=V, Nb, and 
+Ta). The unit of SHC and 𝐺𝐶 is (ℏ/𝑒)(Ω cm)−1 and (Ω cm)−1. 
+ 
+𝜎𝑦𝑧
+𝑥  
+𝜎𝑧𝑥
+𝑦  
+𝜎𝑥𝑦
+𝑧  
+𝐺𝐶 
+SHA 
+V2C 
+-408 
+-369 
+-348 
+1.05×104 
+0.06 
+Nb2C 
+-581 
+-569 
+-565 
+1.61×104 
+0.07 
+Ta2C 
+-1093 
+-1167 
+-1084 
+9.86×103 
+0.22 
+To further understand the origin of large SHC in Ta2C, we calculate the band-
+projected SBC and k-resolved SBC at EF in Fig. 3(a) and 3(b), respectively. It is 
+observed that the large peaks of SBC at EF mainly appear at gapped nodal lines around 
+P1, P2 and L1 in Fig. 2(a). In addition, we also calculate the k-resolved SBC in (001) 
+plane at EF in Fig. 3(d). The SBC locates around the center ring and outer Hexagon 
+region, corresponding to nodal lines near Г point in Fig. 2(c). 
+
+7 
+ 
+ 
+FIG. 3. (a) The band structure of Ta2C with the projection of SBC. (b) k-revolved SBC along 
+high symmetry lines. (c) Energy-dependent SHC of M2C. (d) SBC in (001) planes of BZ. The EF is 
+set to zero in both of (a) and (c). 
+ 
+B. The electronic band structures and SHE in few-layer M2C 
+  The intimate relationship between SHC with nodal-line structure in bulk M2C implies 
+that strong SHE can also exist in monolayer or few-layered M2C. We first calculate the 
+SHC of the monolayered (1L) M2C, shown in Table II. Due to the reduced symmetry 
+of 2D structures, the dominant independent component is 𝜎𝑥𝑦
+𝑧  . The SHC ( 𝜎𝑥𝑦
+𝑧  ) 
+increases gradually as SOC strength becomes larger from V2C to Nb2C and Ta2C. The 
+SHC and SHA of Ta2C reach -191 (ℏ/𝑒)(Ω cm)−1 and 0.05 at EF . Since V2C, Nb2C, 
+and Ta2C shares similar geometry and band structures, therefore, we take Ta2C as an 
+example to study the layer dependence of SHC (𝜎𝑥𝑦
+𝑧  ) by enhancing the thickness 
+
+2.0
+1.0
+(eV)
+1.0
+(a)
+Cnergy (
+0.0
+0.5
+-1.0
+-2.0
+0.0
+() ()
+0.0
+(b)
+-0.5
+-0.5
+logol
+2-1.0
+-1.0
+-1.5
+T
+M
+K
+r
+A
+L
+H
+A
+800
+0.0
+V,C
+(p)
+SHC(h/e) (S/cm)
+400
+Nb,C
+Ta,C
+0
+-0.6
+400
+800
+-1200
+-1.2
+-2
+0
+2
+Energy (eV)8 
+ 
+from monolayer (1L) up to eight layers (8L). The absolute value of SHC at EF versus 
+the number of layers is calculated and shown in Fig. 4. Generally, the SHC shows a 
+linearly increasing trend as layer numbers go from 1 to 5 and rise slightly for 6L-8L 
+Ta2C. The maximum value of SHC is 590 (ℏ/𝑒)(Ω cm)−1, which is larger than 2D 
+systems such as 2L PtSe2 (~27 (ℏ/𝑒)(Ω cm)−1  at EF  [24]), 3L MoTe2 (~200 
+(ℏ/𝑒)(Ω cm)−1  at EF [21]), and 1L SnTe (~245 (ℏ/𝑒)(Ω cm)−1  at 𝐸𝐹 +
+1.09 eV  [38]).  
+TABLE II. Intrinsic SHC for three tensor elements for monolayer M2C (M=V, Nb, and Ta). The 
+unit of SHC and GC is (ℏ/𝑒)(Ω cm)−1 and (Ω cm)−1. 
+ 
+𝜎𝑥𝑦
+𝑧  
+𝜎𝑦𝑧
+𝑥  
+𝜎𝑧𝑥
+𝑦  
+C
+G  
+SHA 
+V2C 
+-24 
+-3 
+-3 
+5.32×103 
+0.009 
+Nb2C 
+-46 
+-6 
+-6 
+1.19×104 
+0.007 
+Ta2C 
+-191 
+-15 
+-14 
+7.06×103 
+0.05 
+ 
+ 
+FIG. 4. Layer-dependent SHC for 1L-8L Ta2C. 
+ 
+To figure out the origin of layer-dependent SHE observed in few-layered Ta2C, we 
+first discuss the electronic structures for 1L Ta2C in Fig. 5(a). Without SOC, the valence 
+band (marked with a red line) crosses the conduction band at two points along the K-Г 
+route. In Fig. 5(e), a careful examination of the energy difference between the two bands 
+reveals that the nodal point near the Г point belongs to the unclosed nodal ring around 
+the Г point. The nodal point near the K point is one of the six equivalent points each on 
+
+600 -
+SHC(h/e) (S/cm)
+500-
+400-
+300-
+200-
+1
+2
+3
+4
+5
+6
+7
+8
+Number of layer9 
+ 
+one Г-K (K’) route due to the symmetry protections of 𝐶3𝑣. In Fig. S2, the inclusion of 
+SOC lifts the degenerate points and opens remarkable band gaps around the EF along 
+the K-Г route. The coexistence of nodal lines and large SOC-induced band gaps aligned 
+near the EF induces large SBC in 1L Ta2C. In Fig. 5(i), the dominant amplitude of SBC 
+mainly concentrates around the central ring and outer raindrop-shaped area. For 3L and 
+5L Ta2C, the increase in thickness induces repeated crossings between the two bands 
+on the Г-K route. Meanwhile, the conduction band sticks closely to the valence band 
+along the M-K route for 5L Ta2C. The increased degeneracy forms another circle 
+around the Г point and many more nodal points along the Г-K route. Correspondingly, 
+the inclusion of SOC leads to band gaps around the nodal lines, thus generating denser 
+distribution of SBC and larger SHC at EF. It should be noted that the energy level of 
+the nodal points along the M-K route is not in the vicinity of EF and thus does not 
+contribute to the SBC distribution in Fig. 5(k). In addition, for 7L Ta2C, the band 
+crossing along the M-K route induce more nodal points in the Г-K route, which, 
+however, do not bring much broader SBC distribution near the EF, thus leading to the 
+slow increment of SHC. 
+ 
+
+10 
+ 
+ 
+FIG. 5. (a)-(d) Electronic band structure without SOC for 1L, 3L, 5L, and 7L Ta2C, respectively. 
+(e)-(h) 2D contour of the energy gap difference between the valence and conduction bands 
+corresponding to (a)-(d). (i)-(l) Calculated SBC in the first BZ corresponding to (a)-(d). 
+ 
+III. DISCUSSION AND CONCLUSIONS 
+We take M2C (M=V, Nb, and Ta) as a representative in the above presentation. As 
+shown in Fig. S3, the essential band crossing between the bands near the EF is also 
+shared by other members of MXenes family. In most materials, like Zr2C, Hf2C, Nb2N, 
+and Ta2N, the band crossing points are below or above the EF, while in other materials 
+such as Zr2N and Hf2N, the band crossing points are just at the EF, which are expected 
+to contribute to strong SHC peaks. 
+
+(a)
+1.5
+(e)
+2.0
+(i)
+0.0
+1.0
+1.5
+0.5
+-0.6
+1L
+0.0
+1.0
+-0.5
+-1.2
+0.5
+-1.0
+1.5
+0.0
+-1.8
+r
+M
+K
+(b)
+1.5
+(f)
+2.0
+(i)
+0.0
+1.0
+Energy (eV)
+1.5
+0.5
+3L
+-0.6
+0.0
+1.0
+0.5
+-1.2
+0.5
+-1.0
+0.0
+T
+-1.8
+M
+K
+(c)
+1.5
+(g)
+(k)
+2.0
+0.0
+1.0
+1.5
+0.5
+5L
+-0.6
+0.0
+1.0
+0.5
+-1.2
+0.5
+1.0
+M
+K
+0.0
+-1.8
+(d)
+1.5
+(h)
+2.0
+(1)
+0.0
+1.0
+Energy (eV)
+1.5
+0.5
+7L
+-0.6
+1.0
+0.5
+-1.2
+0.5
+1.0
+0.0
+-1.8
+M
+K11 
+ 
+In summary, we have predicted large intrinsic SHE in layered transition metal 
+carbides M2C (M=V, Nb, and Ta). Due to strong SOC and contributions of multiple 
+nodal lines in the band structure, the SHC and SHA of bulk Ta2C can reach up to ~1100 
+(ℏ/𝑒)(Ω cm)−1 and ~0.20, respectively. For a few-layered Ta2C, the 𝜎𝑥𝑦
+𝑧  component 
+of SHC exhibits a monotonic increase as the number of layers increases. The maximum 
+value of SHC is 590 (ℏ/𝑒)(Ω cm)−1 for 8L Ta2C, which is larger than reported 2D 
+systems so far and comparable to many 3D topological systems. Therefore, one can 
+effectively tune the SHC by controlling the layer thickness in layered 2D materials. Our 
+results not only elucidate the interplay between layer-dependent SHC and band 
+topology, but also provide theoretical guidance for developing next-generation 
+spintronic devices. 
+ 
+ACKNOWLEDGMENTS 
+This work is supported by National Natural Science Foundation of China (Grant 
+No. 11574118), NSFC (Grants No. 51672023), and NSAF (Grant No. U1930402).  
+ 
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+ 
+
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new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf,len=627
+page_content='1  Layer-number-dependent spin Hall effects in transition metal  monocarbides M2C (M=V, Nb, Ta)  Xi Zuo1, Yulin Feng1, Na Liu1, Bing Huang2, Desheng Liu3*, Bin Cui3*  1College of Physics and Electronic Science, Hubei Normal University, Huangshi  435002, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  2Beijing Computational Science Research Center, Beijing 100193, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  3School of Physics, National Demonstration Center for Experimental Physics  Education, Shandong University, Jinan 250100, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  emails: liuds@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' cuibin@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='cn  Abstract  The recent discovery of strong spin Hall effects (SHE) in 2D layered topological  semimetals has attracted intensive attention due to its exotic electronic properties and  potential applications in spintronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' In this paper, we systematically study the  topological properties and intrinsic SHE of layered transition metal carbides M2C (M=V,  Nb, Ta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The results show that both bulk and monolayer M2C have symmetry-protected  nodal points (NPs) and lines (NLs) originating from the d band crossing near the Fermi  level (EF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The inclusion of SOC breaks the degeneracy of NLs and NPs, contributing  to large spin Hall conductivity (SHC) up to ~1100 and ~200 (ℏ/𝑒)(Ω cm)−1 for bulk  and monolayer Ta2C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Remarkably, we find that magnitude of SHC exhibits  a significant enhancement by increasing the layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' For eight-layer Ta2C, the  maximum value of SHC can reach up to ~600 (ℏ/𝑒)(Ω cm)−1, comparable to many  reported 3D topological materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Analysis of spin Berry curvature reveals that the  large SHC originates from layer-number-dependent nodal line structure near the EF, in  which the repeated crossover between valence and conduction bands creates large  amounts of NPs along the Г-K route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Our findings not only provide a new platform for  experimental research of low-dimensional SHE, but also suggest an effective way of  realizing giant SHE by controlling layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  2  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' INTRODUCTION  Spin Hall effects (SHE), a relativistic phenomenon that a transverse spin current  generated by an applied electric field in the absence of a magnetic field, has become an  important topic in recent years [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Two key factors, intrinsic spin Hall conductivity  (SHC) and spin Hall angle (SHA), influence the performance of SHE-based devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  The intrinsic SHC can be accurately calculated by integrating the spin Berry curvature  (SBC) of the occupied bands in the Brillouin zone (BZ) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' And the SHA of a SHE  system is the ratio of the SHC to the charge conductivity (GC), which represents the  charge-to-spin interconversion efficiency at room temperature [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Therefore, exploring  SHE systems with large SHC and SHA is one of the main goals in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  With the rapid development of topological matters, strong SHE is observed in  topological systems such as topological insulators (TIs) and topological semimetals  (TSMs) [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Due to their spin-momentum-locked surface states, TIs are considered  to be ideal systems for generating pure spin currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' However, various experiments  show that the SHA of the Bi2Se3 family can vary from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='01 to 425 due to the bulk  doping problem induced by the hybridization between surface states and bulk  states [5,8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Recently, increasing attention has been focused on the intrinsic SHE in  TSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' For example, nodal point semimetals ZrSiTe [10], TaAs [11], and LaAlGe [12]  have been predicted to host large intrinsic SHC exceeding 500 (ℏ/𝑒)(Ω cm)−1 due  to large SBC around the spin-orbit coupling (SOC) induced small gapped Dirac or Weyl  nodal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' On the other hand, large intrinsic SHEs have also been reported in nodal  line semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Different from nodal point semimetals, the gapped nodal lines can  induce many band anticrossing points distributed with large and continuous SBC in the  entire BZ, contributing to a remarkable SHC [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' It is predicted that NLSMs such as  W3Ta [14], Ta3As [15], ThAl2 [16], and InBi [17] with multiple nodal lines around the  Fermi level (EF) exhibit a giant SHC exceeding 1000 (ℏ/𝑒)(Ω cm)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' To realize large  intrinsic SHC and SHA as well as to design new devices for spin-charge current  conversion, it is important to explore new TSMs and study the interplay between the  SHC and the band topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  3  Recently, the 2D transition metal dichalcogenides (TMDs) family are reported to  exhibit large SHE due to their tunable SOC, GC, and band topology [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' For  example, the Weyl semimetals MoTe2 and WTe2 are reported to own large SHA of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='32 [21] and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='17 [22], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Spin-orbit torque in few-layers Dirac semimetal  PtTe2 [23] reveals that the SHC exhibits a monotonical increment with a maximum  value of about 1000 (ℏ/𝑒)(Ω cm)−1 as the layer thickness increases from 3 nm to 20  nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Correspondingly, first-principle calculations on bilayer PtSe2[24] and tri-layer  MoTe2 [21] reveal that its intrinsic SHC is about 27 and 200 (ℏ/𝑒)(Ω cm)−1 at the  EF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' This raises the question of whether we can find large intrinsic SHC in  layered 2D materials comparable to familiar heavy metals or 3D topological materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  On the other hand, the influence of layer thickness on the SHE in 2D materials still calls  for further theoretical investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  In this paper, we systematically investigate electronic structures and intrinsic SHE  in 2D layered transition metal carbides M2C (M=V, Nb, Ta, known as MXenes), which  exhibit a fascinating combination of properties such as controllable minimum layer  thickness, large electric conductivity, rich electronic structures, and can be easily  synthesized in the laboratory [25–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Both bulk and monolayer M2C exhibit large SHC  owing to strong SOC and contributions of multiple nodal lines in the band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  Taking Ta2C as an example, we observe a significant enhancement of SHC by varying  the thickness from one layer (1L) to eight layers (8L), which originates from a layer-  number-dependent nodal line structure near the EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The rest of this article is organized  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' II A, we first report the calculated band structures, SHC, and SHA  for bulk M2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' II B, we then present the calculated SHC (𝜎𝑥𝑦  𝑧 ) by varying the  layer number from 1L up to 8L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' To understand the origin of layer-number-dependent  SHC, we also give the electronic band structure, nodal line structure, and k-resolved  SBC at the EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Finally, the conclusion drawn from this work is summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  4  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The electronic band structures and SHE in bulk M2C  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 1(a), the bulk M2C (M=V, Nb, Ta) crystallize in a trigonal crystal  structure with the space group 𝑃3̅𝑚1 (164).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The C atom locates at the corner sites of  the unit cell, while two M atoms locate at equivalent sites with the Wyckoff position of  2d (1/3, 2/3, 1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Each C atom is bonded to six equivalent M atoms to form an edge-  sharing octahedral, and each M is bonded in a distorted T-shaped geometry to three  equivalent C atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The optimized lattice parameters and bond lengths are summarized  in Table S1 in the Supplemental Material (SM) [31], which agree well with previous  experimental and theoretical values [29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The BZ is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (a) Crystal structure of bulk M2C (M=V, Nb, Ta) compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (b) Brillouin zone (BZ) of  a primitive cell of M2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  Based on the optimized structures, the calculated band structures in the absence of  SOC for Ta2C are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 2(a), respectively, and those of other compounds are  displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' S1 in the SM [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The orbital character analysis shows that the Ta 5d  orbitals dominate the bands near the EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' It is interesting that several linear bands cross  each other and form the nodal points P1 and P2 and nodal line L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Here, we use N to  replace the number of occupied bands, shown as the red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Along K-Г  and L-H route, P1 and P2 nodal points are formed by the band crossing between the  singlet B and A states under the C2 point group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' On the other hand, it is observed that  the Nth and (N+1)th bands degenerate and induce the nodal line L1, which belongs to  the doublet E states under C3v point group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The multiple band degeneracies  (a)  (b)5  around the EF imply the nodal-line structures in Ta2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' To confirm our physical intuition,  we perform a systemic nodal-line searching in the first BZ (see Methods section in the  SM [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Here, we first identify all the k points with zero energy gap between the Nth  and (N+1)th bands and plot them with green lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 2(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Then similar  treatment is used in the nodal line searching between the (N-1)th and Nth bands,  shown with blue lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' There are totally 17 nodal lines that can be classified into four  classes: Class I, the closed nodal ring around the Г point in the 𝜎ℎ plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Class II, three  curved nodal lines and one straight nodal line crossing the Г point along the Г-A route;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  Class III, six parabolic-like nodal lines connecting horizontal edges of the BZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Class IV,  six straight nodal lines along vertical edges of the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (a) band structure and projected density of states of Ta2C without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' P1, P2 and L1  represent the two nodal points and the nodal line in the band structure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (b) Band-  crossing-formed nodal lines in the first BZ of Ta2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (c) top view of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The EF is set to zero in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  When the SOC effect is included, the spin-rotation symmetry is destroyed, which  breaks the nodal line structure protected by PT symmetry [32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 3(a), the SOC  lifts the degeneracies of most nodal points in band structures of Ta2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The SOC-induced  bandgap is 25 meV and 360 meV at P1 and P2, and a maximum value of 550 meV at Г  point along Г-A route, which is significantly larger than V2C (from 8 to 47 meV) and  Nb2C (from 11 to 142 meV) (see detailed band structure of M2C systems in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' S1 in  (a)  2  Energy (eV)  total  Ta-d  C-p  1  2  M  K  r  A  L  H  A  PDOS  (b)  (c)6  the SM [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' This means Ta has a much stronger SOC than V and Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Since strong  SOC-induced band anticrossings around the nodal lines can generate large local SBC,  it is expected that M2C systems have strong SHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' To evaluate the SHE in M2C, we have  estimated their SHC values at EF and listed them in Table I (see SM [31] for  computational details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' At first glance, three SHC components are near isotropic for  V2C, Nb2C, and Ta2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Taking 𝜎𝑥𝑦  𝑧   as an example, the magnitude of SHC values  increases rapidly from top to bottom, which is in accord with the variation of SOC  strength from V to Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The SHC at EF reaches -1084 (ℏ/𝑒)(Ω cm)−1 for Ta2C, which  is comparable to 5d transition metal β−W ~ 1255 (ℏ/𝑒)(Ω cm)−1 [34], and recently  reported nodal-line systems, such as InBi ~ 1100 (ℏ/𝑒)(Ω cm)−1 [17], HfH2 ~ 1100  (ℏ/𝑒)(Ω cm)−1 [35] and Ta3Bi ~ 1700 (ℏ/𝑒)(Ω cm)−1 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' We also show energy-  dependent SHC (𝜎𝑥𝑦  𝑧 ) of Ta2C in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Interestingly, for a wider range of EF shifting,  the magnitude of SHC values for Ta2C can still keep larger than 103 (ℏ/𝑒)(Ω cm)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  Based on the definition of SHA described in SM [31], we estimate the GC of M2C and  calculate the SHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Due to their rather small GC values, the intrinsic SHA of Ta2C can  reach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='22, which is larger than that of Pt (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='068) [36] and comparable to that of \uf062\uf02dTa  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='15) [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Intrinsic SHC for three independent tensor elements for bulk M2C (M=V, Nb, and  Ta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The unit of SHC and 𝐺𝐶 is (ℏ/𝑒)(Ω cm)−1 and (Ω cm)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  𝜎𝑦𝑧  𝑥  𝜎𝑧𝑥  𝑦  𝜎𝑥𝑦  𝑧  𝐺𝐶  SHA  V2C  408  369  348  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='05×104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='06  Nb2C  581  569  565  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='61×104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='07  Ta2C  1093  1167  1084  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='86×103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='22  To further understand the origin of large SHC in Ta2C, we calculate the band-  projected SBC and k-resolved SBC at EF in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 3(a) and 3(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' It is  observed that the large peaks of SBC at EF mainly appear at gapped nodal lines around  P1, P2 and L1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' In addition, we also calculate the k-resolved SBC in (001)  plane at EF in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The SBC locates around the center ring and outer Hexagon  region, corresponding to nodal lines near Г point in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (a) The band structure of Ta2C with the projection of SBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (b) k-revolved SBC along  high symmetry lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (c) Energy-dependent SHC of M2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (d) SBC in (001) planes of BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The EF is  set to zero in both of (a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The electronic band structures and SHE in few-layer M2C  The intimate relationship between SHC with nodal-line structure in bulk M2C implies  that strong SHE can also exist in monolayer or few-layered M2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' We first calculate the  SHC of the monolayered (1L) M2C, shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Due to the reduced symmetry  of 2D structures, the dominant independent component is 𝜎𝑥𝑦  𝑧  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The SHC ( 𝜎𝑥𝑦  𝑧  )  increases gradually as SOC strength becomes larger from V2C to Nb2C and Ta2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The  SHC and SHA of Ta2C reach -191 (ℏ/𝑒)(Ω cm)−1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='05 at EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Since V2C, Nb2C,  and Ta2C shares similar geometry and band structures, therefore, we take Ta2C as an  example to study the layer dependence of SHC (𝜎𝑥𝑦  𝑧  ) by enhancing the thickness  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  (eV)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  (a)  Cnergy (  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content='0  () ()  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  logol  2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  T  M  K  r  A  L  H  A  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  V,C  (p)  SHC(h/e) (S/cm)  400  Nb,C  Ta,C  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='6  400  800  1200  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='2  2  0  2  Energy (eV)8  from monolayer (1L) up to eight layers (8L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The absolute value of SHC at EF versus  the number of layers is calculated and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Generally, the SHC shows a  linearly increasing trend as layer numbers go from 1 to 5 and rise slightly for 6L-8L  Ta2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The maximum value of SHC is 590 (ℏ/𝑒)(Ω cm)−1, which is larger than 2D  systems such as 2L PtSe2 (~27 (ℏ/𝑒)(Ω cm)−1  at EF  [24]), 3L MoTe2 (~200  (ℏ/𝑒)(Ω cm)−1  at EF [21]), and 1L SnTe (~245 (ℏ/𝑒)(Ω cm)−1  at 𝐸𝐹 +  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='09 eV  [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Intrinsic SHC for three tensor elements for monolayer M2C (M=V, Nb, and Ta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The  unit of SHC and GC is (ℏ/𝑒)(Ω cm)−1 and (Ω cm)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  𝜎𝑥𝑦  𝑧  𝜎𝑦𝑧  𝑥  𝜎𝑧𝑥  𝑦  C  G  SHA  V2C  24  3  3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='32×103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='009  Nb2C  46  6  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='19×104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='007  Ta2C  191  15  14  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='06×103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='05  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Layer-dependent SHC for 1L-8L Ta2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  To figure out the origin of layer-dependent SHE observed in few-layered Ta2C, we  first discuss the electronic structures for 1L Ta2C in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Without SOC, the valence  band (marked with a red line) crosses the conduction band at two points along the K-Г  route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 5(e), a careful examination of the energy difference between the two bands  reveals that the nodal point near the Г point belongs to the unclosed nodal ring around  the Г point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The nodal point near the K point is one of the six equivalent points each on  600 -  SHC(h/e) (S/cm)  500-  400-  300-  200-  1  2  3  4  5  6  7  8  Number of layer9  one Г-K (K’) route due to the symmetry protections of 𝐶3𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' S2, the inclusion of  SOC lifts the degenerate points and opens remarkable band gaps around the EF along  the K-Г route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The coexistence of nodal lines and large SOC-induced band gaps aligned  near the EF induces large SBC in 1L Ta2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 5(i), the dominant amplitude of SBC  mainly concentrates around the central ring and outer raindrop-shaped area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' For 3L and  5L Ta2C, the increase in thickness induces repeated crossings between the two bands  on the Г-K route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Meanwhile, the conduction band sticks closely to the valence band  along the M-K route for 5L Ta2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The increased degeneracy forms another circle  around the Г point and many more nodal points along the Г-K route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Correspondingly,  the inclusion of SOC leads to band gaps around the nodal lines, thus generating denser  distribution of SBC and larger SHC at EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' It should be noted that the energy level of  the nodal points along the M-K route is not in the vicinity of EF and thus does not  contribute to the SBC distribution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 5(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' In addition, for 7L Ta2C, the band  crossing along the M-K route induce more nodal points in the Г-K route, which,  however, do not bring much broader SBC distribution near the EF, thus leading to the  slow increment of SHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (a)-(d) Electronic band structure without SOC for 1L, 3L, 5L, and 7L Ta2C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  (e)-(h) 2D contour of the energy gap difference between the valence and conduction bands  corresponding to (a)-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' (i)-(l) Calculated SBC in the first BZ corresponding to (a)-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' DISCUSSION AND CONCLUSIONS  We take M2C (M=V, Nb, and Ta) as a representative in the above presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' As  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' S3, the essential band crossing between the bands near the EF is also  shared by other members of MXenes family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' In most materials, like Zr2C, Hf2C, Nb2N,  and Ta2N, the band crossing points are below or above the EF, while in other materials  such as Zr2N and Hf2N, the band crossing points are just at the EF, which are expected  to contribute to strong SHC peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  (a)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  (e)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content='8  r  M  K  (b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  (f)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content='8  M  K  (c)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  (g)  (k)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content='8  (d)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  Energy (eV)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  7L  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='8  M  K11  In summary, we have predicted large intrinsic SHE in layered transition metal  carbides M2C (M=V, Nb, and Ta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Due to strong SOC and contributions of multiple  nodal lines in the band structure, the SHC and SHA of bulk Ta2C can reach up to ~1100  (ℏ/𝑒)(Ω cm)−1 and ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='20, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' For a few-layered Ta2C, the 𝜎𝑥𝑦  𝑧  component  of SHC exhibits a monotonic increase as the number of layers increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' The maximum  value of SHC is 590 (ℏ/𝑒)(Ω cm)−1 for 8L Ta2C, which is larger than reported 2D  systems so far and comparable to many 3D topological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Therefore, one can  effectively tune the SHC by controlling the layer thickness in layered 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Our  results not only elucidate the interplay between layer-dependent SHC and band  topology, but also provide theoretical guidance for developing next-generation  spintronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work is supported by National Natural Science Foundation of China (Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 11574118), NSFC (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 51672023), and NSAF (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' U1930402).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Yuan, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Zhao, Calculation of Intrinsic Spin Hall Conductivity by Wannier  Interpolation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Khang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Ueda, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Hai, A Conductive Topological Insulator with Large Spin Hall  Effect for Ultralow Power Spin–Orbit Torque Switching, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content='  [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Shao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Gurung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Zhang, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Tsymbal, Dirac Nodal Line Metal for Topological  Antiferromagnetic Spintronics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' 122, 077203 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' Kondou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Tomita, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Ikhlas, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Higo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' MacDonald, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Nakatsuji, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Otani,  Giant Field-like Torque by the out-of-Plane Magnetic Spin Hall Effect in a Topological  Antiferromagnet, Nat Commun 12, 6491 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=', Magnetization Switching through Giant Spin-Orbit Torque in a Magnetically Doped  Topological Insulator Heterostructure, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Deorani, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Banerjee, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Koirala, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Brahlek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Oh, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Yang, Topological Surface  States Originated Spin-Orbit Torques in Bi2Se3, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content='  12  [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Guo, Tunable Large Spin Hall and Spin Nernst Effects in the Dirac Semimetals  ZrXY ( X = Si , Ge ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Y = S , Se , Te), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Felser, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Yan, Strong Intrinsic Spin Hall Effect in the TaAs Family of  Weyl Semimetals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' Shen, Origin and Enhancement of the Spin Hall  Angle in the Weyl Semimetals LaAlSi and LaAlGe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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+page_content=' Sławińska, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Cerasoli, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Postorino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Supka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Curtarolo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content=' Fornari, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  Buongiorno Nardelli, Giant Spin Hall Effect in Two-Dimensional Monochalcogenides, 2D Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
+page_content='  6, 025012 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAzT4oBgHgl3EQfefxJ/content/2301.01436v1.pdf'}
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